Patent Publication Number: US-2020276493-A1

Title: Districting Strategy Game

Description:
RELATED APPLICATIONS 
     This application claims priority to U.S. Provisional Application Ser. No. 62/785,824, filed on Dec. 28, 2018, the entirety of which is incorporated by reference herein. 
    
    
     FIELD OF THE INVENTION 
     The present technology relates to the field of games, and more particularly to strategy games. 
     COPYRIGHT NOTICE 
     A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent disclosure, as it appears in the Patent and Trademark Office patent files or records, but otherwise reserves all copyright rights whatsoever. 
     SUMMARY 
     Strategy games of the present technology may be played by at least one player. The playing surface comprises a region that includes a bounded shape having an area divided into a plurality of sectors. Each sector comprises a bounded shape having an area within the region that does not overlap with any other sector. Each sector contains a set of elements, each element of the set of elements having a type and quantity. During play, each player makes one move per turn, according to a set of rules defining types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors, in pursuit of combining the plurality of sectors into a given number of districts in a manner that seeks to achieve a pre-defined goal based on an aggregation of the elements within each district. 
     Methods of playing strategy games of the present technology include providing a playing surface that includes a region comprising a bounded shape having an area divided into a plurality of sectors. Each sector comprises a bounded shape having an area within the region that does not overlap with any other sector, and each sector contains a set of elements, each element of the set of elements having a type and quantity. Methods of playing strategy games of the present technology further include making one move per turn per player, according to a set of rules defining types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors, in pursuit of combining the plurality of sectors into a given number of districts in a manner that seeks to achieve a pre-defined goal based on an aggregation of the elements within each district. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Specific examples have been chosen for purposes of illustration and description, and are shown in the accompanying drawings, forming a part of the specification. Within the Figures, like parts have been given like numbers for ease of reference. It should be understood that the drawings are not necessarily drawn to scale and that they are intended to be merely illustrative. 
         FIG. 1  illustrates a first example of a strategy game of the present technology, having a first region. 
         FIG. 2  illustrates an example of a region that can be included in a second strategy game of the present technology. 
         FIG. 3  illustrates examples of possible district shapes having four sectors that can be formed during play of a strategy game using the region of  FIG. 2 . 
         FIG. 4  illustrates a possible solution of a strategy game using the region of  FIG. 2  in accordance with a first pre-defined goal. 
         FIG. 5  illustrates a possible solution of a strategy game using the region of  FIG. 2  in accordance with a second pre-defined goal. 
         FIG. 6  illustrates a possible solution of a strategy game using the region of  FIG. 2  in accordance with a third pre-defined goal. 
         FIG. 7  illustrates examples of possible district shapes having six sectors that can be formed during play of a strategy game using the region of  FIG. 1 . 
         FIG. 8  illustrates a possible solution of a strategy game using the region of  FIG. 1  in accordance with a first pre-defined goal. 
         FIG. 9  illustrates a possible solution of a strategy game using the region of  FIG. 1  in accordance with a second pre-defined goal. 
         FIG. 10  illustrates a possible solution of a strategy game using the region of  FIG. 1  in accordance with a third pre-defined goal. 
         FIG. 11  illustrates an example of a region that can be included in a third strategy game of the present technology. 
         FIG. 12  illustrates a strategy game including the region of  FIG. 11 . 
         FIG. 13  illustrates the scoreboard of  FIG. 12 . 
         FIG. 14  illustrates one example of sector tiles that can be used in a strategy game of  FIG. 12 . 
         FIG. 15A  illustrates a possible arrangement of markers on a region during play. 
         FIG. 15B  illustrates a second possible arrangement of markers on a region during play. 
         FIG. 16A  illustrates a third possible arrangement of markers on a region during play. 
         FIG. 16B  illustrates a fourth possible arrangement of markers on a region during play. 
         FIG. 17A  illustrates a fifth possible arrangement of markers on a region during play. 
         FIG. 17B  illustrates a sixth possible arrangement of markers on a region during play. 
         FIG. 18  illustrates examples of sector relationships. 
         FIG. 19  illustrates a seventh possible arrangement of markers on a region during play. 
         FIG. 20  illustrates an eighth possible arrangement of markers on a region during play. 
         FIG. 21  illustrates one possible position for the game of  FIG. 11  after 15 turns. 
         FIG. 22  illustrates one possible position for the game of  FIG. 21  after 23 turns. 
         FIG. 23  illustrates one possible position for the game of  FIG. 22  after 27 turns. 
         FIG. 24  illustrates one possible position for the game of  FIG. 23  after 29 turns. 
         FIG. 25  illustrates one possible position for the game of  FIG. 24  after 32 turns. 
         FIG. 26  illustrates one possible position for the game of  FIG. 25  after 34 turns. 
         FIG. 27  illustrates one possible position for the game of  FIG. 26  after 35 turns. 
         FIG. 28  illustrates one possible position for the game of  FIG. 27  after 36 turns. 
         FIG. 29  illustrates one possible position for the game of  FIG. 28  after 37 turns. 
         FIG. 30  illustrates one possible position for a final position of the game of  FIG. 29 . 
         FIG. 31  illustrates a guide for arrangement of a symmetric game of  FIG. 11 . 
         FIG. 32  illustrates a symmetric game setup table for a symmetric game of  FIG. 11 . 
         FIG. 33  illustrates a fourth example of a strategy game of the present technology. 
         FIG. 34  illustrates one possible initial position for a strategy game of  FIG. 33 . 
         FIG. 35  illustrates one possible final position for a strategy game of  FIG. 34 . 
         FIG. 36  illustrates a fifth example of a strategy game of the present technology. 
         FIG. 37  illustrates a scoreboard that may be used with the strategy game of  FIG. 36 . 
         FIG. 38A  illustrates a possible solution by a first player in phase one of a first strategy game of  FIG. 36 . 
         FIG. 38B  illustrates a possible solution by a second player in phase one of the strategy game of  FIG. 38A . 
         FIG. 39A  illustrates a possible solution by a first player in phase two of the strategy game of  FIG. 38A . 
         FIG. 39B  illustrates a possible solution by a second player in phase two of the strategy game of  FIG. 38A . 
         FIG. 40A  illustrates a possible solution by a first player in a second strategy game of  FIG. 36 . 
         FIG. 40B  illustrates a possible solution by a second player in the strategy game of  FIG. 40A . 
         FIG. 41  illustrates examples of sector tiles in a strategy game of the present technology having four elements. 
         FIG. 42A  illustrates an initial position in a region of a sixth example of a strategy game of the present technology. 
         FIG. 42B  illustrates one possible final position for a strategy game of  FIG. 42A . 
         FIG. 43  illustrates a scoreboard that may be used in the strategy game of  FIG. 42A . 
         FIG. 44  illustrates a seventh example of a strategy game of the present technology. 
         FIG. 45  illustrates one possible initial position for a strategy game of  FIG. 44 . 
         FIG. 46  illustrates one possible final position for a strategy game of  FIG. 45 . 
         FIG. 47  illustrates one possible final position for an eighth example of a strategy game of the present technology. 
         FIG. 48  illustrates a scoreboard that may be used in a strategy game of  FIG. 47 . 
         FIG. 49  illustrates a game board that may be used with a ninth example of a strategy game of the present technology. 
         FIG. 50  illustrates one possible initial position for a tenth example of a strategy game of the present technology. 
         FIG. 51  illustrates examples of sector tiles that may be used in a strategy game of  FIG. 50 . 
         FIG. 52  illustrates a scoreboard that may be used in a strategy game of  FIG. 50 . 
         FIG. 53  illustrates a rotationally symmetric sector arrangement for a strategy game of  FIG. 50 . 
         FIG. 54  illustrates one possible region that may be used in an eleventh example of a strategy game of the present technology. 
         FIG. 55  illustrates examples of sector tiles that may be used in a strategy game of  FIG. 54 . 
         FIG. 56  illustrates one possible initial position in a region that may be used in a twelfth example of a strategy game of the present technology. 
         FIG. 57  illustrates part 1 of scoreboard that may be used in the strategy game of  FIG. 56 . 
         FIG. 58  illustrates part 2 of scoreboard that may be used in the strategy game of  FIG. 56 . 
         FIG. 59  illustrates one possible position for the game of  FIG. 56  during play. 
         FIG. 60  illustrates one possible position for the game of  FIG. 59  after 40 turns. 
         FIG. 61  illustrates one possible position for the game of  FIG. 60  after 41 turns. 
         FIG. 62  illustrates one possible position for the game of  FIG. 61  after 42 turns. 
         FIG. 63  illustrates one possible position for the game of  FIG. 62  after 65 turns. 
         FIG. 64  illustrates one possible position for the game of  FIG. 63  after 70 turns. 
         FIG. 65  illustrates one possible position for the game of  FIG. 64  after 75 turns. 
         FIG. 66  illustrates one possible position for the game of  FIG. 65  after 79 turns. 
         FIG. 67  illustrates one possible position for the game of  FIG. 66  after 81 turns. 
         FIG. 68  illustrates one possible position for the game of  FIG. 67  after 83 turns. 
         FIG. 69  illustrates one possible position for the game of  FIG. 68  after 84 turns. 
         FIG. 70  illustrates one possible final position for phase 1 the game of  FIG. 69 . 
         FIG. 71  illustrates one possible initial position in a region that may be used in a thirteenth example of a strategy game of the present technology. 
         FIG. 72  illustrates part 1 of scoreboard that may be used in the strategy game of  FIG. 71 . 
         FIG. 73  illustrates part 2 of scoreboard that may be used in the strategy game of  FIG. 71 . 
         FIG. 74  illustrates one possible final position for the strategy game of  FIG. 71 . 
         FIG. 75  provides a table that includes a key to symbols used in  FIGS. 1-74 . 
     
    
    
     DETAILED DESCRIPTION 
     The present technology provides strategy games involving the division of a region into districts to achieve one or more predefined goals. The use of the prefix “pre” herein means any time prior to beginning the play of the game. Games of the present invention can be provided as board games, sector based games, paper based games, electronic games, or games on any other suitable presentation medium. For example, the playing surface on which a region may be provided may take the form of a board, a set of sectors, a piece of paper, a three-dimensional form, or a screen. 
     Table 1, provided in  FIG. 75 , is a key that explains the meaning of the various numbers and shapes that appear in the Figures. The first 5 rows of Table 1 represent designations that may be used with sectors. The first row of Table 1 is a sector number, and each sector within a region may have a distinct sector number to distinguish the sector from any other sector within the region. The fifth row represents the total population within a sector. The second through fourth rows represent designations of a party majority, for use in examples of strategy games that include political parties, which are represented in the Table as including a red party, a blue party, and a green party. The final row of Table 1 provides a designation for a scoring token, which may be used in examples of strategy games that have a scoreboard. The remainder of the rows of Table 1 provide representations of designations for markers, including home base markers and expansion markers, that may be placed onto a sector during a player&#39;s move to assign that sector to a district. 
       FIG. 1  illustrates one example of a strategy game  100  of the present technology. The strategy game  100  includes a region  102 , which is a hexagon. Generally, in strategy games of the present technology, the region has at least one area or volume bounded by pre-defined external boundaries. The region can be any real or imagined bounded two-dimensional shape or three-dimensional form, such as a polygon or other geometric shape, a polyhedron, or a geographical area. In many examples, the region is presented as a map. While the region is preferably defined by a single bounded shape or form, a region can include multiple bounded shapes or forms. In examples where a region has multiple bounded shapes or forms, it is preferred that at least a portion of each bounded shape or form be predefined to be connected with at least part of one of the other bounded shapes or forms. 
     The region  102  is pre-divided into a plurality of sectors  104 . The sectors are stationary, and do not move during play of the game. Each sector has an area or volume within the region bounded by pre-defined boundaries. Each sector is distinct and does not overlap with any other sector. Each sector can be any real or imagined bounded shape or form—such as a polygon or other geometric shape, a polyhedron or other three-dimensional form, or a geographical area—within the region. For example, each sector  104  is a triangle. For purposes of the strategy games described herein, a sector cannot be further divided. Every part of the region  102  is defined as being part of a sector, and, collectively, the sectors cover the entire area or volume of the region  102 . 
     Each sector  104  contains a set of elements, each set of elements including one or more elements. The set of elements in a sector is a complete list of the elements contained in the sector, and each element of the set of elements has a type and quantity. The quantity of an element may be any amount, and is preferably greater than zero. In some examples, the type of each element is voters that favor a particular political party. In other examples, the type of each element may be a resource (i.e., something useful), a hazard (i.e., something harmful), or scrap (i.e., something neither useful nor harmful). Each type of element may have the same value to each player. Alternatively, each element may have a value that is player-specific. That is, the same element may be a resource (i.e., something useful) to one player but a hazard (i.e., something harmful) or scrap (i.e., something neither useful nor harmful) to another player. 
     Referring to  FIG. 1 , the illustrated strategy game  100  is a game in which the type of element in each sector is voters that favor a particular political party, and the quantity is the margin of voters within the sector that favor the indicated political party. For example, voters in sector  106  favor the Blue party by a margin of  2 , and voters in sector  108  favor the Red party by a margin of 9. The voter margin may represent single voters, hundreds of voters, thousands of voters, or any other suitable amount of voters, as appropriate for a given game. 
     The strategy game  100  shown in  FIG. 1  also includes a set of markers  110 , wherein the set of markers comprises a plurality of marker subsets  112 , each marker subset representing a district. Specifically, each marker within each marker subset  112  is configured to be placed on a sector to assign the sector as being part of the district represented by the marker subset  112 . Each marker subset  112  comprises a plurality of markers, and may include a home base marker  114  and at least one, preferably more than one, expansion marker  116 . In strategy game  100 , there are fifty-four sectors  104 , and nine marker subsets  112 , each marker subset  112  representing one of nine districts will be created during the game. In order for each sector  104  to have a marker on it at the end of the game, there are preferably at least six markers in each marker subset  112 . In this example, there may be one home base marker  114  and at least five expansion markers  116  in each marker subset  112 . 
     The strategy game  100  shown in  FIG. 1  also includes a scoreboard  118 . The scoreboard  118  has one row for each of the nine districts that will be created during the game, each row containing a scoring spectrum, such as the illustrated spectrum of which party (Red or Blue) is favored and the margin by which the party is favored within the district. Each row has a scoring token  120  that may be used keep track of the favored party and total voter margin within each district. 
     Games of the present technology may be played by at least one player. Some embodiments are designed to be played by a single player, while other embodiments are designed to be played by a plurality of players, such as at least two players. The term “player” as used herein can mean one individual, or a team of individuals. In many games of the present technology, two or more players take alternating turns, and each player must make one move per turn. In some embodiments of the present technology there is only one player who faces an individual challenge. In some games of the present technology, two or more players independently consider the exact same challenge, each using a separate copy of identical game components. In such examples, each player may take turns independently of the other players, using their own set of the game components. Turns may or may not be under time constraints. The player who does the best job of attaining the pre-defined goal wins the game. 
     Generally, in order to play a game of the present technology, one or more players are provided with a region that has been divided into non-overlapping sectors that (i) may not be further divided, (ii) do not overlap, and (iii) together cover the area or volume of the large region. At the outset, the one or more players are informed of each sector&#39;s precise shape, location, and set of elements, and such information may be depicted graphically. In most examples, there are no districts yet created within the region at the start of play, and each sector is initially considered to be unassigned. 
     During play, each player makes one move per turn by assigning a sector to a district, according to a set of rules defining types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors, in pursuit of combining the plurality of sectors into a given number of districts in a manner that seeks to achieve a pre-defined goal based on an aggregation of the elements within each district. Examples of pre-defined goals include, but are not limited to: (a) maximizing a portion of the given number of districts that contain at least a certain level of at least one of the elements of the set of elements; (b) minimizing a portion of the given number of districts that contain at least a certain level of at least one of the elements of the set of elements; and (c) maximizing a number of points earned by at least one of the players, where the number of points earned by the at least one player depends on the aggregation of the elements within each district. In examples of strategy games of the present technology that include at least two political parties, such as strategy game  100  of  FIG. 1 , examples of pre-defined goals include, but are not limited to: (a) maximizing a portion of the given number of districts controlled by one of the political parties, and (b) equalizing a portion of the given number of districts controlled by each of the political parties. 
     Rules of strategy games of the present technology define types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors. For example, the rules may provide at least two general categories of moves that a player can make:
         1. Establish a new district by assigning the first sector that belongs to it.   2. Expand a district by assigning an unassigned sector to an already established district.       

     In some examples, the rules include additional categories of moves that are permitted. For example, three additional categories of moves could be:
         3. Reassign a sector from one district to another district.   4. Break up one or more adjacent districts by un-assigning all sectors that belong to them.   5. Freeze a district. No player may modify the district during the next several turns.       

     As another example, rules regarding how districts may be formed may include:
         A. Every sector within the region must be assigned to exactly one district.   B. Each district must be a single connected piece.   C. Each district may be required to have at least, or at most, a certain number of sectors, or may be required to include a specific pre-defined number of sectors.       

     In many games of the present technology, play concludes when there are no more permitted moves, or when every sector within the region has been assigned to a district. If there are two or more players, the winning player is the player that does the best job of achieving the pre-defined goal. If there is one player, the player wins if the pre-defined goal is achieved, and otherwise loses. 
     Computer games incorporating the concept may be played in at least two modes. In mode 1, a computer plays the role of (i.e., makes the decisions for) one or more players. In this mode, the computer makes use of sophisticated artificial intelligence techniques that are programmed into it ahead of time by a team of expert computer scientists. In mode 2, the computer provides visualization, data storage, and communication services to facilitate play but does not participate as a decision maker during play. 
     Taxonomy of Strategy Games of the Present Technology: 
     Strategy games of present technology include hundreds of recreational and non-recreational games and puzzles. A taxonomy for the exemplary strategy games is provided below, and is based on a nine-part code. This code summarizes the main aspects of a given example of the present technology. Each part of the code contains one or more capital letters or integers and is separated from the other parts of the code by forward slashes. The generic code for an example of the current technology is as follows: 
     Part1/Part2/Part3/Part4/Part5/Part6/Part7/Part8/Part9 
     Part 1 of the code is either the letter “A” or “D.” It is “A” if the example is analog in nature; it is “D” if the example is digital in nature. 
     Part 2 is either “Z” or “G.” It is “Z” if the example is a single-player puzzle (e.g., an individual challenge like a Sudoku puzzle). It is “G” if the example is a multi-player game. 
     Part 3 refers to the shape of each sector and the number of sectors. It consists of a letter followed by an integer with no interceding punctuation. It begins with “S” if each sector is a square; “T” if each sector is an equilateral triangle; “H” if each sector is a regular hexagon; “C” if each sector is a complex, two-dimensional shape such as the shape of a real-world county or country; and “0” if each sector is another shape (e.g., a three-dimensional form). The integer Y that follows the letter indicates how many sectors are in the game. If the letter is (S, T, H), the number of sectors is (36*Y, 54*Y, 37*Y) respectively or slightly less than this. If the letter is “C” or “0,” the value Y gives the exact number of sectors. 
     Part 4 is either “P” or “N.” It is “P” if the example focuses on politics. It is “N” if the example does not focus on politics. 
     Part 5 is an integer that gives the number of element types that are found within the sectors. Its value often ranges from 2-6. 
     Part 6 is either the letter “V” or an integer. If it is the letter “V,” the number of districts to be formed is variable and is unknown at the start. Otherwise, the number of districts to be formed is known at the start and equals the value in this part of the code. 
     Part 7 specifies the game paradigm. It only applies to multi-player games, thus it exists only if part 2 of the code is “G.” Part 7 is “U” if the game involves alternating, turn-based play. It is “I” if the game involves simultaneous independent play in which each player takes turns independently of the others. Games with simultaneous independent play can have any number of players, whereas games with alternating, turn-based play typically have no more than 6 players. 
     Parts 8 and 9 only apply to multi-player games with alternating, turn-based play; these parts of the code exist only if part 2 of the code is “G” and part 7 of the code is “U.” 
     Part 8 indicates the number of players in the game. It is expressed as a range—with two integers separated by a hyphen—if different numbers of players can play the game. It is a single integer if the game is designed for a specific number of players (e.g. for two players only). 
     Part 9 contains one or more of the letters “E,” “X,” “R,” “B,” and “F.” These five letters respectively refer to five categories of allowed moves—“Establish,” “Expand,” “Reassign,” “Break up,” and “Freeze”—which are briefly described in a previous paragraph. This part of the code contains the letters that correspond to the categories of moves that are allowed in the game. 
     The description of each example provided below begins with a discussion of its taxonomic code. This code gives the reader a quick understanding of the example&#39;s main aspects. One or more parts of a code may contain the question mark symbol “?” if those aspects are unspecified. 
     EXAMPLES 
     Several non-limiting examples of strategy games of the present technology are provided below. While the examples use numbers, letters, and generic shapes to distinguish between different districts, element types, and sectors, it should be noted that other methods of distinction could be used. For example, colors or specialized graphics could be used. 
     Example 1 
     Strategy games of the Example 1 have a taxonomic code A/Z/S1/P/2/9. They are analog, single-player puzzles with square sectors and a political focus in which two types of elements are present in the sectors and nine districts are formed. A nearly unlimited number of possible instances of this kind of puzzle can be created, one of which is illustrated in  FIG. 2  as strategy game  200 . A collection of instances of this kind of strategy game can be assembled in a booklet. 
     Two aspects distinguish this kind of puzzle from most other types of logic puzzles. First, there may be multiple solutions to a given puzzle; a unique solution is not guaranteed. Second, the “partial solution” concept does not apply. In other words, if one partially finishes a puzzle, there is no guarantee that the partial solution will give rise to a complete solution. A logical “guess and check” approach is recommended for solving this kind of puzzle. 
     In strategy game  200 , the player is given a map of a square shaped region  202  that has been divided into 36 square shaped sectors  204 , which are arranged in six rows and six columns. The player is tasked with dividing the region  200  into a given number of political districts—i.e. to draw lines that define the boundaries of the districts—in order to achieve the stated objective. 
     Two types of elements—two political parties—occupy the region. One element is the Red Party and the other element is the Blue Party. Each sector  204  may represent a community, and each community has a number which is the community&#39;s voter margin. A black number in a white circle means that there are more Red Party supporters than Blue Party supporters in the community (see the key in Table 1). In such a case, the community favors the Red Party. A white number in a black circle means that there are more Blue Party supporters than Red Party supporters in the community. In such a case, the community favors the Blue Party. The number itself is the margin (in thousands of voters) by which the community supports one party over the other. For example, sector  206  has a black 3 in a white circle, which may mean that there are 3000 more Red Party supporters than Blue Party supporters in that community. In this case, the community&#39;s voter margin is “+3 Red.” In sector  208 , there is a white 4 in a black circle, which may mean that there are 4000 more Blue Party supporters than Red Party supporters in that community. The voter margin in sector  208  is “+4 Blue.” In Sector  210 , there is a zero, which means that the community equally supports the two parties. In such a case, the community&#39;s voter margin is 0. 
     In each puzzle of Example 1, the player is asked to divide the region into 9 political districts. In other words, the player is asked to draw lines that define the boundaries of 9 political districts. 
     The rules for forming political districts are as follows.
         1. Each sector must belong in its entirety to one and only one district.   2. No two districts may overlap.   3. Each district must consist of four sectors that form a single connected piece.       

       FIG. 3  shows examples of shapes  302 ,  304 ,  306 ,  308 , and  310 , in which a district may be formed. Each shape of  FIG. 3  is formed from four sectors  204  that form a single connected piece. Rotations and reflections of shapes  302 - 310  would also be acceptable. 
     When a district is formed, the player must pay attention to its voter margin (i.e. margin). A district&#39;s voter margin indicates which party has more voters in the district. A district&#39;s voter margin depends on the voter margins of the sectors in the district. It equals the difference between the sum of the black and white numbers in the district. The voter margin favors the Red Party if the sum of the black numbers in white circles exceeds the sum of the white numbers in black circles; it favors the Blue Party if the opposite is true; and it is zero if the sum of the black numbers equals the sum of the white numbers in a district. 
     For example, if a district has four sectors with voter margins “+4 Red,” “+6 Blue,” “0,” and “+7 Red,” then the district&#39;s voter margin is “+5 Red” (=4+7+0−6). In other words, there are 5000 more Red Party supporters than Blue Party supporters in the district. The party with more voters in a district is said to control the district. Neither party controls a district—a district is tied—if the district&#39;s voter margin is 0. 
     In each puzzle, the player may be asked to pursue one of three goals:
         A. Create political districts that maximize the advantage of the Red Party   B. Create political districts that maximize the advantage of the Blue Party   C. Create political districts that equalize the advantage of the two parties       

     Goals A, B, and C relate to the voter margins of the districts that are formed. In exact terms, Goal A is to, first and foremost, maximize the number of districts controlled by the Red Party and, secondarily, maximize the margin by which the Red Party controls its least safe district. Goal B is to do the same except to the benefit of the Blue Party. Goal C is to (i) equalize the number of districts controlled by each party, (ii) equalize the margin by which each party controls its least safe district, and (iii) maximize the number of tied districts that have a voter margin of 0. A party&#39;s least safe district is the district in which it has the smallest majority. 
     Each puzzle may have an easy version and a hard version. The easy version asks the player to pursue the goal at hand—A, B, or C—to a modest extent. The hard version asks the player to pursue the same goal to the maximum possible extent. 
     There are different solutions to the strategy game  200  for each goal A, B, and C. For example,  FIG. 4  shows a solution in which the region  202  has been divided into nine districts  212 - 228  to achieve goal A. In the solution of  FIG. 4 , the Red party controls seven of the districts by a margin of at least +2. These include districts  212 ,  214 ,  218 ,  220 ,  222 ,  224 , and  228 . In  FIG. 5 , the region  202  has been divided into nine districts  230 - 246  to achieve goal B. In the solution of  FIG. 5 , the Blue party controls seven of the districts by a margin of at least +2. These include districts  230 ,  232 ,  234 ,  240 ,  242 ,  244 , and  246 . In  FIG. 6 , the region  200  has been divided into nine districts  248 - 264  to achieve goal C. In the solution of  FIG. 6 , each party controls two districts, five districts are tied, and the margin by which each party controls its least safe district is +9. Districts  248 ,  250 ,  252 ,  254 , and  262  are tied. Districts  256  and  260  are controlled by the Red Party and have voter margins of “+9 Red” and “+12 Red” respectively. Districts  258  and  264  are controlled by the Blue Party and have voter margins of “+12 Blue” and “+9 Blue” respectively. 
     Example 2 
     Strategy games of the Example 2 have a taxonomic code A/Z/T1/P/2/9. They are analog, single-player puzzles with triangular sectors and a political focus in which two types of elements are present in the sectors and nine districts are formed. 
     In some examples of this kind of puzzle, the region may be a hexagonal region that has 54 triangular sectors (i.e. communities). Each community may have the same population. An example of such a region is region  102  in  FIG. 1 . 
     In the example shown in  FIG. 1 , supporters of two political parties—Red and Blue—occupy the region  102 . Each sector  104  represents a community, and has a number which represents the community&#39;s voter margin. A black number in a white circle means that there are more Red Party supporters than Blue Party supporters in the community. A white number in a black circle means that there are more Blue Party supporters than Red Party supporters in the community. The number within each sector is the margin (in thousands of voters) by which the community supports one party over the other. 
     In each puzzle of Example 2, the player is asked to divide the region into 9 political districts. In other words, the player is asked to draw lines that define the boundaries of 9 political districts. 
     The rules for forming political districts are as follows.
         1. Each community must belong in its entirety to one and only one district.   2. No two districts may overlap.   3. Each district must consist of six sectors that form a single connected piece.       

       FIG. 7  shows examples of shapes  402 - 424  in which a district may be formed. Each shape of  FIG. 7  is formed from six sectors  104  that form a single connected piece. Rotations and reflections of shapes  402 - 424  would also be acceptable. 
     When a district is formed, the player must pay attention to its voter margin. Just as in Example 1, a district&#39;s voter margin equals the difference between the sum of the black and white numbers in the district. The voter margin favors the Red Party if the sum of the black numbers in white circles exceeds the sum of the white numbers in black circles; it favors the Blue Party if the opposite is true; and it is zero if the sum of the black numbers equals the sum of the white numbers in a district. 
     In each puzzle, the player may be asked to pursue one of three goals:
         A. Create political districts that maximize the advantage of the Red Party   B. Create political districts that maximize the advantage of the Blue Party   C. Create political districts that equalize the advantage of the two parties       

     Each puzzle may have an easy version and a hard version. The easy version asks the player to pursue the goal at hand—A, B, or C—to a modest extent. The hard version asks the player to pursue the same goal to the maximum possible extent. 
     There are different solutions to the strategy game  100  for each goal A, B, and C. For example,  FIG. 8  shows a solution in which the region  102  has been divided into nine districts  122 - 138  to achieve goal A. In the solution of  FIG. 8 , the Red party controls eight of the districts—all except district  134 —by a margin of at least +1. In  FIG. 9 , the region  102  has been divided into nine districts  140 - 156  to achieve goal B. In the solution of  FIG. 9 , the Blue party controls eight of the districts—all except district  146 —by a margin of at least +1. In  FIG. 10 , the region  102  has been divided into nine districts  158 - 174  to achieve goal C. In the solution of  FIG. 10 , each party controls three districts, three districts are tied, and the margin by which each party controls its least safe district is +2. Districts  162 ,  172 , and  174  are tied. Districts  160 ,  164 , and  166  are controlled by the Red Party and have voter margins of “+2 Red,” “+6 Red,” and “+17 Red” respectively. Districts  158 ,  168 , and  170  are controlled by the Blue Party and have voter margins of “+12 Blue,” “+2 Blue,” and “+11 Blue” respectively. 
     Example 3 
     Strategy games of the Example 3 have a taxonomic code A/G/S1/P/2/9/U/2/EXR. They are analog, multi-player games with square sectors and a political focus in which two types of elements—namely two political parties—are present and nine districts are formed. The game proceeds according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed. 
       FIGS. 11-32  illustrate a strategy game  500 , having a region  502 . In strategy game  500 , two players representing opposing political parties—Red and Blue—vie for political control of a square state (region  502 ) by competitively creating nine political districts out of 36 square sectors  504  in alternating, turn-based fashion.  FIG. 11  shows the initial arrangement of the region  502 , prior to being divided into districts by the game play.  FIG. 30  shows one possible final arrangement of the region  502 , after the region  502  is divided into nine districts by game play. 
     In strategy game  500 , each sector represents a community. The region represents a state and has an American-style, two-party political system in which one person is elected to represent each political district. At the start, the districts have not been formed and the players know the location and political composition of each sector (i.e., which party its citizens favor and by how much). During the first phase of the game, players build the political districts by assigning sectors to political districts one sector at a time, in alternating turns. They may also reassign sectors from large districts to adjacent smaller districts in order to better equalize the district sizes. During the optional second phase of the game, the voter margin in each district is converted into a numerical likelihood of each party winning the district, and an election is simulated, which may be done by rolling dice. The winner is the player whose party controls more districts than his/her opponent. A tie is possible if players skip phase 2 of the game. 
     In the final position, shown in  FIG. 30 , home base markers  600  and expansion markers  602 , of the types shown in Table 1, are shown that have been placed by the players during the game. Bold lines indicate boundaries between the nine districts  510 - 526 . 
     The table below shows the final result of this game. The Blue Party wins this example game by a score of 5 districts to 3 districts (with one tied district): 
     
       
         
           
               
               
               
               
             
               
                   
                   
               
               
                   
                   
                 Party in 
                 By How 
               
               
                   
                 District 
                 Control 
                 Much 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 Brown 
                 Neither 
                 0 
                 (=6 + 3 − 5 − 4) 
               
               
                   
                 Red 
                 Blue 
                 7 
                 (=8 + 6 + 0 − 7) 
               
               
                   
                 Orange 
                 Blue 
                 3 
                 (=5 + 3 − 3 − 2) 
               
               
                   
                 Yellow 
                 Red 
                 8 
                 (=9 + 5 − 2 − 4) 
               
               
                   
                 Green 
                 Red 
                 5 
                 (=7 + 6 − 8) 
               
               
                   
                 Blue 
                 Blue 
                 2 
                 (=9 + 2 − 1 − 8) 
               
               
                   
                 Purple 
                 Red 
                 9 
                 (=1 + 2 + 8 − 1 − 1) 
               
               
                   
                 Pink 
                 Blue 
                 1 
                 (=7 + 4 − 6 − 4) 
               
               
                   
                 Gray 
                 Blue 
                 9 
                 (=0 + 5 + 7 − 3) 
               
               
                   
                   
               
            
           
         
       
     
     Game Components 
     In strategy games of this example, the region  502  is laid out and includes boundaries for the sectors. To allow multiple varied games to be played, however, the elements of the sectors are not pre-printed on the region. Instead, sector tiles  610  ( FIG. 14 ) are provided, which may be shuffled or otherwise reorganized, and laid onto the region at the start of a game. The set of elements  616  for each sector tile  610  is provided on the sector tile.  FIGS. 12-14  illustrate game components that may be used in strategy game  500 :
         A region  502 , configured to receive sector tiles  610 . In this example, the region is a 6×6 square, forming a 36 square grid, each square of the grid being numbered for reference from 1 to 36, and being configured to receive a sector tile  610 .   36 square sector tiles  610 , examples of two types of which are shown in  FIG. 14  as sector tiles  612  and  614 . The set of elements  616  on each sector tile  610  are shown as being marked twice on each sector tile  610 , in different orientations for visibility from different angles, but it should be understood that the set of elements on a sector tile  610  may be shown in any suitable manner, such as a single representation as shown in  FIG. 11 . The set of elements that may be used for the 36 sector tiles  610  for strategy game  500  are listed below, and the quantity of each sector type is shown in parentheses:       

     
       
         
           
               
               
               
               
               
               
             
               
                   
               
             
            
               
                 +9 Red (1) 
                 +5 Red (2) 
                 +1 Red 
                 (2) 
                 +3 Blue (2) 
                 +7 Blue (2) 
               
               
                 +8 Red (2) 
                 +4 Red (2) 
                 0 
                 (2) 
                 +4 Blue (2) 
                 +8 Blue (2) 
               
               
                 +7 Red (2) 
                 +3 Red (2) 
                 +1 Blue 
                 (2) 
                 +5 Blue (2) 
                 +9 Blue (1) 
               
               
                 +6 Red (2) 
                 +2 Red (2) 
                 +2 Blue 
                 (2) 
                 +6 Blue (2) 
               
               
                   
               
            
           
         
       
         
         
           
             9 home base markers  600  (one for each marker subset representing one of nine districts: Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) 
             162 expansion markers  602  (eighteen for each marker subset representing one of the nine districts: Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray) 
             9 scoring tokens  606  (one for each of the nine districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray), each which may be cube-shaped with faces showing the numbers 10, 20, 30, 40, and 50 or any other suitable shape 
             2 ten-sided dice  618  (1 black and 1 white), each showing the values 0-9 
             Scoreboard  604  ( FIG. 13 ), which may have nine rows  608 , one for each of the nine districts. Each row  608  may contain a scoring spectrum that indicates which party (Red or Blue) is favored and the margin by which the party is favored within the district. 
           
         
       
    
     Game Setup 
     Players may decide (yes or no) if a symmetric game will be played and (yes or no) if phase 2 of the game will be played. The “no-no” option is recommended for beginners. The “yes-no” option is a game of pure skill, whereas the “no-yes” option maximizes the role of luck in the game. Players then decide who plays Red, who plays Blue, and who takes the first turn. 
     The game components may be laid out at the start of the game is shown in  FIG. 12 . The sector tiles  610  may be drawn one at a time and placed face up in a grid square of the region  502  to form a sector  504 . The sector tiles  610  may be placed onto the region sequentially on grid squares 1-36, or in any other suitable manner. The resulting region  502  with its sectors  504  may look like the region represented in  FIG. 11 . 
     Sector Tiles 
     Each sector tile  610  has a set of elements marked at least once thereon that show the element type (Red Party or Blue Party being favored) and the quantity (voter margin). A black number in a white circle, such as first element set  506  in  FIG. 11 , means that there are more Red Party supporters than Blue Party supporters in the community (see the key in Table 1). In this case, we say the community favors the Red Party. A white number in a black circle, such as second element set  508  in  FIG. 11 , means that there are more Blue Party supporters than Red Party supporters in the community. In this case, we say the community favors the Blue Party. The number itself is the margin (which may be in thousands of voters) by which the community supports one party over the other. For example, the black 3 in the white circle of second element set  508  means that there are 3000 more Red Party supporters than Blue Party supporters in the community. In this case, we say that the community&#39;s voter margin is “+3 Red.” The white 7 in the black circle of first element set  506  means that there are 7000 more Blue Party supporters than Red Party supporters in the community. In this case, we say that the community&#39;s voter margin is “+7 Blue.” A zero means that the community equally supports the two parties. In this case, we say that the community&#39;s voter margin is 0. In some examples, every community is designated as having the same total population. 
     As discussed above, in this example there are a total of thirty six sector tiles  610 . The seventeen sector tiles favoring the Red party are identical with respect to their voter margins to the seventeen sector tiles favoring the Blue party, and two sectors have a voter margin of 0. Hence, the overall voter margin in the state is 0; the same number of voters support each party statewide. 
     In this example, the thirty six sector tiles  610 —which remain in their initial positions as sectors  504  once placed for the game—are used as building blocks to form nine political districts that will cover the region  502 . The nine districts are identified by color: Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray. Initially, no sector  504  belongs to any district. During the game, players use colored markers to assign communities to political districts. Each community eventually belongs to exactly one political district. Since nine districts will be created from thirty six sectors  504 , at the end of the game the size of the average district—the number of sectors it has—will be four. However, the rules may permit variation in the size of a district, so some districts may be smaller or larger than others. 
     The voter margin of a district depends on the voter margins of the sectors  504  that comprise it. The voter margin of a district equals the difference between the sum of the black and white numbers in the district. The voter margin favors the Red Party if the sum of the black numbers in white circles exceeds the sum of the white numbers in black circles; it favors the Blue Party if the opposite is true; and it is zero if the sum of the black numbers equals the sum of the white numbers in a district. A district&#39;s voter margin indicates which party has more voters in the district. For example, if a district has four communities with voter margins “+4 Red,” “+6 Blue,” “0,” and “+7 Red,” then the district&#39;s voter margin is “+5 Red” (=4+7+0−6). In other words, there are 5000 more Red Party supporters than Blue Party supporters in the district. The party with more voters in a district is said to control the district. Neither party controls a district—a district is tied—if the district&#39;s voter margin is 0. 
     Scoreboard and Scoring Tokens 
     During the game, the current voter margin of each district is indicated by the position and orientation of its scoring token  606  on the scoreboard  604 . In particular, each district&#39;s scoring token  606  must always be placed so that (1) the value in the square it occupies plus (2) the number on the side of the scoring token that faces up equals the district&#39;s current voter margin. A scoring token may be cubed shaped, and may have its faces marked in the following manner: unmarked, 10, 20, 30, 40, 50. The side of a scoring token  606  that should be face up depends upon the range of the voter margin, and may correspond to unmarked: 0-12, 10:13-22, 20:23-32, 30:33-42, 40:43-52, 50:53-62. The scoring token  606  for a district may be placed on a square within its row  608  when a district&#39;s voter margin favors the (Blue, Red) Party respectively. For example, consider a moment in the game when three communities with voter margins “+5 Blue,” “+1 Red,” and “+4 Blue” have been assigned to the Green District. In this case, the Green District&#39;s voter margin is “+8 Blue” (=5+4−1), so the green scoring token should be placed on square “Green District Voter Margin=+8 Blue” with its unmarked side facing up. If a community with voter margin “+9 Blue” were added to this district, its new voter margin would be “+17 Blue” (=9+5+4−1), and the green scoring token would be moved to square “Green District Voter Margin=+7 Blue” with its “10” side facing up. Alternatively, if a community with voter margin “+9 Red” were added to this district, its new voter margin would be “+1 Red” (=9+1−5−4), and the green scoring token would be moved to square “+1 Red” with its unmarked side facing up. 
     Playing the Game 
     Play may include the following three phases, although the second phase is optional.
         1. Build political districts   2. Run an election (optional)   3. Identify the winner       

     Phase 1: Build Political Districts 
     Summary 
     The first phase is the main phase of the game. During this phase, players may take turns assigning sectors  504  to political districts, one sector at a time, until every sector belongs to a political district. The assignment of a sector  504  to a political district is accomplished by placing a home base marker  600  or expansion marker  602  on a vacant sector  504 . Players may also reassign sectors from large districts to adjacent smaller districts to better equalize the district sizes. This is done by changing the color of the marker on a sector  504 . At the end of this phase, there will be 9 non-overlapping political districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray—that cover the region  502 . 
     Each district evolves in the same way. Initially, it is formless. At some point, it is established when its home base marker  600  is placed on a vacant sector  504 . (A vacant sector is a sector with no marker on it.) It is then expanded whenever one of its expansion markers  602  is placed on a vacant sector  504  that is adjacent to a sector  504  that already belongs to the district. 
     The process of building political districts is relatively unrestricted. There is no general requirement for the sequence in which, or locations where, districts are constructed. Once begun, the construction of a district may be temporarily halted while players take turns establishing, expanding, and/or resizing other districts. There is no district size requirement. However, the rules encourage the creation of districts having four sectors  504 . 
     Importantly, all marker subsets  620  (consisting of the home base marker and expansion markers for a given color) and all sectors  504  are available to all players. No player “owns” any marker subset or sector  504 . As long as the rules below are followed, any player may contribute to building any district during any turn. No matter which player established a district, any other player may expand the district or reassign a sector  504  from that district to another district. 
     Details 
     Note: During play, the voter margins of all sectors  504  are visible to both players. In  FIGS. 15-20 , however, the grid numbers 1-36 are shown instead of the sector voter margins, for ease of reference. 
     Players may take alternating turns beginning with the starting player. During a player&#39;s turn, he/she (A) makes one move and then (B) records the move on the scoreboard. Forfeiting a turn (i.e. passing on a turn) is not allowed. 
     All moves must be of type 1, 1A, 2, 2A, 3, or 3A (described below). Moves of type 1 and 1A establish a new district. These moves are in category E. Moves of type 2 and 2A expand an existing (i.e. already established) district. These moves are in category X. Moves of type 3 and 3A resize two adjacent districts. These moves are in category R. “A” means “alternate move.” 
     Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than nine districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 9 th  district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. In many games, stage 2 is skipped and play proceeds directly from stage 1 to stage 3. Play concludes when no legal moves exist. 
     The six types of legal moves are as follows.  FIGS. 15-20  illustrate different examples when certain move types are allowed by the rules. Explanations for the asterisked terms are provided at the end of these descriptions.
         1 Establish a new district by placing its home base marker  600  on a vacant sector  504 . This move must meet two requirements. (a) The sector must be 2 or more (horizontal+vertical) steps away from each previously placed home base marker. The two sectors shown in (i), (ii), and (iii) in  FIG. 18  are 1, 2, and 2 steps away from each other respectively. Sectors  20  and  29  in  FIG. 12  are four steps away from each other. (b) There must be space to grow this district to a size of 4 connected* sectors.
           The position shown in  FIG. 15A  illustrates moves of type 1. Here, placing a home base marker on sector  10 ,  11 , or  17  is not allowed because it violates requirement (a). Also, placing a home base marker on sector  30 ,  35 , or  36  is not allowed because it violates requirement (b). In this position there are only two possible moves of type 1: place a home base marker on sector  12  or sector  31 .   
           1A Establish a new district by placing its home base marker  600  on a vacant sector  504 . This move must meet two requirements. (c) The sector must be in the largest open space on the board. An open space is a set of connected* vacant sectors. (d) The sector must be the farthest (in number of steps) from a previously placed home base marker (among the sectors satisfying requirement (c)).
           The position shown in  FIG. 15B  illustrates moves of type 1A. Here, placing a home base marker on sector  3  or  17  is not allowed because it violates requirement (c). Also, placing a home base marker on sector  30  or  35  is not allowed because it violates requirement (d). In this position there are only three possible moves of type 1A: place a home base marker on sector  14 ,  19 , or  36 .   
           2 Expand an existing district by placing one of its expansion markers  602  on a vacant sector  504 . This move must meet three requirements. (e) The district must remain connected.* (f) The district&#39;s new size after this move—including the new sector and any sectors that are captured**—may not exceed 4 sectors. (g) No district may be trapped.***
           The position shown in  FIG. 16A  illustrates moves of type 2. Here, placing an orange expansion marker  602  on sector  3  or  23  is not allowed because (e) is violated. Also, placing a yellow expansion marker  602  on sector  10  is not allowed because (f) is violated. Also, placing a purple expansion marker  602  on sector  7  or  13  is not allowed because sector(s) are captured and (f) is violated. Finally, placing a gray expansion marker  602  on sector  30  is not allowed because the Green District would be trapped and (g) would be violated.   
           2A Expand an existing district by placing one of its expansion markers  602  on a vacant sector  504 . This move must meet three requirements. (e) The district must remain connected.* (h) Only the smallest expandable district may be expanded. A district is expandable if there is at least one vacant sector adjacent to it. (i) No sectors may be captured.**
           The position shown in  FIG. 16B  illustrates moves of type 2A. Here, placing a blue expansion marker on sector  6  is not allowed because (e) is violated. Also, placing an orange expansion marker on sector  35  is not allowed because (h) is violated. Finally, placing a blue expansion marker on sector  35  is not allowed because (i) would be violated. In this position there are only five possible moves of type 2A: place a brown expansion marker on sector  6 ; place a gray expansion marker on sector  6  or  24 ; or place a blue expansion marker on sector  24  or  36 .   
           3 Reassign a community from one district (say District X) to another (say District Y) by removing the District X marker from a sector and replacing it with a District Y expansion marker. This move must meet five requirements. (j) District Y must exist prior to this move. (k) District Y must not be expandable prior to this move. (l) District X must be at least 2 sectors larger than District Y prior to this move. (m) Districts X and Y must each remain connected.* (n) The District X marker that is removed must be an expansion marker; it may not be a home base marker.
           The position shown in  FIG. 17A  illustrates moves of type 3. Here, reassigning sector  13  to the Red District is not allowed because the Red District has not been established and (j) is violated. Reassigning sector  16  to the Gray District is not allowed because the Gray District is expandable and (k) is violated. Reassigning sector  9  to the Green District is not allowed because (l) is violated. Reassigning sector  23  or  35  to the Pink District is not allowed because (m) is violated. Reassigning sector  24  or  36  to the Pink District is not allowed because (n) is violated. In this position there are only two possible moves of type 3: reassign sector  16  to the Green District or reassign sector  28  to the Pink District.   
           3A This move has the same requirements as move type 3 except that (n) is not required
           The position shown in  FIG. 17B  illustrates moves of type 3A. In this position, six moves of type 3A are available: reassign sector  26  or  31  to the Red District; reassign sector  26  to the Purple District; and reassign sector  16 ,  23 , or  27  to the Pink District.   
               * See subsection entitled “Connectedness” below   * See subsection entitled “Captured sectors” below   ** See subsection entitled “Trapped districts” below   

     Connectedness 
     Two sectors are adjacent—and connected—if and only if they share a common edge. For example, the two-sector area shown in (i) in  FIG. 18  is connected, but the two-sector areas shown in (ii) and (iii) in  FIG. 18  are NOT connected. 
     In this game, every political district must be connected at all times. That is, at all times and for any two sectors that belong to a given district (say District X), there must be a path within District X—a sequence of adjacent sectors that all belong to District X—connecting those two sectors. 
     Captured Sectors 
     A set of connected, vacant sectors is captured if it is (i) surrounded by a single district or (ii) surrounded by the edge of the board on one side and a single district on the other side. In  FIG. 19 , sector  1  is captured by the Green District (consisting of sectors  2  and  7 - 8 ); sectors  5 - 6  are captured by the Blue District; sectors  31 - 36  are captured by the Red District; and sector  16  is captured by the Blue District. No other sectors are captured. 
     A move of type 2 which captures exactly one sector is allowed if the district&#39;s new size—including the sector on which the marker is placed and the sector that is captured—is no greater than four sectors. All other moves that capture sectors are forbidden. For example, if sector  1  is vacant, it is permissible to add sector  2  to a district consisting of sectors  7 - 8 . In this case, sector  1  is captured and the new district consists of sectors  1 - 2  and  7 - 8 . However, if sectors  5 - 6  are vacant, adding sector  12  to a district consisting of sectors  4  and  10 - 11  is not allowed. 
     A sector that is captured during a legal move of type 2 is immediately assigned to the district that has captured it. An expansion marker is immediately placed on this sector, and the scoreboard is updated appropriately. 
     Trapped Districts 
     A district is trapped if (i) it (and the open spaces beside it) is either surrounded by a single district or is surrounded by the edge of the board on one side and a single district on the other side and (ii) its size (in sectors) plus the sizes of the open spaces beside it is less than four. 
     In  FIG. 20 , the Gray District (consisting of sector  1 ) is trapped by the Green District (consisting of sectors  2  and  7 - 8 ) and the Pink and Orange Districts are trapped by the Blue District. The Yellow District is not trapped because it can still grow to a size of four sectors. 
     A move of type 2 which traps a district is forbidden. For example, if the Gray District consists of sector  1  and the Green District consists of sectors  7 - 8 , then an expansion of the Green District to sector  2  is not allowed. Also, placing a blue expansion marker on sector  15  to achieve the position in  FIG. 20  is not allowed. 
     End of Phase 1 
     Phase 1 ends when no legal moves exist. When this happens, exactly one marker will occupy each sector, and the state will be partitioned into nine political districts that average four sectors each. 
     Phase 2: Run an Election 
     Summary 
     This optional phase of the game accounts for the surprises that can happen in real-world elections. Sometimes the candidate whose party has the majority of voters in a district is defeated by his/her opponent. This may happen if a candidate lacks  charisma , public speaking skills, good looks, or other personal qualities or if the candidate takes unpopular stands on issues such as education, health care, the economy, infrastructure, foreign affairs, the environment, etc. In this phase of the game, the voter margin in each district is converted into a numerical likelihood of each party winning the district, and an election in each district is simulated by rolling dice  618 . 
     Details 
     Each district is considered one at a time beginning with the Brown District. 
     First, using the table below, the voter margin for the party with more voters in the district is converted into a numerical likelihood of that party winning an election in the district. For example, a “+8 Red” voter margin in the Yellow District converts to a 97% chance for the Red Party to win an election in the Yellow District. 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                 District 
                   
               
               
                   
                 Voter 
                 Winning 
               
               
                   
                 Margin 
                 Likelihood 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                   
                 0 
                 50% 
               
               
                   
                 +1 
                 60% 
               
               
                   
                 +2 
                 69% 
               
               
                   
                 +3 
                 77% 
               
               
                   
                 +4 
                 84% 
               
               
                   
                 +5 
                 90% 
               
               
                   
                 +6 
                 93% 
               
               
                   
                 +7 
                 95% 
               
               
                   
                 +8 
                 97% 
               
               
                   
                 +9 
                 99% 
               
               
                   
                 +10 or more 
                 100%  
               
               
                   
                   
               
            
           
         
       
     
     Second, a random number from 1-100 is produced by simultaneously rolling the two 10-sided dice. The result shown on the black (white) die is the value of the tens (ones) digit of the random number. For example, if the black (white) die shows 7 (1), the result is 71. If the black (white) die shows 0 (8), the result is 8. The only exception to the above rule is that a roll of “zero-zero” gives the result of 100. 
     Third, the random number is compared to the winning percentage (e.g.  97  for the above case). If the random number is less than or equal to the winning percentage, the party with more voters in the district wins the district election. If the random number is greater than the winning percentage, the party with fewer voters in the district wins the district election. In the above example, the Red Party wins the Yellow District election if the random number is from 1-97, and the Blue Party wins the Yellow District election if the random number is from 98-100. If both parties have a 50% chance of winning a district, the Blue Party wins if the random number is from 1-50 and the Red Party wins if the random number is from 51-100. If a party has a 100% chance of winning a district, it automatically wins that district without a dice roll. After the winner of an election is identified, the scoring token that matches the district color is placed on the “Blue Wins” or “Red Wins” square in that district&#39;s portion of the scoreboard  604 . 
     The above procedure is repeated for each of the nine districts. 
     Phase 3: Identify the Winner 
     In the game&#39;s final phase, the overall winner is identified. 
     If phase 2 is played, the winner is the player whose party wins five or more district elections. If phase 2 is not played, players identify the party that controls each district, i.e. the party with more voters in each district. This is done by looking at the positions of the scoring tokens on the scoreboard. The winner is the player whose party controls more districts than his/her opponent. If the players control an equal number of districts, the result is a tie. 
     Example of Play 
       FIGS. 21-30  provide an example of play. After the setup is finished, assume that the initial sector arrangement is as shown in  FIG. 11 . During play, sector grid numbers are not visible to the players. In  FIGS. 21-29 , however, sector grid numbers 1-36 are shown for ease of reference. 
     During stage 1 of play, only moves of type 1, 2, and 3 are allowed. After 15 turns, assume the position in  FIG. 21  is reached. An guide to the specific home base markers  600  and expansion markers  602  shown is provided in Table 1. 
     The available moves of type 1 in this position are as follows:
         Establish a new district on sector  1 ,  13 ,  15 ,  18 ,  21 ,  22 ,  24 ,  27 ,  34 , or  36         

     The available moves of type 2 in this position are as follows:
         Expand Red District to sector  5     Expand Orange District to sector  19  or  31     Expand Yellow District to sector  17  or  18     Expand Blue District to sector  22 ,  23 ,  24 ,  27 , or  36     Expand Purple District to sector  1 ,  3 ,  7 ,  9 , or  14     Expand Pink District to sector  3 ,  5 ,  9 ,  15 ,  17 , or  22     Expand Gray District to sector  14 ,  21 ,  27 ,  31 , or  33         

     No moves of type 3 are available in this position. 
     Notes:
         Sector  1  is captured and immediately added to the Purple District if the Purple District is expanded to sector  7 .   An expansion of the Blue District to sector  34  or  35  is not allowed because it violates requirement (f).   An expansion of the Yellow District to sector  5  is not allowed because the Red District would be trapped and requirement (g) would be violated.   An expansion of the Gray District to sector  19  is not allowed because the Orange District would be trapped and requirement (g) would be violated.       

     Eight moves later in the game, after a total of 23 moves, the new position is shown in  FIG. 22 . The game is still in stage 1 because a move of type 1 is available. 
     The available moves of type 1 in this position are as follows:
         Establish Green District on sector  21 ,  22 , or  27         

     The available moves of type 2 in this position are as follows:
         Expand Brown District to sector  33  or  35     Expand Orange District to sector  19         

     No moves of type 3 are available in this position. 
     Notes:
         Reassigning sector  4  from the Pink District to the Red District is not allowed because sector  4  is occupied by a home base marker (see requirement (n)).   Similarly, reassigning sector  11  from the Yellow to Red District is not allowed.   Reassigning sector  12  from the Yellow District to the Red District is not allowed because the Yellow District would not be connected (see requirement (m)).       

     Four moves later, after a total of 27 moves, the new position is shown in  FIG. 23 . Here, (i) there is no way to make a move of type 1 that satisfies its criteria and (ii) fewer than nine districts have been established. Thus, stage 2 of play may begin. The next move must be of type 1A, 2, or 3. 
     The available moves of type 1A in this position are as follows:
         Establish Green District on sector  22         

     The available moves of type 2 in this position are as follows:
         Expand Orange District to sector  13         

     No moves of type 3 are available in this position. 
     Notes:
         A move of type 1A requires that a new district be established on a sector that is in the largest open space on the board. Among the sectors satisfying this requirement, a sector that ties for being the most steps away from a previously placed home base marker must be selected. In the current position, the largest open space consists of three sectors:  17 ,  22 , and  23 . Among these sectors, only one—sector  22 —is two or more steps away from all previously placed home base markers. So there is only one legal move of type 1A in this position.       

     Two moves later, after a total of 29 moves, the new position is shown in  FIG. 24 . Here, all nine districts have been established, so we are in stage 3 of play. In stage 3, the next move must be of type 2 or 3A whenever a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. 
     The available moves of type 2 in this position are as follows:
         Expand Orange District to sector  14     Expand Green District to sector  23         

     The available moves of type 3A in this position are as follows:
         Reassign sector  4  from Pink District to Red District   Reassign sector  11  from Yellow District to Red District       

     Notes:
         The Red District is the only district that is not expandable. Thus, it is the only district that could possibly “steal” a community from another district (see requirement (k)).       

     Three moves later, after a total of 32 moves, the new position is shown in  FIG. 25 . In this position, no move of type 2 exists. Thus, the next move must be of type 2A or 3A. 
     The available moves of type 2A in this position are as follows:
         Expand Brown District to sector  35     Expand Blue District to sector  35     Expand Purple District to sector  3  or  9     Expand Pink District to sector  3  or  9         

     The available moves of type 3A in this position are as follows:
         Reassign sector  4  from Pink District to Red District   Reassign sector  11  from Yellow District to Red District       

     Notes:
         Four districts—Brown, Blue, Purple, Pink—currently tie for being the smallest expandable district (see requirement (h)).       

     Two moves later. After a total of 34 moves, the new position is shown in  FIG. 26 . In this position, a move of type 2 is available. Thus, the next move must be of type 2 or 3A. 
     The available moves of type 2 in this position are as follows:
         Expand Red District to sector  3         

     The available moves of type 3A in this position are as follows:
         Reassign sector  9  from Purple District to Pink District       

     One move later, after a total of 35 moves, the new position is shown in  FIG. 27 . Here, no move of type 2 exists. Thus, the next move must be of type 2A or 3A. 
     The available moves of type 2A in this position are as follows:
         Expand Brown District to sector  35     Expand Blue District to sector  35         

     The available moves of type 3A in this position are as follows:
         Reassign sector  9  from Purple District to Pink District       

     One move later, after a total of 36 moves, the new position is shown in  FIG. 28 . There are no vacant sectors, so all future moves will be of type 3A. Play continues until no such moves exist. 
     The available moves in this position are as follows:
         Reassign sector  21  from Brown District to Green District   Reassign sector  21  from Brown District to Pink District   Reassign sector  9  from Purple District to Pink District       

     One move later, after a total of 37 moves, the new position is shown in  FIG. 29 . No legal moves exist in this position, so phase 1 of play concludes. The districts that have been formed are now final. 
     The final position at the end of phase 1 is shown in  FIG. 30 , with the districts  510 - 526  marked. The markers played and the community voter margins are shown. 
     The final district voter margins are shown on the scoreboard (see table below). 
     If phase 2 is not played, the game immediately ends, and the Blue Party wins by a score of 5 districts to 3 districts (with one tied district). 
     
       
         
           
               
               
               
               
             
               
                   
                   
               
               
                   
                   
                 No. 
                 Voter 
               
               
                   
                 District 
                 Communities 
                 Margin 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 1. Brown 
                 4 
                 0 
                 (=6 + 3 − 5 − 4) 
               
               
                   
                 2. Red 
                 4 
                 +7 Blue 
                 (=0 + 8 + 6 − 7) 
               
               
                   
                 3. Orange 
                 4 
                 +3 Blue 
                 (=5 + 3 − 3 − 2) 
               
               
                   
                 4. Yellow 
                 4 
                 +8 Red 
                 (=5 + 9 − 2 − 4) 
               
               
                   
                 5. Green 
                 3 
                 +5 Red 
                 (=6 + 7 − 8) 
               
               
                   
                 6. Blue 
                 4 
                 +2 Blue 
                 (=9 + 2 − 1 − 8) 
               
               
                   
                 7. Purple 
                 5 
                 +9 Red 
                 (=1 + 2 + 8 − 1 − 1) 
               
               
                   
                 8. Pink 
                 4 
                 +1 Blue 
                 (=7 + 4 − 6 − 4) 
               
               
                   
                 9. Gray 
                 4 
                 +9 Blue 
                 (=7 + 0 + 5 − 3) 
               
               
                   
                   
               
            
           
         
       
     
     If phase 2 is played, dice  618  are rolled to determine the winning party in each district. In the game at hand, the final voter margin of the (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (0, +7 Blue, +3 Blue, +8 Red, +5 Red, +2 Blue, +9 Red, +1 Blue, +9 Blue). Using a preceding table, these margins translate to winning likelihoods of (50%, 95%, 77%, 97%, 90%, 69%, 99%, 60%, 99%) for the parties with the majority of voters in these districts respectively. Note that each party has a 50% chance of winning the Brown District, and no party automatically wins a district with 100% probability. 
     Dice  618  are then thrown to determine the election results. The results are summarized in the table below. Despite being at a disadvantage going into the election, the Red Party “gets lucky” and wins the elections in five out of nine districts. The Red Party wins the game by a score of 5 districts to 4 districts. 
     
       
         
           
               
               
               
               
               
               
             
               
                   
                   
               
               
                   
                   
                 Voter 
                 Winning 
                 Dice 
                 Election 
               
               
                   
                 District 
                 Margin 
                 Likelihood 
                 Roll 
                 Result 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Brown 
                 0 
                 50% for Blue 
                 68 
                 Red Wins 
               
               
                   
                 Red 
                 +7 Blue 
                 95% for Blue 
                 95 
                 Blue Wins 
               
               
                   
                 Orange 
                 +3 Blue 
                 77% for Blue 
                 4 
                 Blue Wins 
               
               
                   
                 Yellow 
                 +8 Red  
                 97% for Red  
                 25 
                 Red Wins 
               
               
                   
                 Green 
                 +5 Red  
                 90% for Red  
                 41 
                 Red Wins 
               
               
                   
                 Blue 
                 +2 Blue 
                 69% for Blue 
                 13 
                 Blue Wins 
               
               
                   
                 Purple 
                 +9 Red  
                 99% for Red  
                 92 
                 Red Wins 
               
               
                   
                 Pink 
                 +1 Blue 
                 60% for Blue 
                 61 
                 Red Wins 
               
               
                   
                 Gray 
                 +9 Blue 
                 99% for Blue 
                 30 
                 Blue Wins 
               
               
                   
                   
               
            
           
         
       
     
     Rules for a Symmetric Game 
     A starting region with a large connected portion of high-numbered sectors favoring the Blue Party but no large connected portion of high-numbered sectors favoring the Red Party is biased in favor of the Red Party. In such a setting, the player representing the Red Party will more easily be able to concentrate or “pack” the voting power of the opposing party into a small number of districts than the player representing the Blue Party. Thus, the Red Party is more likely to win the game. 
     The purpose of a symmetric game is to remove bias from the initial sector arrangement and give each party—Red and Blue—a fair chance of winning the game. This is particularly important in a tournament setting. 
     A symmetric game has three additional rules compared to a regular game. Rule 1 creates a symmetric initial sector arrangement, and rules 2 and 3 minimize the possibility of a symmetric position during play. The three rules are as follows.
         1. During the game setup, the sector arrangement must be counter-symmetric with respect to an imaginary dot in the center of the region (see  FIG. 12 ). In a counter-symmetric sector arrangement the two sectors comprising every pair of diametrically-opposed sectors—every two sectors that are on exact opposite sides of the state—have the same voter margins but they favor different parties. This arrangement guarantees that the starting map is unbiased, favoring neither party.
             FIG. 31  shows a counter-symmetric sector arrangement. Three portions of the region  502  are shown: the central square  622 ; the inner ring  624  which encircles it; and the outer ring  626  which encircles the inner ring. Bold lines distinguish these three portions of the region. Note that the sectors on opposite sides of the central square have the same number but opposite voter margins. Sector  15  is “+6 Red” whereas sector  22  is “+6 Blue,” and sector  16  is “+7 Blue” whereas sector  21  is “+7 Red” (see  FIG. 12  for the sector number key). The sectors on opposite sides of the inner ring  624  also have the same number but opposite voter margins. The same holds true for the sectors in the outer ring  626 . This map&#39;s symmetry gives each player a fair chance of winning the game.   A random counter-symmetric initial sector arrangement can be efficiently created using the “Symmetric Game Setup Table”  628  shown in  FIG. 32  and any thirty six expansion markers. The procedure works as follows. First, all markers are removed from the Symmetric Game Setup Table. The thirty-six sector tiles are then mixed and organized face down into a single deck. Sector tiles are drawn from the deck one at a time. When a sector tile is drawn, players look at the portion of the Symmetric Game Setup Table 628 that matches the sector&#39;s voter margin and color (e.g. “+5” and “Blue”). If the total number of markers in this portion of the table is greater than or equal to the total number of markers in the portion of the table with the same voter margin but opposite color, the sector tile is placed face up in the first unoccupied location according to the sector sequence in  FIG. 12 . Otherwise the sector tile is placed face up in an unoccupied location that is diametrically opposed to where an opposing sector tile (with the opposite voter margin) has already been placed. Then a marker is placed on a dot in the Symmetric Game Setup Table that matches the sector&#39;s voter margin and color. This continues until all 36 sector tiles are drawn and placed face up.   
           2. During the second and third moves of phase 1 (the 1 st  move made by the player who goes second, and the 2 nd  move made by the player who goes first), no marker may be placed on a sector that is diametrically opposed to a sector on which a marker has already been placed.   3. During phase 1, a move (of any type) that creates a district that (a) has size four and (b) coincides with the central square is never allowed.       

     Tournament Play 
     This example of the present technology a game of pure skill if (A) players decide who plays first prior to the start of the game, (B) a symmetric game is played, and (C) phase 2 of the game is skipped. This form of the game, like international chess and the Japanese game go, is highly suited to tournament play. Unlike chess and go, the initial board position in this game is always different, so every game has a unique opening. 
     Handicap Play 
     This game is suited to handicap play. If the players&#39; skill levels differ, the playing field can be leveled by changing the voter margin of one or more sectors. For example, if the stronger player represents the Blue Party, the players may agree, before any sector tiles  610  are placed, to change the voter margin of the first “+8 Blue” sector tile that is placed from “+8 Blue” to 0. Alternatively, the weaker player may be allowed to make more than 50% of the moves—for example 5 of every 9 moves. 
     Game Alternative #1: Form Seven Districts (Each of Size 5) 
     In one variation of strategy game  500 , seven districts are formed instead of nine districts. In this variation, only seven marker subsets  620 , representing seven district colors, are used, and the average size of a district at the end of the game is about five sectors  504 . At the end of the game, the region will be divided into seven political districts. This variation of the game may be played according to the same rules above except that the requirements for moves of type 1, 1A, and 2 are slightly different as described below:
         A move of type 1 must meet two requirements. (a) The sector on which the home base marker is placed may not be in the ring surrounding any previously placed home base marker. (b) There must be space to grow the new district to a size of 5 connected sectors. In  FIG. 12 , the ring surrounding sector  1  consists of sectors  2  and  7 - 8 . The ring surrounding sector  10  consists of sectors  3 - 5 ,  9 ,  11 , and  15 - 17 . The ring surrounding sector  5  consists of sectors  4 ,  6 , and  10 - 12 .   A move of type 1A must meet two requirements. (c) The sector on which the home base marker is placed must be in the largest open space on the board. (d) Among the sectors satisfying the requirement c, the sector must be one that is not in a ring surrounding a previously placed home base marker. If no such sector exists, the sector must satisfy requirement c and be the farthest (in number of horizontal+vertical steps) from a previously placed home base marker.   A move of type 2 must meet three requirements. (e) The district must remain connected. (f) The district&#39;s new size—including the new sector and any captured sectors—must not be greater than 5 sectors. (g) No district may be trapped. A district is trapped if (i) it (and the open spaces beside it) is either surrounded by a single district or is surrounded by the edge of the board on one side and a single district on the other side and (ii) its size (in sectors) plus the sizes of the open spaces beside it is less than 5.       

     Example 4 
     Strategy games of Example 4 have a taxonomic code A/G/S2/P/2/12/U/2/EXR. They are analog, multi-player games with 72 square sectors and a political focus in which two types of elements—namely two political parties—are present and twelve districts are formed. The games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed. 
     One example is strategy game  700  as shown in  FIGS. 33-35 , which is a larger version of the game described in Example 3. Strategy game  700  includes a region  702 , which as illustrated is set up as an 8×9 grid divided into sector placeholders  718  numbered 1-72. Sector tiles  716  are provided, which may be shuffled or otherwise reorganized, and laid out on the region with one sector tile  716  per sector placeholder  718  to form 72 sectors  704  in the region  702  at the start of a game, as shown in  FIG. 34 . The set of elements  724  for each sector  704  is provided on each sector tile  716 . 
     Synopsis 
     Strategy game  700  is very similar to strategy game  500  described in Example 3. The main differences are as follows. First, in strategy game  700 , there are 72 sectors—exactly twice as many sectors of each kind as in strategy game  500 . Second, at the start of strategy game  700  the sector tiles  716  are randomly placed in a 9×8 rectangular arrangement within region  702  to form sectors  704 . Third, twelve districts—the average size of which at the end of the game may be six sectors  704 —will be formed. A set of markers  726  containing a total of 12 marker subsets  710  may be used, each marker subset  710  representing a district and consisting of one home base marker  712  and eighteen expansion markers  714 . As shown there are nine rectangular home base markers (one of each for Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray) and 162 square expansion markers (18 of each of the same colors as the rectangular home base markers), and then three additional home base markers that are diamond shaped (one each for the Brown2, Red2, and Orange2 Districts) and three additional sets of expansion markers that are triangular in shape (18 of each of the same colors as the diamond shaped home base markers). Each district is distinguishable by the color and/or shape of the markers used to form it. Fourth, the precise rules for making moves of types 1, 1A, 2, and 2A are slightly different in this game to encourage most districts to have a size of six sectors  704  at the end of the strategy game  700 . 
     In the illustrated example, strategy game  700  includes two scoreboards, a first scoreboard  706  and a second scoreboard  708 . Each scoreboard has a plurality of rows  722  and looks like  FIG. 13 . The total number of rows in both scoreboards is at least 12, and thus one row may be used to track the voter margin of each district. There are also twelve scoring tokens  720 , one for each district. In other examples of strategy game  700 , there may only be one scoreboard, which would have one row  722  for each of the twelve districts to be formed during the game. 
     As illustrated,  FIG. 34  shows the region  702  at the start of strategy game  700 , and  FIG. 35  shown one possible solution at the end of game play. In  FIG. 35 , the home base markers  712  and expansion markers  714  have been placed in a manner that establishes twelve districts  728 - 750 . 
     The table below shows the final result of this game. The Blue Party wins this example game by a score of 6 districts to 5 districts (with one tied district). 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                   
                 Voter 
               
               
                   
                 District 
                 Margin 
               
               
                   
                   
               
             
            
               
                   
                 Brown 
                 +2 Blue 
               
               
                   
                 Red 
                 +13 Blue  
               
               
                   
                 Orange 
                 +5 Red  
               
               
                   
                 Yellow 
                 +4 Red  
               
               
                   
                 Green 
                 +9 Red  
               
               
                   
                 Blue 
                 +2 Blue 
               
               
                   
                 Purple 
                 +5 Blue 
               
               
                   
                 Pink 
                 +4 Blue 
               
               
                   
                 Gray 
                 +2 Red  
               
               
                   
                 Brown2 
                 0 
               
               
                   
                 Red2 
                 +4 Blue 
               
               
                   
                 Orange2 
                 +10 Red   
               
               
                   
                   
               
            
           
         
       
     
     Playing the Game 
     Strategy game  700  may have the same three phases as, and may be played in a manner that is nearly identical to, Example 3. 
     Phase 1: Build Political Districts 
     During a player&#39;s turn, he/she (A) makes one move and (B) records the move on the appropriate scoreboard  706  or  708 . All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 80 moves—40 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent. 
     Play may be divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 12 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 12 th  district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. In many games, stage 2 is skipped and play proceeds directly from stage 1 to stage 3. Play concludes when no legal moves exist. Forfeiting a turn (i.e. passing on a turn) is not allowed. 
     The rules may provide six types of legal moves, such as those listed below. The terms “captured” and “connected” have the same meaning as described in Example 3. Moves of type 3 and 3A are identical to Example 3.
         1 Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (a) The sector may not be in the ring surrounding any previously placed home base marker. In  FIG. 33 , the ring surrounding sector  1  consists of sectors  2  and  9 - 10 . The ring surrounding sector  10  consists of sectors  1 - 3 ,  9 ,  11 , and  17 - 19 . The ring surrounding sector  5  consists of sectors  4 ,  6 , and  12 - 14 . (b) There must be space to grow this district to a size of 6 connected sectors.   1A Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (c) The sector must be in the largest open space on the board. An open space is a set of connected vacant sectors. (d) Among the sectors satisfying requirement c, the sector must be one that is not in a ring surrounding a previously placed home base marker. If no such sector exists, the sector must satisfy requirement c and be the farthest (in number of horizontal+vertical steps) from a previously placed home base marker. Sectors  2  and  21  in  FIG. 33  are five steps away from each other.   2 Expand an established district by placing one of its expansion markers on a vacant sector. This move must meet three requirements. (e) The district must remain connected. (f) The district&#39;s new size—including this new sector and any captured sectors—must not be greater than 6 sectors. (g) No district may be trapped. A district is trapped if (i) it (and the open spaces beside it) is either completely surrounded by a single district or is surrounded by the edge of the board on one side and a single district on the other side and (ii) its size (in sectors) plus the sizes of the open spaces beside it is less than 6.   2A Expand an established district by placing one of its expansion markers on a vacant sector. This move must meet four requirements. (e) The district must remain connected. (h) Only the smallest expandable district may be expanded. A district is expandable if there is at least one vacant sector adjacent to it. (i) No sectors may be captured. (o) The district&#39;s new size must not be greater than 19 sectors. This last requirement relates to limited marker quantities.   3 Reassign a community from one district (say District X) to another (say District Y) by removing the District X marker from a sector and replacing it with a District Y expansion marker. This move must meet five requirements. (j) District Y must exist prior to this move. (k) District Y must not be expandable prior to this move. (l) District X must be at least 2 sectors larger than District Y prior to this move. (m) Districts X and Y must each remain connected after this move. (n) The District X marker that is removed must be an expansion marker; it may not be a home base marker.   3A This move has the same requirements as move type 3 except that (n) is not required.       

     End of Phase 1 
     Phase 1 ends when no legal moves exist. When this happens, exactly one marker will occupy each sector, and the state will be partitioned into 12 political districts that average 6 sectors each.  FIG. 35  shows a possible position at the end of phase 1. 
     Phase 2: Run an Election 
     This phase of the game is nearly identical to Example 3 except that a different table (shown below) is used to convert a district&#39;s voter margin into the probability that the party with more voters in the district wins an election in the district. 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                 District 
                   
               
               
                   
                 Voter 
                 Winning 
               
               
                   
                 Margin 
                 Likelihood 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                   
                 0 
                 50% 
               
               
                   
                 +1 
                 59% 
               
               
                   
                 +2 
                 67% 
               
               
                   
                 +3 
                 74% 
               
               
                   
                 +4 
                 80% 
               
               
                   
                 +5 
                 85% 
               
               
                   
                 +6 
                 89% 
               
               
                   
                 +7 
                 92% 
               
               
                   
                 +8 
                 94% 
               
               
                   
                 +9 
                 96% 
               
               
                   
                 +10 
                 98% 
               
               
                   
                 +11 
                 99% 
               
               
                   
                 +12 or more 
                 100%  
               
               
                   
                   
               
            
           
         
       
     
     Phase 3: Identify the Winner 
     This phase of the game is exactly the same as in Example 3. If phase 2 is played, the winner is the player whose party wins seven or more district elections. If each party wins six district elections, the result is a tie. If phase 2 is not played, players look at the scoring tokens on the scoreboard to identify the party that controls each district, i.e. the party with more voters in each district. The winner is the player whose party controls more districts than his/her opponent. If the players control an equal number of districts, the result is a tie. 
     Game Alternative #1: Form 18 Districts (with an Average Size of 4 Sectors) 
     One variation of strategy game  700  includes the formation of 18 districts—instead of 12—in the same 9×8 region  702 . In this variation, exactly twice as many home base, expansion, and scoring tokens are used compared to Example 3. The size of the average district at the end of this variation of the game is 4 sectors  704 . This variation may be played according to the same rules above except that the requirements for moves of type 1, 1A, and 2 may be slightly different as described below:
         A move of type 1 must meet two requirements. (a) The sector on which a home base marker is placed must be at least 2 horizontal+vertical steps away from all previously placed home base markers. (b) There must be space to grow the new district to a size of 4 connected sectors. In  FIG. 33 , sectors  20  and  54  are six steps away from each other.   A move of type 1A must meet two requirements. (c) The sector on which a home base marker is placed must be in the largest open space on the board. (d) Among the sectors satisfying the requirement c, the sector must tie for being the farthest (in number of horizontal+vertical steps) from a home base marker.   A move of type 2 must meet three requirements. (e) The district must remain connected. (f) The district&#39;s new size—including the new sector and any captured sectors—must not be greater than 4 sectors. (g) No district may be trapped. A district is trapped if (i) it (and the open spaces beside it) is surrounded by the edge of the board on one side and a single district on the other side and (ii) its size (in sectors) plus the sizes of the open spaces beside it is less than 4.       

     Rules for a Symmetric Game 
     As with strategy game  500 , strategy game  700  can be played as a symmetric game. The purpose of a symmetric game is to remove bias from the initial sector arrangement and give each party—Red and Blue—a fair chance of winning the game. 
     When strategy game  700  is played as a symmetric game, there are two additional rules that may be used compared to a regular game. Rule 1 creates a symmetric initial sector arrangement, and rule 2 reduces the possibility of a symmetric position during play. The two rules are as follows.
         1. During the game setup, the sector arrangement must be counter-symmetric with respect to an imaginary dot in the center of the state. This arrangement guarantees that the starting map is unbiased, favoring neither party. A procedure for doing this may be similar or nearly identical to that described in the subsection “Rules for a symmetric game” in the description of Example 3.   2. During the 2 nd , 3 rd , 4 th , and 5 th  moves of phase 1 (i.e. the 1 st  and 2 nd  moves made by the player who goes second, and the 2 nd  and 3 rd  moves made by the player who goes first), no marker may be placed on a sector that is diametrically opposed to a sector on which another marker has already been placed.       

     Example 5 
     Strategy games of Example 5 have a taxonomic code A/G/S3/P/2/15/U/2/EXR. They are analog, multi-player games with 90 square sectors and a political focus in which two types of elements—namely two political parties—are present and 15 districts are formed. These games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed. 
     Strategy games of Example 5 are is very similar to the games described in Examples 3-4. In one example, two players—Red and Blue—vie for political control of a 9×10 rectangular state by competitively creating 15 political districts (whose average size is 6) out of 90 square communities. The game can be played with any 90 sectors in which the sets of red and blue sectors are identical—for example 5 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (80 sector tiles); 3 each of sector tiles “+9 Red” and “+9 Blue” (six sector tiles); and 4 sector tiles with a voter margin of 0. Players take alternating turns beginning with the starting player. During a player&#39;s turn, he/she (A) makes one move and then (B) records the move on two scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 100 moves—50 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent. 
     Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 15 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 15 th  district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist. 
     The six types of moves allowed in the rules for games of this type are summarized below.
         1 Establish a district on a sector that is not in the ring surrounding any other home base marker. There must be space to grow this district to size 6 or more.   1A Establish a district on a sector that is not in the ring surrounding any other home base marker (among the sectors in the largest open space on the board). If this is not possible, establish a district on a sector that is farthest (in number of steps) from a home base marker (among the sectors in the largest open space on the board).   2 Expand a district so (e) it remains connected, (f) its new size (including any captured sectors) is 6 or less, and (g) no district is trapped.   2A Expand the smallest expandable district so (e) it remains connected, (i) no sectors are captured, and (o) its new size is 19 or less.   3 (Same as in Example 3).   3A (Same as in Example 3).       

     Example 6 
     Strategy games of Example 6 are larger versions of the games described in Examples 3-5. These games have a taxonomic code A/G/S4/P/2/15/U/2/EXR. They are analog, multi-player games with 121 square sectors and a political focus in which two types of elements—namely two political parties—are present and 15 districts are formed. These games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed. 
     In at least one example, two players—Red and Blue—vie for political control of an 11×11 square region by competitively creating 15 political districts (whose average size is just above 8) out of 121 square sectors, each of which represents a community. The game can be played with any 121 sectors in which the sets of red and blue sectors are identical—for example 7 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (112 sector tiles); 3 each of sector tiles “+9 Red” and “+9 Blue” (six sector tiles); and 3 sector tiles with a voter margin of 0. Players take alternating turns beginning with the starting player. During a player&#39;s turn, he/she (A) makes one move and then (B) records the move on two scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 140 moves—70 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent. 
     Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 15 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 15 th  district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist. 
     The six types of legal moves are summarized below.
         1 Establish a district on a sector that is at least 3 (horizontal+vertical) steps away from all other home base markers. There must be space to grow this district to size 8 or more.   1A Establish a district on a sector that ties for being the farthest (in number of horizontal+vertical steps) from a home base marker (among the sectors in the largest open space on the board).   2 Expand a district so (e) it remains connected, (f) its new size (including any captured sectors) is 8 or less, and (g) no district is trapped.   2A Expand the smallest expandable district so (e) it remains connected, (i) no sectors are captured, and (o) its new size is 19 or less.   3 (Same as in Examples 3-5)   3A (Same as in Examples 3-5)       

     Example 7 
     Strategy games of Example 7 are larger versions of the games described in Examples 3-6. They have a taxonomic code A/G/S5/P/2/21/U/2/EXR. They are analog, multi-player games with 169 square sectors and a political focus in which two types of elements—namely two political parties—are present and 21 districts are formed. Games of this type proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed. 
     In one example, two players—Red and Blue—vie for political control of a 13×13 region by competitively creating 21 political districts (whose average size is just above 8) out of 169 square sectors. The game can be played with any 169 sectors in which the sets of red and blue sectors are identical—for example 10 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (160 sector tiles); 3 each of sector tiles “+9 Red” and “+9 Blue” (six sector tiles); and 3 sector tiles with a voter margin of 0. Players take turns beginning with the starting player. During a player&#39;s turn, he/she (A) makes one move and then (B) records the move on three scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 180 moves—90 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent. 
     Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 21 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 21 st  district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist. 
     The six types of legal moves are summarized below.
         1 Establish a district on a sector that is at least 3 (horizontal+vertical) steps away from all other home base markers. There must be space to grow this district to size 8 or more.   1A Establish a district on a sector that ties for being the farthest (in number of horizontal+vertical steps) from a home base marker (among the sectors in the largest open space on the board).   2 Expand a district so (e) it remains connected, (f) its new size (including any captured sectors) is 8 or less, and (g) no district is trapped.   2A Expand the smallest expandable district so (e) it remains connected, (i) no sectors are captured, and (o) its new size is 19 or less   3 (Same as in Examples 3-6)   3A (Same as in Examples 3-6)       

     Example 8 
     This is a larger version of the games described in Examples 3-7. This game has taxonomic code A/G/S6/P/2/21/U/2/EXR. It is an analog, multi-player game with 210 square sectors and a political focus in which two types of elements—namely two political parties—are present and 21 districts are formed. The game proceeds according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed. 
     This game is very similar to Examples 3-7. In this game, two players—Red and Blue—vie for political control of a 15×14 rectangular state by competitively creating 21 political districts (whose average size is 10) out of 210 square communities. The game can be played with any 210 sectors in which the sets of red and blue sectors are identical—for example 12 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (192 sector tiles); 6 each of sector tiles “+9 Red” and “+9 Blue” (12 sector tiles); and 6 sector tiles with a voter margin of 0. Players take turns beginning with the starting player. During a player&#39;s turn, he/she (A) makes one move and then (B) records the move on three scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 240 moves—120 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent. 
     Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 21 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 21 st  district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist. 
     The six types of legal moves are summarized below.
         1 Establish a district on a sector that is at least 3 (horizontal+vertical) steps away from all other home base markers. There must be space to grow this district to size 10 or more.   1A Establish a district on a sector that ties for being the farthest (in number of horizontal+vertical steps) from a home base marker (among the sectors in the largest open space on the board).   2 Expand a district so (e) it remains connected, (f) its new size (including any captured sectors) is 10 or less, and (g) no district is trapped.   2A Expand the smallest expandable district so (e) it remains connected, (i) no sectors are captured, and (o) its new size is 37 or less.   3 (Same as in Examples 3-7)   3A (Same as in Examples 3-7)       

     Example 9 
     Strategy games of Example 9 have a taxonomic code A/G/S1/P/2/9/I. They are analog, multi-player games with 36 square sectors and a political focus in which two types of elements—namely two political parties—are present and 9 districts are formed. The game paradigm is simultaneous independent play, in which each player makes one move per turn in sequential turns, independently of the other players. Any number of players—two or more—may play. 
     Game Summary 
     In one example, illustrated in  FIGS. 36-39B  as strategy game  800 , players compete, optionally under time constraints, to see who can best create the nine political districts of a square region  802 , which represents a state. The region  802  consists of 36 square sector placeholders  806  that are formed into 36 sectors  804  at the start of play, each sector  804  representing a community. The region  802  has an American-style, two-party political system in which one person is elected to represent each political district. At the outset, the districts are formless and the players know the political status of each community (i.e. which party its citizens favor and by how much). During the first phase of the game, players simultaneously and independently work on identical copies of the region  802  to create political districts that achieve the pre-defined goal of maximizing the political advantage of the Red Party. During the (optional) second phase of the game, players simultaneously and independently work on identical copies of the same map to create political districts that achieve the pre-defined goal of maximizing the political advantage of the Blue Party. The winner is the player who does the best job of achieving the pre-defined goals during the game. 
     Components 
       FIGS. 36-38  show the game components that may be used to play strategy game  800 . A timer  814 , which may be digital or analog, may be used in examples of strategy game  800  that are played under time constraints. In addition, each player should have a copy of the same game set which contains the following items:
         A region  802     36 sector tiles  820 , which may have the same element sets as the sector tiles in Example 3   A set of expansion markers  822  divided into nine marker subsets consisting of a plurality of expansion markers  808 . In this instance, 76 expansion markers in the following amounts and colors: 8 Brown, 10 Red, 8 Orange, 8 Yellow, 8 Green, 10 Blue, 8 Purple, 8 Pink, 8 Gray   9 scoring tokens  812  (one for each of the nine districts to be formed)   A scoreboard  810 , which may be identical to scoreboard  604     A game board  818  ( FIG. 37 ), which may have a duplicate region  824  divided into 36 duplicate sector placeholders  826 . The game board  818  may also have a plurality of rows  828  configured to allow a player to track aspects of districts formed on the game board  818 .
 
Setup (about 10 Minutes)
       

     The players decide (yes or no) if phase 2 of the game will be played, and they agree upon a time limit for each phase of the game. The “yes” option with a 10-minute time limit is recommended. (Such a game lasts about 50 minutes.) 
     One player may be selected as the leader. All players except the leader may organize their 36 sector tiles  820  into 19 face-up piles—one pile for each number+color combination—so that specific sector numbers and colors can be quickly located. The leader may spread their sector tiles  820  out face down, mix them, and organize them face down into a single deck. The leader may then draw the 36 sector tiles  820  from the deck one at a time and place them face up with one on each sector placeholder  806  of the region  802  to form sectors  804 . Each time a sector is drawn, the leader may announce its (a) position in the sequence, (b) color, and (c) number—for example “Sector  1 : Red 4,” “Sector  2 : Blue 2,” “Sector  3 : Zero,” etc.—so that every other player may find the same sector tile  820  from his/her game set and place it in the same location in his/her region  802 . When this process ends, each player has a copy of the leader&#39;s sector arrangement in his/her region  802 . One possible arrangement of sectors  804  in the region  802  is identical to that shown in  FIG. 11 . 
     Scoreboard and Scoring Tokens 
     Each player may use his/her scoreboard  810  and scoring tokens  812  to keep track of the voter margins of the districts that he/she creates during play. 
     At the end of each phase of the game, the leader or other players may visit each player&#39;s playing area to ensure that the positions and orientations of his/her scoring tokens properly show the voter margins of the districts that he/she has created. Each district&#39;s scoring token  812  should be placed so that (1) the value in the square it occupies on the scoreboard plus (2) the number on the side of the scoring token that faces up equals the district&#39;s voter margin. When the scoring tokens  812  are cubes—having sides that are unmarked and marked 10, 20, 30, 40, 50 respectively—the (unmarked, 10, 20, 30, 40, 50) side of a scoring token should face up if the voter margin of its district is in the range (0-12, 13-22, 23-32, 33-42, 43-52, 53-62) respectively. The scoring token  812  is placed on the scoreboard to reflect the voter margin in each district. For example, if Player 1&#39;s Green District contains sectors with voter margins “+6 Blue,” “+1 Red,” “+9 Blue,” and “+7 Blue,” then the Green District&#39;s voter margin is “+21 Blue” (=6+9+7−1) and Player 1&#39;s green scoring token should be placed on the square “Green District Voter Margin=+11 Blue” with its “10” side facing up. If Player 2&#39;s Gray District contains sectors with voter margins “+3 Blue,” “+1 Red,” “+9 Red,” and “+5 Blue,” then the Gray District&#39;s voter margin is “+2 Red” (=1+9−3−5) and Player 2&#39;s gray scoring token should be placed on the square “Gray District Voter Margin=+2 Red” with its unmarked side facing up. 
     Playing the Game 
     Play may consist of the following two phases. The second phase is optional.
         1. Build political districts that maximize the Red Party&#39;s advantage   2. Build political districts that maximize the Blue Party&#39;s advantage
 
Phase 1: Build Political Districts that Maximize the Red Party&#39;s Advantage
       

     The timer  814  is set to the time limit agreed upon by the players, if time limits are being used. Play begins when the timer starts or the players agree to begin. 
     During the first phase of the game, each player independently uses his/her colored markers to form 9 political districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray—on his/her 36 sectors  804 . Each district is formed by placing four markers, one per turn, that match the district color on four adjacent sectors. Each player&#39;s main goal in this phase is to create 9 political districts—i.e. a district plan—in which the Red Party controls as many districts as possible. A party controls a district if it has the majority—strictly more than half—of the voters in a district. Each player&#39;s secondary goal is to make the “voter margin in the district that the Red Party controls by the least amount” as high as possible. 
     The rules may require that each player&#39;s district plan must satisfy the following two requirements:
         A. Each sector must belong to exactly one political district. That is, there must be exactly one expansion marker  808  on each sector  804 .   B. Each political district must consist of four connected sectors  804 .       

     Each player is free to use his/her scoreboard, game board, and markers as desired. It is recommended that each player (a) use the expansion markers  808  beside his/her 36 sectors  804  and scoreboard to create and evaluate potential district plans and (b) use the duplicate region  824  on his/her game board  818  to store the best district plan that he/she has found. At the end of this phase, each player&#39;s final district plan must be displayed by a set of 36 markers (four per color) that are placed either on his/her region  802  or on the duplicate region  824  on his/her game board  818 . 
     When play concludes, the final district plan made by each player is scored. The scoring of each player&#39;s final district plan is done by (a) computing the district voter margins, (b) placing scoring tokens appropriately on the scoreboard, and (c) computing the following values:
         1. The number of districts controlled by the Red Party.   2. The lowest voter margin in the districts controlled by the Red Party (“Lowest Voter Margin in Red Districts”).       

     If a player&#39;s district plan violates requirement A or B above, he/she receives scores of 0 and 1 for items 1 and 2 respectively. 
     Illustrative Example 
     One example of the result of phase 1 of a two-player version of game  800  is shown in  FIGS. 38A and 38B . Bold lines around the districts created by the expansion markers  808  indicate the final district plans in phase 1 for each player. 
     At this point, Player 1&#39;s scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+3 Red, +2 Red, +10 Red, +18 Blue, +16 Blue, +12 Red, +5 Blue, +4 Red, +8 Red). The Red Party controls 6 districts, and the lowest voter margin in those six districts is “+2 Red.” Player 2&#39;s scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+2 Red, +7 Blue, +7 Red, +2 Red, +3 Red, +3 Red, +5 Red, +19 Blue, +4 Red). The Red Party controls 7 districts, and the lowest voter margin in those seven districts is “+2 Red.” Overall, Player 2 has done better in this phase of the game because his/her district plan gives the Red Party control of more districts than Player 1&#39;s district plan. Each player uses two red markers to mark his/her scores for phase 1 in the first two rows  828  of his/her game board  818  as shown below. 
                                                 Game Board Item   Player 1   Player 2                                                        1. No. Districts Controlled by Red Party   6   7           2. Lowest Voter Margin in Red Districts   2   2                        
Phase 2: Build Political Districts that Maximize the Blue Party&#39;s Advantage
 
     At the end of phase 1, players remove all markers from their scoreboards and game boards, except the two red markers used to mark their final scores for phase 1 on their game boards. The timer, if used, is then set to the time limit agreed upon by the players. Play of phase 2 is then started. 
     During phase 2, play proceeds exactly as in phase 1 except that now each player&#39;s (i) main goal is to create a district plan in which the Blue Party controls as many districts as possible and (ii) secondary goal is to make the “voter margin in the district that the Blue Party controls by the least” as high as possible. The rules for making districts are just as in phase 1. 
     When the timer goes off, all players cease their activities, disengage from their playing areas, and assemble as a group in the middle of the room. Working as a team, the players together (a) compute the district voter margins, (b) place scoring tokens appropriately on the scoreboard, and (c) compute the following for each player&#39;s final district plan:
         3. The number of districts controlled by the Blue Party.   4. The lowest voter margin in the districts controlled by the Blue Party (“Lowest Voter Margin in Blue Districts”).       

     If a player&#39;s district plan violates requirement A or B above, he/she receives scores of 0 and 1 for items 3 and 4 respectively. 
     The above quantities are remembered by placing two blue markers on the appropriate squares in rows 3 and 4 of each player&#39;s game board. 
     Illustrative Example 
     One example of the result of phase 2 of the two-player version of game  800  is shown in  FIGS. 39A and 39B . Bold lines around the districts created by the expansion markers  808  indicate the final district plans in phase 2 for each player. 
     At this point, Player 1&#39;s scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+5 Blue, +1 Blue, +10 Blue, +2 Blue, +1 Blue, +6 Red, +5 Blue, +4 Blue, +22 Red). The Blue Party controls 7 districts, and the lowest voter margin in those seven districts is “+1 Blue.” Player 2&#39;s scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (“+10 Red,” “+15 Blue,” “+6 Blue,” “+3 Blue,” “+25 Red,” “+3 Blue,” 0, “+3 Blue,” “+5 Blue”). The Blue Party controls 6 districts, and the lowest voter margin in those six districts is “+3 Blue.” Overall, Player 1 has done better in this phase of the game because his/her district plan gives the Blue Party control of more districts than Player 2&#39;s district plan. Each player uses two blue markers to mark his/her scores for phase 2 in rows 3-4 of his/her game board  818  as shown below. 
     
       
         
           
               
               
               
               
             
               
                   
                   
               
               
                   
                 Game Board Item 
                 Player 1 
                 Player 2 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 3. No. Districts Controlled by Blue Party 
                 7 
                 6 
               
               
                   
                 4. Lowest Voter Margin in Blue Districts 
                 1 
                 3 
               
               
                   
                   
               
            
           
         
       
     
     Identifying the Winner(s) 
     The winner of the game is identified, by process of elimination, by looking at the markers in the rows  828  of each player&#39;s game board  818 . These markers may show the scores for the following items:
         1. Number of districts controlled by the Red Party in phase 1   2. Lowest voter margin in the districts controlled by the Red Party in phase 1   3. Number of districts controlled by the Blue Party in phase 2 (if phase 2 played)   4. Lowest voter margin in the districts controlled by the Blue Party in phase 2 (if phase 2 played)       

     If phase 2 is not played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest score for item 1 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest score for item 2 above is eliminated. Any player who is not eliminated wins the game. 
     If phase 2 is played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest sum of scores for items 1+3 is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest sum of scores for items 2+4 is eliminated. Any player not eliminated wins the game. In the preceding illustrative example, the players&#39; scores for items 1+3 have the same sum, so the sum of the scores for items 2+4 is the tiebreaker. The sum of Player 1&#39;s scores for items 2+4 is 3. The sum of Player 2&#39;s scores for items 2+4 is 5, so Player 2 wins. 
     Example 10 
     Strategy games of Example 10 are similar to those of Example 9 and have a taxonomic code A/G/S1/P/2/9/I. They are analog, multi-player games with 36 square sectors and a political focus in which two types of elements—namely two political parties—are present and 9 districts are formed. These games proceed according to simultaneous independent play, so any number of players may play. 
     In one example, the game is set up exactly as described in Example 9, but the pre-defined goal of each player is to create the most balanced set of political districts. In particular, each player&#39;s pre-defined goal may be to create a district plan that (i) equalizes the number of districts controlled by each party, (ii) equalizes the margin by which each party controls its least safe district, and (iii) maximizes the number of tied districts that have a voter margin of 0. Item (i) has priority over (ii), and (ii) has priority over (iii). Each player&#39;s district plan must satisfy requirements A-B (see description of Example 9). 
     At the end of play, each player&#39;s final district plan must be displayed and scored. Scoring may include (a) computing the district voter margins, (b) placing scoring tokens appropriately on the scoreboard, and (c) computing the following for each player&#39;s final district plan:
         1. The number of districts controlled by the Red Party.   2. The lowest voter margin in the districts controlled by the Red Party.   3. The number of districts controlled by the Blue Party.   4. The lowest voter margin in the districts controlled by the Blue Party.   5. The magnitude of the difference between items 1 and 3 above (the “Red-Blue Control Differential”).   6. The magnitude of the difference between items 2 and 4 above (the “Lowest Voter Margin Differential”).   7. The number of tied districts (with a voter margin of 0).       

     The above items may be remembered by placing two red, two blue, and three gray markers in appropriate places on the rows of each player&#39;s game board  828 . 
     Identifying the Winner(s) 
     The winner is identified by process of elimination. First, every player whose district plan violates one of the requirements A-B (see description of Example 9) is eliminated. Second, among the remaining players, every player whose district plan does not tie for having the lowest score for item 5 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the lowest score for item 6 above is eliminated. Finally, among the remaining players, every player whose district plan does not tie for having the highest score for item 7 above is eliminated. Any player who is not eliminated wins the game. If all players&#39; district plans violate one of the requirements A-B (see description of Example 9), all players lose. 
     Example of Play 
       FIGS. 40A and 40B  illustrate one possible conclusion of a strategy game of Example 10, with Player 1&#39;s final district plan shown in  FIG. 40A  and Player 2&#39;s final district plan shown in  FIG. 40B . Bold lines indicate the districts formed by the layout of expansion markers  808  for each player. 
     Player 1&#39;s scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+19 Blue, +5 Blue, 0, +2 Blue, +12 Red, +8 Blue, +4 Red, +7 Red, +11 Red). In player 1&#39;s district plan, the Red and Blue Party each control 4 districts; the lowest voter margin in the districts controlled by the Red Party is 4; the lowest voter margin in the districts controlled by the Blue Party is 2; and one district is tied. Player 2&#39;s scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (0, +3 Blue, +15 Blue, +14 Blue, +3 Red, +25 Red, +15 Blue, +9 Red, +10 Red). In player 2&#39;s district plan, the Red and Blue Party each control 4 districts; the lowest voter margin in the districts controlled by the Red Party is 3; the lowest voter margin in the districts controlled by the Blue Party is 3; and one district is tied. 
     The scoring of items 1-7 above takes place in rows 1-7 of the game board and is summarized in the table below. Both players&#39; district plans satisfy requirements A-B, and the players have the same score for item 5, so the score for item 6 is the tiebreaker. Player 2 has a lower score for item 6, so Player 2 wins the game. 
     
       
         
           
               
               
               
               
             
               
                   
                   
               
               
                   
                   
                 Player 1 
                 Player 2 
               
               
                   
                 Item 
                 Score 
                 Score 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Item 1 
                 4 
                 4 
               
               
                   
                 Item 2 
                 4 
                 3 
               
               
                   
                 Item 3 
                 4 
                 4 
               
               
                   
                 Item 4 
                 2 
                 3 
               
               
                   
                 Item 5 
                 0 
                 0 
               
               
                   
                 Item 6 
                 2 
                 0 
               
               
                   
                 Item 7 
                 1 
                 1 
               
               
                   
                   
               
            
           
         
       
     
     Example 11 
     Strategy games of Example 11 are generally for 2-4 players and are similar to games of Example 3. They have taxonomic code A/G/S1/P/4/9/U/2-4/EXR. They are analog, multi-player games with 36 square sectors in which four types of elements—such as four political parties—are present and nine districts are formed. These games proceed according to alternating, turn-based play, and moves in categories “E,” “X,” and “R” are allowed. 
     Game Summary 
       FIGS. 41-43  illustrate components of a strategy game  900 . Game components not shown in  FIGS. 41-43  may be identical to those shown in  FIG. 12 . In this game, 2-4 players representing opposing political parties (Red Lightning Bolts, Orange Suns, Green Diamonds, and Blue Moons) vie for political control of the region  902 , which may represent a Martian colony, by competitively creating nine political districts out of 36 square sectors  904  in alternating turn-based fashion. At the outset, the districts are formless and the players know how many voters support each party in each community. During the first phase of the game, players build the political districts by assigning communities to political districts one community at a time. They may also reassign communities from large districts to adjacent smaller districts in order to better equalize the district sizes. During the (optional) second phase of the game, the political status of each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice. In the game&#39;s final phase, the parties that control the districts are identified, and 3 (1) points are awarded to a party that has sole (joint) control of a district. The player whose party has more points than any other player&#39;s party is the winner. If parties represented by two or more players tie for having the most points, those players jointly win. 
       FIG. 41  shows examples of two sector tiles  910  that may be used in this game. Each sector tile  910  represents a community and has a set of elements  912  including four icons in its center. Each icon represents an element, such as one person, or a set number of people, who support a political party. There are four non-limiting examples of icons shown, though any suitable icons may be used—a lightning bolt (red), sun (orange), diamond (green), and moon (blue)—corresponding to the four political parties. The number of identical icons represents the quantity  914  of the particular element. For example, the community represented by the first sector tile  910  on the left of  FIG. 41  contains four people each of whom supports a different party. The community represented by the second sector tile  910  on the right of  FIG. 41  contains two people who support the Red Lightning Bolts and two people who support the Green Diamonds. 
       FIGS. 42A and 42B  provide a visual summary of one possible version of strategy game  900 .  FIG. 42A  shows one possible starting position.  FIG. 42B  shows one possible final position for the game, with home base markers  906  and expansion markers  908  forming nine districts (indicated by bold lines). The table below shows the total number of voters supporting each party in each district of  FIG. 42B  at the end of the game. A superscript W indicates that a party wins a district outright. A superscript T indicates that a party ties for winning a district. 
     
       
         
           
               
               
               
               
               
               
             
               
                   
                   
               
               
                   
                   
                 Red 
                 Orange 
                 Green 
                 Blue 
               
               
                   
                   
                 Party 
                 Party 
                 Party 
                 Party 
               
               
                   
                 District 
                 Voters 
                 Voters 
                 Voters 
                 Voters 
               
               
                   
                   
               
             
            
               
                   
                 Brown 
                 4 
                   5 W   
                 4 
                 3 
               
               
                   
                 Red 
                 4 
                  5 T   
                 2 
                  5 T   
               
               
                   
                 Orange 
                  4 T   
                  4 T   
                  4 T   
                  4 T   
               
               
                   
                 Yellow 
                 3 
                 4 
                 3 
                   6 W   
               
               
                   
                 Green 
                 3 
                 3 
                   4 W   
                 2 
               
               
                   
                 Blue 
                 3 
                 2 
                 5 
                   6 W   
               
               
                   
                 Purple 
                   6 W   
                 5 
                 5 
                 4 
               
               
                   
                 Pink 
                 3 
                  5 T   
                  5 T   
                 3 
               
               
                   
                 Gray 
                   6 W   
                 3 
                 4 
                 3 
               
               
                   
                   
               
            
           
         
       
     
     The table below shows the total number of districts that each party wins outright (W) and ties for winning (T). It also shows the total points earned by each party assuming that 3 points are earned when a party wins a district outright and 1 point is earned when a party ties for winning a district. In the final tally, the Blue Party wins this game with 8 points (2 outright wins+2 ties). 
     
       
         
           
               
               
               
               
               
             
               
                   
                   
               
               
                   
                   
                 # Outright 
                 # (T) 
                   
               
               
                   
                 Party 
                 Wins (W) 
                 Ties 
                 Points 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 Red 
                 2 
                 1 
                 7 (=3*2 + 1) 
               
               
                   
                 Orange 
                 1 
                 3 
                 6 (=3*1 + 3) 
               
               
                   
                 Green 
                 1 
                 2 
                 5 (=3*1 + 2) 
               
               
                   
                 Blue 
                 2 
                 2 
                 8 (=3*2 + 2) 
               
               
                   
                   
               
            
           
         
       
     
     Components 
     The components for this game are similar to those used in Example 3. They are as follows.
         A region  902 , which may be a square in shape and be divided into 36 sector placeholders.   36 sector tiles  910  (number of sectors with quantity of each icon [Red Lightning Bolt, Orange Sun, Green Diamond, Blue Moon] is in parentheses below)       

     
       
         
           
               
               
               
               
               
               
               
             
               
                   
               
             
            
               
                 [1, 1, 1, 1] (2) 
                 [2, 1, 1, 0] (1) 
                 [1, 2, 1, 0] (1) 
                 [1, 1, 2, 0] (1) 
                 [2, 1, 0, 1] (1) 
                 [1, 2, 0, 1] (1) 
                 [1, 1, 0, 2] (1) 
               
               
                 [4, 0, 0, 0] (1) 
                 [2, 0, 1, 1] (1) 
                 [1, 0, 2, 1] (1) 
                 [1, 0, 1, 2] (1) 
                 [0, 2, 1, 1] (1) 
                 [0, 1, 2, 1] (1) 
                 [0, 1, 1, 2] (1) 
               
               
                 [0, 4, 0, 0] (1) 
                 [2, 2, 0, 0] (1) 
                 [2, 0, 2, 0] (1) 
                 [2, 0, 0, 2] (1) 
                 [0, 2, 2, 0] (1) 
                 [0, 2, 0, 2] (1) 
                 [0, 0, 2, 2] (1) 
               
               
                 [0, 0, 4, 0] (1) 
                 [3, 1, 0, 0] (1) 
                 [1, 3, 0, 0] (1) 
                 [3, 0, 1, 0] (1) 
                 [1, 0, 3, 0] (1) 
                 [3, 0, 0, 1] (1) 
                 [1, 0, 0, 3] (1) 
               
               
                 [0, 0, 0, 4] (1) 
                 [0, 3, 1, 0] (1) 
                 [0, 1, 3, 0] (1) 
                 [0, 3, 0, 1] (1) 
                 [0, 1, 0, 3] (1) 
                 [0, 0, 3, 1] (1) 
                 [0, 0, 1, 3] (1) 
               
               
                   
               
            
           
         
       
         
         
           
             9 rectangular home base markers  906  (one for each of 9 districts: Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) 
             162 expansion markers  908  ( 18  for each of the nine districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray) 
             36 scoring tokens  606  (nine in each of the colors red, orange, green, and blue) 
             2 ten-sided dice  618  (1 black, 1 white), each showing the values 0-9 
             Scoreboard  916  ( FIG. 43 ) 
           
         
       
    
     Game Setup 
     The setup is similar that in Example 3. The main difference is that, in this game, four scoring tokens—one for each color—are stacked on the position “No. Voters for Each Party=0” in each district&#39;s portion of the scoreboard. 
     Scoreboard and Scoring Tokens 
       FIG. 43  shows a scoreboard  916  that may be used in strategy game  900 . Scoreboard  916  has nine rows  918 , one for each district to be formed. During the game, the number of voters that support each political party in each district is indicated by the position of a scoring token that matches the party&#39;s color in the row  918  for that district. For example, consider the final position shown in  FIG. 42B . In this position, the total number of (Red Lightning Bolts, Orange Suns, Green Diamonds, and Blue Moons) in the Brown District is (4, 5, 4, 3) respectively. To indicate this, a (red, orange, green, blue) scoring token should be placed at position “No. Voters for Each Party”=(4, 5, 4, 3) respectively in the Brown District row  918  of the scoreboard  916 . The positions of the scoring tokens may be updated after every turn. 
     Playing the Game 
     Play may include of the following three phases. The second phase is optional.
         1. Build political districts   2. Run an election (optional)   3. Identify the winner       

     Phase 1: Build Political Districts 
     This phase proceeds exactly as in Example 3. The only difference is that the turns alternate among up to four players instead of two. 
     Phase 2: Run an Election 
     Phase 2 is somewhat different in a four-party version of strategy game  900  as compared to strategy game  500  in Example 3. In this phase of the game, the political status in each district is converted into a numerical likelihood of each party winning the district, and an election in each district is simulated by rolling dice. In particular, the three-step procedure below (A-B-C) is performed for each district beginning with the Brown District. 
     (A) The table below is used to convert the district&#39;s political status into a probability of each party winning an election in the district. This is done by (1) ranking the parties according to voter support in the district (i.e. deciding which party is in 1 st , 2 nd , 3 rd  and 4 th  place); (2) computing the difference in voter support between the parties; and (3) finding the appropriate row in the table below. If two or more parties have the same voter tally, their ranking is inconsequential but the players must still (arbitrarily) rank them. For example, if the political status of the Green district in a four-player game is (3 Red Voters, 3 Orange Voters, 4 Green Voters, 2 Blue Voters), then the Green (Red, Orange, Blue) Party is in 1 st  (2 nd , 3 1d , 4 th ) place, the difference between 1 st  and 2 nd  place is 1 voter, the difference between 2 nd  and 3 rd  place is 0 voters, and the difference between 3 rd  and 4 th  place is 1 voter. In this case, the Green (Red, Orange, Blue) Party has a 60% (20%, 20%, 0%) chance of winning an election in the Green District. In a 2-3 player game, parties not represented by an active player are not ignored. These parties can still win districts and earn points at the end of the game. However, such parties are not allowed to win the game. Only a party represented by an active player may win the game. 
     
       
         
           
               
               
             
               
                   
               
               
                   
                 Winning Percentage for Party in 
               
               
                 District Political Status 
                 1 st /2 nd /3 rd /4 th  Place 
               
               
                   
               
             
            
               
                 Four parties have same number of voters 
                 25/25/25/25 
               
               
                 Three parties tied with the most voters 
                 33/33/33/0 
               
               
                 Two parties tied with the most voters and their 
                 45/45/10/0 
               
               
                 voter count exceeds party in 3 rd  place by 1 
               
               
                 Two parties tied with the most voters and their 
                 50/50/0/0 
               
               
                 voter count exceeds party in 3 rd  place by at least 2 
               
               
                 One party is alone in the lead and two parties have 
                 60/20/20/0 
               
               
                 one less voter than the leader 
               
               
                 One party is alone in the lead and one party has 
                 70/30/0/0 
               
               
                 one less voter than the leader 
               
               
                 One party is alone in the lead and its voter count 
                 100/0/0/0 
               
               
                 exceeds party in 2 nd  place by at least 2 
               
               
                   
               
            
           
         
       
     
     (B) Next, a random number from 1-100 is produced by simultaneously rolling the two 10-sided dice. The value on the black (white) die is the tens (ones) digit of the random number. For example, if the black (white) die shows 7 (1), the result is 71. If the black (white) die shows 0 (8), the result is 8. The only exception to the above rule is that a roll of “zero-zero” gives the result of 100. In the unlikely event that the result is 100 and three parties are tied for having the most voters in the district, the dice should be re-rolled until a result below 100 is obtained. 
     (C) The winner of the district&#39;s election is then determined by comparing the random number to the parties&#39; winning percentages in the district. If the random number is less than or equal to the winning percentage of the 1 st  place party, the 1 st  place party wins the district election. Otherwise, if the random number is less than or equal to the sum of the winning percentages of the 1 st  and 2 nd  place parties, the 2 nd  place party wins the district election. Otherwise, if the random number is less than or equal to the sum of the winning percentages of the 1 st , 2 nd , and 3 rd  place parties, the 3 rd  place party wins the district election. If the 1 st , 2 nd , and 3 rd  place parties have the same number of voters in a district and the result is 100, the dice are re-rolled until a value below 100 is obtained. Otherwise, if the random number is greater than the sum of the winning percentages of the 1 st , 2 nd  and 3 rd  place parties, the 4 th  place party wins the district election. In the example described in step A, the Green Party wins the Green District election if the random number is any value from 1-60; the Red Party wins the Green District election if the random number is any value from 61-80; and the Orange Party wins the Green District election if the random number is any value from 81-100. All scoring tokens are then removed from that district&#39;s portion of the scoreboard, and a single marker matching the color of the party that wins the election that is placed on the “Red Wins,” “Orange Wins,” “Green Wins,” or “Blue Wins” square in that district&#39;s portion of the scoreboard. 
     The above procedure is repeated for each of the nine political districts. 
     Illustrative Example 
     We consider the (4-player) game whose final board position is shown in  FIG. 42B . The scoreboard for this position is shown in the section “Game Summary” above. Note that a 1 st  place party automatically wins a district election (with 100% chance) if it has at least two more voters than the 2 nd  place party. Thus, in the game at hand, the Blue Party automatically wins the Yellow District, and the Red Party automatically wins the Gray District. 
     Dice  618  are then rolled to determine the winners of the seven districts in which there is 
     not an automatic winner. The election results are summarized in the table below. In the table, the dice rolls are shown in the “Dice Roll” column, and “R,” “O,” “G,” and “B” refer to the Red, Orange, Green, and Blue Party respectively. The overall result is that the Blue (Red, Green) Party wins the elections in three (two, four) districts. 
     
       
         
           
               
               
               
               
               
               
               
               
               
             
               
                   
               
               
                   
                 Red 
                 Orange 
                 Green 
                 Blue 
                 Party 
                 Winning 
                   
                   
               
               
                   
                 Party 
                 Party 
                 Party 
                 Party 
                 Ranking 
                 Percentage 
                 Dice 
                 Election 
               
               
                 District 
                 Voters 
                 Voters 
                 Voters 
                 Voters 
                 (1 st , 2 nd , 3 rd , 4 th ) 
                 1 st /2 nd /3 rd /4 th   
                 Roll 
                 Result 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 Brown 
                 4 
                 5 
                 4 
                 3 
                 O, R, G, B 
                 60/20/20/0 
                 81 
                 Green Wins 
               
               
                 Red 
                 4 
                 5 
                 2 
                 5 
                 O, B, R, G 
                 45/45/10/0 
                 45 
                 Orange Wins 
               
               
                 Orange 
                 4 
                 4 
                 4 
                 4 
                 R, O, G, B 
                 25/25/25/25 
                 34 
                 Orange Wins 
               
               
                 Yellow 
                 3 
                 4 
                 3 
                 6 
                 B, O, R, G 
                 100/0/0/0 
                 — 
                 Blue Wins 
               
               
                 Green 
                 3 
                 3 
                 4 
                 2 
                 G, R, O, B 
                 60/20/20/0 
                  2 
                 Green Wins 
               
               
                 Blue 
                 3 
                 2 
                 5 
                 6 
                 B, G, R, O 
                 70/30/0/0 
                 100  
                 Green Wins 
               
               
                 Purple 
                 6 
                 5 
                 5 
                 4 
                 R, O, G, B 
                 60/20/20/0 
                 57 
                 Red Wins 
               
               
                 Pink 
                 3 
                 5 
                 5 
                 3 
                 O, G, R, B 
                 50/50/0/0 
                 66 
                 Green Wins 
               
               
                 Gray 
                 6 
                 3 
                 4 
                 3 
                 R, G, O, B 
                 100/0/0/0 
                 — 
                 Red Wins 
               
               
                   
               
            
           
         
       
     
     Phase 3: Identify the Winner(s) 
     In the game&#39;s final phase, the overall winner is identified. If phase 2 was played, the winner is the player whose party wins the most district elections. If more than one party ties for winning the most district elections, these parties together win the game and the result is a tie. For example, the Green Party wins the game shown in the table above. 
     If phase 2 was not played, the scoreboard is used to identify the party that controls each district. Two or more parties jointly control a district if they tie for having the greatest number of voters in the district. If there is no tie, the party with the greatest number of voters in the district solely controls the district. Each party receives three points for each district that it solely controls and one point for each district that it jointly controls. The player whose party has more points than any other player&#39;s party is the winner. If parties represented by two or more players tie for having the most points, those players jointly win. A party not represented by a player may not win the game. Refer to section “Game Summary” above to see who wins the game whose final position is shown in  FIG. 42B  if phase 2 is not played. 
     Example 12 
     This example encompasses several games that are larger versions of the game described in Example 11. These larger versions are played with more sectors than Example 11 but are otherwise very similar to Example 11. The relationship of these games to Example 11 is analogous to the relationship of Examples 4-8 to Example 3. 
     The taxonomic codes for five possible games included in this example are listed below. All games are analog, multi-player games with square sectors and a political focus in which four types of elements—namely four political parties—are present, the game proceeds according to alternating turn-based play, there are 2-4 players, and moves in categories “E,” “X,” and “R” are allowed.
         A/G/S2/P/4/12/U/2-4/EXR   A/G/S3/P/4/15/U/2-4/EXR   A/G/S4/P/4/15/U/2-4/EXR   A/G/S5/P/4/21/U/2-4/EXR   A/G/S6/P/4/21/U/2-4/EXR       

     The aforementioned five games get progressively larger with 72, 90, 121, 169, and 210 sectors respectively. 
     The move types allowed in the aforementioned five games are identical to the move types described in Examples 4-8 respectively. 
     The main difference between these games and Examples 4-8 is that up to four players can play these games. 
     Play of any of the above games proceeds in a manner similar to Example 11. During the first phase of the game, players build the political districts by assigning sectors to political districts one sector at a time. They may also reassign sectors from large districts to adjacent smaller districts in order to better equalize the district sizes. During the (optional) second phase of the game, the political status of each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice. In the game&#39;s final phase, the parties that control the districts are identified, and 3 (1) points are awarded to a party that has sole (joint) control of a district. The player whose party has more points than any other player&#39;s party is the winner. If parties represented by two or more players tie for having the most points, those players jointly win. 
     Example 13 
     Strategy games of Example 13 are a combination of the simultaneous independent play undertaken in Example 9 and the four-party environment considered in Example 11. Such games have taxonomic code A/G/S1/P/4/9/I. They are analog, multi-player games with 36 square sectors and a political focus in which four types of elements—namely four political parties—are present and nine districts are formed. The games proceed according to simultaneous independent play, so any number of players may participate. 
     Components 
     In one example, the game components are highly similar to the components used in Examples 9 and 11 and shown in  FIGS. 36-37 and 41-43 . A timer  814  may be needed to play this game, if time constraints are being used. In addition, each player should have a copy of the same game set which contains the following items:
         36 sector tiles  910  having the same markings as the sectors in Example 11   80 expansion markers  908  (8 Brown, 10 Red, 10 Orange, 8 Yellow, 10 Green, 10 Blue, 8 Purple, 8 Pink, 8 Gray)   36 scoring tokens  812  (4 for each of the nine district colors)   A scoreboard  916  (same as in  FIG. 43 )   A game board  818  (similar to  FIG. 37  but with more scoring rows on the left side)
 
Setup (about 10 minutes)
       

     The setup is very similar to that in Example 9, but with each player laying out a region  902  having sectors  904 . Overall, each player creates a copy of the exact same 6×6 sector arrangement in his/her playing region  902  and organizes piles of markers within his/her playing area to prepare for what follows. 
     Playing the Game 
     Play may include four phases, each having a different pre-defined goal, as listed below. Phases 2-4 are optional.
         1. Build political districts that maximize the Red Party&#39;s advantage   2. Build political districts that maximize the Orange Party&#39;s advantage   3. Build political districts that maximize the Green Party&#39;s advantage   4. Build political districts that maximize the Blue Party&#39;s advantage       

     Each phase proceeds like a phase described in Example 9. Players use their markers, scoreboard, and game board as desired to try to achieve the pre-defined goal. The main goal in each phase is to create 9 political districts—i.e. a district plan—in which the concerned party controls as many districts as possible. A party controls a district if it has strictly more voters in a district than any other party. Each player&#39;s secondary goal is to maximize the total amount—summed over the districts controlled by the concerned party—by which the concerned party leads its closest adversary in the districts that it controls. 
     At the end of each phase of the game, each player tracks his/her score with respect to the pre-defined goals above by placing markers on squares in relevant rows  828  of his/her game board  818 . Penalties are assessed if a player&#39;s district plan violates requirement A or B (see Example 9). 
     Identifying the Winner(s) 
     The winner of the game is identified, by process of elimination, by looking at the markers on the rows  828  of each player&#39;s game board  818 . These markers show the scores for up to eight items:
         1. Number of districts controlled by the Red Party in phase 1   2. Total amount by which the Red Party controls its districts in phase 1   3. Number of districts controlled by the Orange Party in phase 2 (if phase 2 played)   4. Total amount by which the Orange Party controls its districts in phase 2 (if played)   5. Number of districts controlled by the Green Party in phase 3 (if phase 3 played)   6. Total amount by which the Green Party controls its districts in phase 3 (if played)   7. Number of districts controlled by the Blue Party in phase 4 (if phase 4 played)   8. Total amount by which the Blue Party controls its districts in phase 4 (if played)       

     If all phases are played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest sum of scores for items 1+3+5+7 is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest sum of scores for items 2+4+6+8 is eliminated. Any player not eliminated wins the game. 
     Example 14 
     Strategy games of Example 14 combine the (optionally) time-limited, simultaneous independent play undertaken in Example 10 and the four-party environment considered in Example 11. Such games have a taxonomic code A/G/S1/P/4/9/I. They are analog, multi-player games with 36 square sectors and a political focus in which four types of elements—namely four political parties—are present and nine districts are formed. These games proceed according to simultaneous independent play, so any number of players may participate. 
     Components 
     In one example, the game components are nearly identical to those in Example 13. The only difference is that a few additional markers are needed to track the final score on the game board. 
     Setup (about 10 Minutes) 
     The setup is very similar to that in Example 9. Overall, each player creates a copy of the exact same 6×6 sector arrangement in his/her region. 
     Scoreboard and Scoring Tokens 
     The scoreboard and scoring tokens may be the same as in Example 11. 
     Playing the Game 
     Play proceeds just as in Example 10. As long as time has not expired, players may use their markers, scoreboard, and game board as desired to try to achieve the desired goal. Each player&#39;s main goal is to create 9 political districts—i.e. a district plan—in which all four parties control the same number of districts. A party controls a district if it has strictly more voters in a district than any other party. Each player&#39;s secondary goal is to equalize the total amount—summed over the districts controlled by each party—by which each party leads its closest adversary in the districts that it controls. Each player&#39;s tertiary goal is to maximize the number of districts in which all four parties have the same number of voters. 
     At the end of play (e.g., when time expires), each player tracks his/her score with respect to the 3 goals above by placing markers on squares in the left part of his/her game board. Penalties are assessed if a player&#39;s district plan violates requirement A or B (see Example 9). 
     Identifying the Winner(s) 
     The winner of the game is identified by looking at the markers on the left side of each player&#39;s game board. These markers show the scores for eleven items:
         1. Number of districts controlled by the Red Party   2. Total amount by which the Red Party controls its districts   3. Number of districts controlled by the Orange Party   4. Total amount by which the Orange Party controls its districts   5. Number of districts controlled by the Green Party   6. Total amount by which the Green Party controls its districts   7. Number of districts controlled by the Blue Party   8. Total amount by which the Blue Party controls its districts   9. The magnitude of the difference between the highest value among items 1, 3, 5, and 7 and the lowest value among items 1, 3, 5, and 7.   10. The magnitude of the difference between the highest value among items 2, 4, 6, and 8 and the lowest value among items 2, 4, 6, and 8.   11. The number of districts in which all parties have the same number of voters.       

     The winner is identified by process of elimination. First, every player whose district plan violates one of the requirements A-B (see description of Example 9) is eliminated. Second, among the remaining players, every player whose district plan does not tie for having the lowest score for item 9 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the lowest score for item 10 above is eliminated. Finally, among the remaining players, every player whose district plan does not tie for having the highest score for item 11 above is eliminated. Any player who is not eliminated wins the game. If all players&#39; district plans violate one of the requirements A-B (see description of Example 9), all players lose. 
     Example 15 
     Strategy games of Example 15 have taxonomic code A/G/T1/P/2/9/U/2/EXR and are triangular-sector games similar to the square-sector games of Example 3. They are analog, multi-player games with 54 sectors in the shape of an equilateral triangle. They have two types of elements—such as two political parties—and nine districts are formed. The games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed. 
     Game Summary 
       FIGS. 44-46  illustrate one example of a strategy game  1600 . In this game, two players representing opposing political parties—Red and Blue—vie for political control of a hexagonal region  1602  by competitively creating nine political districts out of 54 triangular sectors in alternating, turn-based fashion. The region  1602  may be divided into 54 sector placeholders  1608 , and sector tiles  1606  may be laid out, on each sector placeholder  1608 , to form sectors  1604  as part of the game set-up. Each sector  1604  may represent a community, and the set of elements on the sector tile  1606  for each sector  1604  includes a type (e.g., favored political party) and quantity (e.g., voter margin by which the indicated party is favored). The strategy game  1600  may also include a set of markers  1610 , divided into marker subsets  1612  each of which represents a district. Each marker subset may include a home base marker  1614  and at least one expansion marker  1616 . The number of marker subsets  1612  preferably equals the number of districts to be formed during the game. The total number of home base markers  1614  and expansion markers  1616  for each marker subset  1612  should be sufficient to form districts of appropriate size for the game. The strategy game  1600  may further include a scoreboard  1618  (same as  FIG. 13 ), which may have one row  1624  for each district to be formed, and one scoring token  1620  for each district to be formed. The strategy game  1600  may also include two dice  1622 , which may be ten sided dice, each having a different color such as one black and one white. 
     During the first phase of strategy game  1600 , players build the political districts by assigning sectors to political districts one sector at a time. They may also reassign sectors from large districts to adjacent smaller districts in order to better equalize the district sizes. During the optional second phase of strategy game  1600 , the political margin in each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice  1622 . The winner is the player whose party controls more districts than his/her opponent. A tie is possible if players skip phase 2 of the game. 
       FIG. 45  shows one possible initial position for a strategy game  1600 , in which the sector tiles  1606  have been placed on the region, with one on each sector placeholder  1608 , to form sectors  1604 .  FIG. 46  shows one possible final position for a game having the initial position shown in  FIG. 45 , with the home base markers  1614  and expansion markers  1616  placed on the region  1602 , one per sector  1604 , to form nine districts (indicated by bold lines). The table below shows the final result for the final position shown in  FIG. 46 . If phase 2 is not played, the Red Party wins the game by a score of 5 districts to 3 districts (with one tied district): 
     
       
         
           
               
               
               
               
             
               
                   
                   
               
               
                   
                   
                 Party in 
                 By How 
               
               
                   
                 District 
                 Control 
                 Much 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Brown 
                 Blue 
                 12 
               
               
                   
                 Red 
                 Red 
                 8 
               
               
                   
                 Orange 
                 Neither 
                 0 
               
               
                   
                 Yellow 
                 Red 
                 2 
               
               
                   
                 Green 
                 Red 
                 1 
               
               
                   
                 Blue 
                 Blue 
                 11 
               
               
                   
                 Purple 
                 Blue 
                 8 
               
               
                   
                 Pink 
                 Red 
                 6 
               
               
                   
                 Gray 
                 Red 
                 14 
               
               
                   
                   
               
            
           
         
       
     
     Game Components 
     The game components are as follows:
         54 triangular sector tiles  1606  (quantity of each sector tile type is shown in parentheses below):       

     
       
         
           
               
               
               
               
               
               
             
               
                   
               
             
            
               
                 +9 Red (3) 
                 +5 Red (3) 
                 +1 Red 
                 (2) 
                 +3 Blue (3) 
                 +7 Blue (3) 
               
               
                 +8 Red (3) 
                 +4 Red (3) 
                 0 
                 (2) 
                 +4 Blue (3) 
                 +8 Blue (3) 
               
               
                 +7 Red (3) 
                 +3 Red (3) 
                 +1 Blue 
                 (2) 
                 +5 Blue (3) 
                 +9 Blue (3) 
               
               
                 +6 Red (3) 
                 +2 Red (3) 
                 +2 Blue 
                 (3) 
                 +6 Blue (3) 
               
               
                   
               
            
           
         
       
         
         
           
             9 home base markers  1614  (one for each of 9 districts: Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) 
             180 expansion markers  1616  (20 for each of the nine districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray) 
             9 scoring tokens  1620  (one for each of 9 districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray), each being cube-shaped with faces showing the numbers 10, 20, 30, 40, and 50 
             2 ten-sided dice  1622  (1 black, 1 white), each showing the values 0-9 
             A scoreboard  1618   
           
         
       
    
     Playing the Game 
     The game has three phases and plays in a manner similar to Example 3. 
     In phase 1, players take alternating turns beginning with the starting player. During a player&#39;s turn, he/she (A) makes one move and then (B) records the move on two scoreboards. Forfeiting a turn (i.e. passing on a turn) is not allowed. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 60 moves—30 by each player—are made in phase 1. 
     Phase 1 is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 9 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 9 th  district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Phase 1 concludes when no legal moves exist. 
     The rules may allow districts to be formed by the following six types of moves. The meanings of the phrases “connectedness,” “captured tile,” and “trapped district” are analogous to those in Example 3.
         1 Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (a′) The sector must be at least three steps away from all previously placed home base markers. (b′) There must be space to grow this district to size 6 or more. Sectors  1  and  7  in  FIG. 44  are five steps away from each other.   1A Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (c′) The sector must be in the largest open space on the board. An open space is a set of connected vacant sectors. (d′) Among the sectors satisfying this criterion, the sector must be the farthest, in number of steps, from a previously placed home base marker.   2 Expand a district by placing one of its expansion markers on a vacant sector. This move must meet three requirements. (e′) The district must remain connected. (f) The district&#39;s new size—including this new sector and any sectors that are captured—must not be greater than 6 sectors. (g′) No district may be trapped.   2A Expand a district by placing one of its expansion markers on a vacant sector. This move must meet three requirements. (e′) The district must remain connected. (h′) Only the smallest expandable district may be expanded. (i′) No sectors may be captured.   3 (Same as in Examples 3-8 and 11-12)   3A (Same as in Examples 3-8 and 11-12)       

     Phases 2-3 are played almost exactly as in Example 3. 
     Rules for a Symmetric Game 
     A symmetric version of this game may be played if players are concerned about bias in the initial sector arrangement. The purpose of a symmetric game is to remove bias from the initial sector arrangement and thereby give each party—Red and Blue—a fair chance of winning the game. 
     A symmetric game has three additional rules compared to a non-symmetric game. Rule 1 guarantees a symmetrical initial sector arrangement, whereas rules 2 and 3 minimize the possibility of a symmetric board position during play. The three rules are as follows.
         1. During the game setup, the sector tile arrangement must be counter-symmetric with respect to the dot in the center of the region. A counter-symmetric sector arrangement is one in which the two sectors comprising every pair of diametrically-opposed sectors—i.e. every two sectors that are on exact opposite sides of the region—have opposite voter margins—for example “+4 Red” and “+4 Blue.” The procedure may be nearly identical to that described in the section “Rules for a symmetric game” in Example 3.   2. During the second and third moves of phase 1 (i.e. the 1 st  move made by the player who goes second, and the 2 nd  move made by the player who goes first), no marker may be placed on sector that is diametrically opposed to a sector on which a marker has already been placed.   3. During phase 1, any move that would result in a district that (a) has size six and (b) coincides with the hexagon in the center of the region is strictly forbidden.       

     Example 16 
     Strategy games of Example 16 are larger versions of the games described in Example 15. These games have taxonomic code A/G/T2/P/2/12/U/2/EXR. They are analog, multi-player games with 96 triangular sectors in which 12 districts are formed. Such games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed. 
     In one example, a strategy game may be very similar to Example 15. In such a game, two players—Red and Blue—vie for political control of the hexagonal region by competitively creating 12 political districts (whose average size is 8) out of 96 triangular sectors. The sectors may be pre-established on the region, or formed by placing one sector tile on each of 96 sector placeholders on the region. The game can be played with any 96 triangular sectors, though it is preferred that the sets of sectors favoring red and blue be identical—for example 6 each of sector tiles “+2 Red” to “+9 Red” and “+2 Blue” to “+9 Blue” (96 sectors total). Players take alternating turns beginning with the starting player. During a player&#39;s turn, he/she (A) makes one move and then (B) records the move on two scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 100 moves-50 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent. 
     Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 12 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 12 th  district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist. 
     The six types of legal moves are summarized below.
         1 Establish a district on a sector that is at least 3 steps away from all other home base markers. There must be space to grow this district to size 8 or more.   1A Establish a district on a sector that ties for being the farthest, in number of steps, from a home base marker (among the sectors in the largest open space on the board).   2 Expand a district so (e′) it remains connected, (f) its new size (including any captured sectors) is 8 or less, and (g′) no district is trapped.   2A Expand the smallest expandable district so (e′) it remains connected, (i′) no sectors are captured, and (o′) its new size is 21 or less.   3 (Same as in Examples 3-8, 11-12, and 15)   3A (Same as in Examples 3-8, 11-12, and 15)       

     Example 17 
     Strategy games of Example 17 are even larger versions of the games described in Example 15. These games have taxonomic code A/G/T3/P/2/15/U/2/EXR. They are analog, multi-player games with 150 triangular sectors, and may have a political focus in which two types of elements—namely two political parties—are present and 15 districts are formed. These games may proceed according to alternating, turn-based play; may have two players; and moves in categories “E,” “X,” and “R” may be allowed. 
     Game Summary 
       FIGS. 47-48  illustrate elements of a strategy game  1000 . In this game, two players representing opposing political parties—Red and Blue—vie for political control of a hexagonal region  1002  by competitively creating fifteen political districts out of 150 triangular sectors  1004  in alternating, turn-based fashion. The region  1002  may be divided into 150 sector placeholders, and sector tiles  1006  may be laid out, on each sector placeholder, to form sectors  1004  as part of the game set-up. Alternatively, each sector  1004  may be pre-established on the region  1002 . Each sector  1004  may represent a community, and the set of elements  1008  in each sector  1004  includes a type (e.g., favored political party) and quantity (e.g., voter margin by which the indicated party if favored). The strategy game  1000  may also include a set of markers, divided into marker subsets that each represents a district. Each marker subset may include a home base marker  1010  and at least one expansion marker  1012 . The number of marker subsets preferably equals the number of districts to be formed during the game. The total number of home base markers  1010  and expansion markers  1012  for each marker subset should be sufficient to form districts of appropriate size for the game. The strategy game  1000  may further include a scoreboard  1014 , which may have one row  1016  for each district to be formed, and scoring tokens sufficient to track the score during play. The strategy game  1000  may also include two dice (which may be identical to dice  1622 ). 
     During the first phase of the game, players build the political districts by assigning sectors  1004  to political districts, one sector per turn. They may also reassign sectors during a turn from large districts to small districts in order to better equalize the district sizes. During the (optional) second phase of the game, the political margin in each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice. The winner is the player whose party controls more districts than his/her opponent. A tie is possible if players skip the game&#39;s second phase. 
     Game Components 
     The game components may be as follows:
         150 triangular sector tiles  1006  (quantity of each sector type is shown in parentheses below):       

     
       
         
           
               
               
               
               
               
               
             
               
                   
               
             
            
               
                 +9 Red (8) 
                 +5 Red (8) 
                 +1 Red 
                 (8) 
                 +3 Blue (8) 
                 +7 Blue (8) 
               
               
                 +8 Red (8) 
                 +4 Red (8) 
                 0 
                 (6) 
                 +4 Blue (8) 
                 +8 Blue (8) 
               
               
                 +7 Red (8) 
                 +3 Red (8) 
                 +1 Blue 
                 (8) 
                 +5 Blue (8) 
                 +9 Blue (8) 
               
               
                 +6 Red (8) 
                 +2 Red (8) 
                 +2 Blue 
                 (8) 
                 +6 Blue (8) 
               
               
                   
               
            
           
         
       
         
         
           
             15 home base markers  1010  (one for each of the 15 districts—Brown, Red, Orange, Yellow, Light Green, Dark Green, Light Blue, Dark Blue, Purple, Pink, Light Gray, Dark Gray, Black, White, Gold) 
             450 expansion markers  1012  (30 for each of the aforementioned 15 districts) 
             60 scoring tokens (four for each of the aforementioned 15 districts) 
             2 ten-sided dice (1 black, 1 white), each showing the values 0-9 
             Scoreboard  1014  ( FIG. 48 ) 
           
         
       
    
     Playing the Game 
     The play is very similar to Example 15 but is more challenging owing to the many sectors. Play may include the following three phases. The second phase is optional.
         1. Build political districts   2. Run an election (optional)   3. Identify the winner       

     Phase 1: Build Political Districts 
     Summary 
     During the first phase of strategy game  1000 , players take turns assigning/reassigning sectors to political districts, one sector at a time, until every sector belongs to a political district. The assignment of a sector  1004  to a political district is accomplished by placing a home base marker  1010  or expansion marker  1012  on a sector tile. Reassignment of a sector  1004  from a district to another district may be accomplished by changing the color of the marker on a sector tile. At the end of the first phase, 15 political districts (e.g., Brown, Red, Orange, Yellow, Light Green, Dark Green, Light Blue, Dark Blue, Purple, Pink, Light Gray, Dark Gray, Black, White, and Gold) will be formed on the region  1002 . 
     Each district evolves in the same general way. Initially, it is formless. At some point, it is established when its home base marker  1010  is placed on a vacant sector  1004 . It is then expanded whenever one of its expansion markers  1012  is placed on a vacant sector that is adjacent to a sector that already belongs to the district. Later, it may be resized so its size is more similar to neighboring districts by reassignment of sectors  1004 . 
     The process of building political districts is relatively unrestricted. There is no general requirement for the sequence in which, or locations where, districts are constructed. Once begun, the construction of a district may be temporarily halted while players take turns establishing, expanding, and/or resizing other districts. There may not be any district size requirement. However, the rules may encourage or require the creation of districts of size 10 (meaning that each district is formed from 10 sectors  1004 ). 
     Importantly, all marker subsets and all sectors are available to all players. No player “owns” any marker subset or sector. As long as the rules below are followed, any player may contribute to building any district during any turn. No matter which player established a district, any other player may expand the district or reassign a sector from that district to another district. 
       FIG. 47  illustrates one possible final position at the conclusion of game play, with fifteen districts formed by the home base markers  1010  and expansion markers  1012  (the boundaries of which are indicated by bold lines). 
     Details 
     Players take turns beginning with the starting player. During a player&#39;s turn, he/she (A) makes one move and then (B) records the move on the scoreboard. 
     All moves must be of type 1, 1A, 2, 2A, 3, or 3A. Moves of type 1 and 1A establish a new district. Moves of type 2 and 2A expand an existing (i.e. already established) district. Moves of type 3 and 3A resize two adjacent districts. “A” means “alternate move.” 
     Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 15 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 15 th  district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. In many games, stage 2 is skipped and play proceeds directly from stage 1 to stage 3. Phase 1 concludes when no legal moves exist. 
     The six types of legal moves are as follows. The meanings of the phrases “connectedness,” “captured tile,” and “trapped district” are analogous to those in Example 3.
         1 Establish a district on a sector that is at least 4 steps away from all other home base markers. There must be space to grow this district to size 10 or more connected sectors.   1A Establish a district on a sector that ties for being the farthest, in number of steps, from a home base marker (among the sectors in the largest open space on the board).   2 Expand a district so (e′) it remains connected, (f) its new size (including any captured sectors) is 10 or less, and (g′) no district is trapped.   2A Expand the smallest expandable district so (e′) it remains connected, (i′) no sectors are captured, and (o′) its new size is 31 or less.   3 Reassign a sector from one district to another. (Same as Examples 3-8, 11-12, 15-16)   3A Reassign a sector from one district to another. (Same as Examples 3-8, 11-12, 15-16)       

     Phase 2: Run an Election 
     The second phase of strategy game  1000  is optional. In the second phase, the political margin in each district is converted into a numerical likelihood of each party winning the district, and an election in each district is simulated by rolling the dice. 
     Each district shown in the final position (e.g., in  FIG. 47 ) is individually considered. First, using the table below, the political margin for the party with more voters in the district is converted into a numerical likelihood of that party winning an election in the district. 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                 Political 
                 Winning 
               
               
                   
                 Margin 
                 Likelihood 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                   
                 0 
                 50% 
               
               
                   
                 +1 
                 59% 
               
               
                   
                 +2 
                 67% 
               
               
                   
                 +3 
                 74% 
               
               
                   
                 +4 
                 80% 
               
               
                   
                 +5 
                 85% 
               
               
                   
                 +6 
                 89% 
               
               
                   
                 +7 
                 92% 
               
               
                   
                 +8 
                 94% 
               
               
                   
                 +9 
                 96% 
               
               
                   
                 +10 
                 98% 
               
               
                   
                 +11 
                 99% 
               
               
                   
                 +12 or more 
                 100%  
               
               
                   
                   
               
            
           
         
       
     
     Next, a random number from 1-100 is produced by simultaneously rolling the two 10-sided dice. See the section “Phase 2: Run an election” in Example 3 for more details. 
     The random number is then compared to the winning percentage. If the random number is less than or equal to the winning percentage, the party with more voters in the district wins the district election. If the random number is greater than the winning percentage, the party with fewer voters in the district wins the district election. If both parties have a 50% chance of winning, the Blue Party wins if the random number is 1-50 and the Red Party wins if the random number is 51-100. After the winner of an election is identified, the unused scoring token that matches the district color is placed on the “Blue Wins” or “Red Wins” square in that district&#39;s row in the scoreboard. 
     The above procedure is repeated for each of the 15 political districts. 
     Phase 3: Identify the Winner 
     In the game&#39;s final phase, the winner is identified. If phase 2 is played, the winner is the player whose party wins eight or more district elections. 
     If phase 2 is not played, the scoreboard is used to identify the party that controls each district, i.e. the party with more voters in each district. The winner is the player whose party controls more districts than his/her opponent. If the two players control an equal number of districts, the result is a tie. 
     Rules for a Symmetric Game 
     As with many previous examples, strategy game  1000  can be played as a symmetric game. The purpose of a symmetric game is to remove bias from the initial sector arrangement and give each party—Red and Blue—a fair chance of winning the game. 
     A symmetric game for strategy game  1000  has three additional rules compared to a regular game. Rule 1 creates a symmetric initial sector arrangement, and rules 2 and 3 reduce the possibility of a symmetric position during play. The three rules are as follows.
         1. During the game setup, the sector arrangement must be counter-symmetric with respect to an imaginary dot in the center of the region. The procedure for doing this may be nearly identical to that described with respect to Example 3.   2. During the 2 nd , 3 rd , 4 th , and 5 th  moves of phase 1 (i.e. the 1 st  and 2 nd  moves made by the player who goes second, and the 2 nd  and 3 rd  moves made by the player who goes first), no marker may be placed on a sector that is diametrically opposed to a sector on which a marker has already been placed.   3. During phase 1, no move may result in a district that (a) has size ten, (b) has a political margin of zero, and (c) entirely covers the small hexagon at the center of the region  1002 .       

     Example 18 
     Strategy games of this Example 18 are even larger versions of the games with hexagonal regions and triangular sectors described in Examples 15-17. These games have taxonomic code A/G/T4/P/2/27/U/2/EXR. They are analog, multi-player games with 216 triangular sectors, and may have a political focus in which two types of elements—namely two political parties—are present and 27 districts are formed. These games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed. 
     In one example, two players—Red and Blue—vie for political control of giant hexagonal region by competitively creating 27 political districts (whose average size is 8) out of 216 triangular communities. The game can be played with any 216 triangular sectors in which the sets of red and blue sectors are identical—for example 12 each of sector tiles “+2 Red” to “+9 Red” and “+2 Blue” to “+9 Blue” (192 sector tiles); 8 each of sector tiles “+1 Red” and “+1 Blue” (16 sector tiles); and 8 sector tiles with a voter margin of 0. Players take turns beginning with the starting player. During a player&#39;s turn, he/she (A) makes one move and then (B) records the move on three scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 240 moves—120 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent. 
     Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 27 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 27 th  district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist. 
     This game has the same sector shape and same final average district size as Example 16. Thus, the details for the six types of legal moves—listed below—are identical to Example 16.
         1 Establish a district on a sector that is at least 3 steps away from all other home base markers. There must be space to grow this district to size 8. (Same as Example 16)   1A Establish a district on a sector that ties for being the farthest, in number of steps, from a home base marker (among the sectors in the largest open space on the board). (Same as Example 16)   2 Expand a district so (e′) it remains connected, (f) its new size (including any captured sectors) is 8 or less, and (g′) no district is trapped. (Same as Example 16)   2A Expand the smallest expandable district so (e′) it remains connected, (i′) no sectors are captured, and (o′) its new size is 21 or less. (Same as Example 16)   3 (Same as Examples 3-8, 11-12, 15-17)   3A (Same as Examples 3-8, 11-12, 15-17)       

     Example 19 
     Strategy games of Example 19 combine the simultaneous independent play undertaken in Example 9 and the triangular sectors used in Example 15. These games have taxonomic code A/G/T1/P/2/9/I. They are analog, multi-player games with 54 triangular sectors and a political focus in which two types of elements (e.g., two political parties) are present and nine districts are formed. The game paradigm is simultaneous independent play, so any number of players may participate. 
     Components 
     The game components are highly similar to the components used in Example 15, shown in  FIGS. 44-46 , with the addition of a game board  1100  as shown in  FIG. 49 , and a timer (such as timer  814  of Example 8) if time constraints will be used. Accordingly each player should have a copy of the same game set which contains the following items:
         Region  1602     54 sector tiles  1606  having the same markings as the sectors in Example 15   112 expansion markers  1616  (12 Brown, 14 Red, 12 Orange, 12 Yellow, 12 Green, 14 Blue, 12 Purple, 12 Pink, 12 Gray). These are simply referred to as “markers.”   9 scoring tokens (same as in Example 15)   Scoreboard  1618  (same as  FIG. 13 )   Game board  1100 , which may have a duplicate region  1102  divided into 54 duplicate sector placeholders  1106 . The game board  1100  may also have a plurality of rows  1104  configured to allow a player to track aspects of districts formed on the game board  1100 .       

     Playing the Game 
     Play may include of the following two phases. Phase 2 is optional.
         1. Build political districts that maximize the Red Party&#39;s advantage   2. Build political districts that maximize the Blue Party&#39;s advantage       

     Each phase proceeds like a phase described in Example 9. Players use their markers, scoreboard, and game board as desired to try to achieve the desired goal. The main goal in each phase is to create 9 political districts of equal size—i.e. a district plan—in which the concerned party controls as many districts as possible. Each player&#39;s secondary goal is to make the “voter margin in the district that the concerned party controls by the least amount” as high as possible. In the district plan, (A) each sector  1604  must be assigned to exactly one district and (B) each district must consist of six connected tiles as in  FIG. 7 . 
     At the end of each phase of the game, each player tracks his/her score with respect to the two goals above by placing markers on the relevant square of the relevant row  1104  of his/her game board  1100 . Penalties are assessed if a player&#39;s district plan violates requirement A or B above. 
     Identifying the Winner(s) 
     The winner of the game is identified, by process of elimination, by looking at the markers on the left side of each player&#39;s game board. These markers show the scores for up to four items:
         1. Number of districts controlled by the Red Party in phase 1   2. Lowest voter margin in the districts controlled by the Red Party in phase 1   3. Number of districts controlled by the Blue Party in phase 2 (if phase 2 played)   4. Lowest voter margin in the districts controlled by the Blue Party in phase 2 (if played)       

     If phase 2 is not played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest score for item 1 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest score for item 2 above is eliminated. Any player who is not eliminated wins the game. 
     If phase 2 is played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest sum of scores for items 1+3 is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest sum of scores for items 2+4 is eliminated. Any player not eliminated wins the game. 
     Example 20 
     Strategy games of Example 20 combine the simultaneous independent play undertaken in Example 10 and the triangular sectors used in Example 15. These games have taxonomic code A/G/T1/P/2/9/I. They are analog, multi-player games with 54 triangular sectors and a political focus in which two types of elements—two political parties—are present and 9 districts are formed. The playing paradigm is simultaneous independent play, so any number of players may play. 
     Game Summary 
     In one example, the game is set up exactly as described in Example 19, but the pre-defined goal of each player is to create the most balanced set of political districts. The region has an American-style, two-party political system in which one person is elected to represent each political district. At the outset, the districts are formless and the players know the political status of each sector (i.e. which party its citizens favor and by how much). During the game, players simultaneously and independently work on identical copies of the region map to create political districts that equalize the political advantage of the two parties, Red and Blue. The winner is the player who creates the most balanced set of political districts. 
     The game components, setup, sectors, scoreboard, and scoring methods are identical to Example 19 except that each player uses an additional 3 gray markers to score three additional items on his/her game board. 
     Playing the Game 
     Play proceeds as in Example 19, except with the pre-defined goal to create a district plan that (i) equalizes the number of districts controlled by each party, (ii) equalizes the margin by which each party controls its least safe district, and (iii) maximizes the number of tied districts that have a voter margin of 0. Item (i) has priority over (ii), and (ii) has priority over (iii). Each player&#39;s district plan must satisfy requirements A-B as stated in the section “Playing the game” in the description of Example 19. 
     At the conclusion of play, the players (a) compute the district voter margins, (b) place scoring tokens appropriately on the scoreboard, and (c) compute the following for each player&#39;s final district plan:
         1. The number of districts controlled by the Red Party.   2. The lowest voter margin in the districts controlled by the Red Party.   3. The number of districts controlled by the Blue Party.   4. The lowest voter margin in the districts controlled by the Blue Party.   5. The magnitude of the difference between items 1 and 3 above (the “Red-Blue Control Differential”).   6. The magnitude of the difference between items 2 and 4 above (the “Lowest Voter Margin Differential”).   7. The number of tied districts (with a voter margin of 0).       

     Identifying the Winner(s) 
     The winner is identified by process of elimination. First, every player whose district plan violates one of the requirements A-B (see description of Example 19) is eliminated. Second, among the remaining players, every player whose district plan does not tie for having the lowest score for item 5 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the lowest score for item 6 above is eliminated. Finally, among the remaining players, every player whose district plan does not tie for having the highest score for item 7 above is eliminated. Any player who is not eliminated wins the game. If all players&#39; district plans violate one of the requirements A-B (see description of Example 19), all players lose. 
     Example 21 
     Strategy games of Example 21 combine the alternating, turn-based play for more than two players from Example 11 with the triangular sectors of Example 15. The taxonomic code for these games is A/G/T1/P/3/9/U/2-3/EXR. They are analog, multi-player games with 54 triangular sectors and a political focus in which three types of elements—three political parties—are present and nine districts are formed. The games proceed according to alternating, turn-based play; there are 2-3 players; and moves in categories “E,” “X,” and “R” are allowed. 
     Game Summary 
       FIGS. 50-53  illustrate elements of one example of this kind of game, a strategy game  1200 . In this game, 2-3 players representing opposing political parties—Red, Green, and Blue—vie for political control of a hexagonal region  1202  by competitively creating nine political districts out of 54 triangular sectors  1204  in turn-based fashion ( FIG. 50 ). The sectors  1204  may be pre-established on the region  1202 , or they may be established during game set-up by placing sector tiles  1206  on sector placeholders (not shown) on the region  1202 . 
       FIG. 51  shows two examples of sector tiles  1208  and  1210  for this game. The first sector tile  1208  has a first set of elements  1212 , and the second sector tile  1214  has a second set of elements  1214 . As shown in  FIG. 51 , each sector tile  1208  and  110  contains three identical representations of the set of elements in that tile. Alternatively, each set of elements may be represented once, as shown on sector tiles  1206 , or any suitable number of times in any arrangement suitable to be viewed by the players. Each set of elements includes one symbol for each element type (e.g., Red, Green, and Blue Parties) and one number showing the quantity (e.g., how many people in the sector support the Red, Green, and Blue Parties respectively). There are 6 people in each sector. The distribution of element sets for one example of a set of sector tiles is provided below, with the number of [Blue, Red, Green] Party supporters in a sector indicated in square brackets and the number of sector tiles of that type shown in parentheses: 
     
       
         
           
               
               
               
               
               
               
               
             
               
                   
               
             
            
               
                 [6, 0, 0] (2) 
                 [5, 0, 1] (2) 
                 [0, 1, 5] (2) 
                 [2, 0, 4] (2) 
                 [3, 0, 3] (2) 
                 [1, 1, 4] (2) 
                 [2, 1, 3] (2) 
               
               
                 [0, 6, 0] (2) 
                 [1, 5, 0] (2) 
                 [4, 2, 0] (2) 
                 [0, 4, 2] (2) 
                 [0, 3, 3] (2) 
                 [3, 2, 1] (2) 
                 [1, 3, 2] (2) 
               
               
                 [0, 0, 6] (2) 
                 [1, 0, 5] (2) 
                 [4, 0, 2] (2) 
                 [0, 2, 4] (2) 
                 [4, 1, 1] (2) 
                 [3, 1, 2] (2) 
                 [1, 2, 3] (2) 
               
               
                 [5, 1, 0] (2) 
                 [0, 5, 1] (2) 
                 [2, 4, 0] (2) 
                 [3, 3, 0] (2) 
                 [1, 4, 1] (2) 
                 [2, 3, 1] (2) 
               
               
                   
               
            
           
         
       
     
     The mechanics of this game are slightly different than in previous examples. In this game, the same types of moves—1, 1A, 2, 2A, 3, and 3A—are allowed during play, but there are additional restrictions regarding their timing. In particular, all moves of type 1, 1A, 2, and 2A must be completed during phase 1 of this game—i.e. the board must be completely full—before the first move of type 3 or 3A is allowed to be made in phase 2. The district size threshold in moves of type 2 also differs from earlier examples. 
       FIG. 50  shows a possible initial board position of strategy game  1200 . 
       FIG. 52  shows a scoreboard  1216  that may be used in strategy game  1200 . The scoring method in this game is nearly identical to that in Example 11. Each row  1218  of the scoreboard is used for tracking the political status of a different district. 
     Play may include the following four phases. Phases 2-3 are optional.
         1. Build political districts using moves of type 1, 1A, 2, and 2A   2. Rebalance district sizes using moves of type 3A   3. Run an election (optional)   4. Identify the winner       

     Phase 1: Build Political Districts Using Moves of Type 1, 1A, 2, and 2A 
     This phase proceeds almost exactly like phase 1 in Example 15. The main differences are that in this game (a) turns alternate among up to three players instead of two players and (b) only moves of type 1, 1A, 2, and 2A are available. 
     All moves made in phase 1 must be of type 1, 1A, 2, or 2A below. 
     Play during the first phase is divided into four stages. During stage 1, only moves of type 1 and 2 are allowed. Play enters stage 2 if (i) no more moves of type 1 exist and (ii) fewer than nine districts have been established. During stage 2, only moves of type 1A and 2 are allowed. Play enters stage 3 immediately after the 9 th  district is established. During stage 3, only moves of type 2 are allowed. Play enters stage 4 if no more moves of type 2 exist. During stage 4, only moves of type 2A are allowed. This phase of the game ends when every sector has been assigned to a political district. 
     The four types of legal moves during this phase of the game are as follows. Note that the “5-sector district size restriction” and “no tile capturing restriction” in moves of type 2 is slightly different than in Example 15 and other previous examples.
         1 Establish a new district (by placing its home base marker) on a sector with the restriction that (i) the sector is at least three steps away from each previously placed home base marker and (ii) there is space to grow this district to a size of six connected sectors.   1A Establish a new district (by placing its home base marker) on a sector that is within the largest connected open space on the board. An open space is an area where no sector has been assigned to a district. Among the sectors satisfying the above criterion, select a sector that ties for being the most steps away from a previously placed home base marker.   2 Expand a district (by placing one of its expansion markers on a sector) so that (i) no sectors are captured, (ii) no district is trapped, (iii) the district&#39;s new size is no greater than 5 sectors, and (iv) the district remains connected.   2A Expand the smallest expandable district (by placing one of its expansion markers on a sector) with the restriction that (i) no sectors are captured and (ii) the district remains connected.       

     Phase 2: Rebalance District Sizes Using Moves of Type 3A 
     This optional phase of the game is motivated by the need to keep the populations of real-world political districts nearly equal. In this phase of the game, players take turns modifying the sector-to-district assignments in order to better equalize the district sizes (which are a proxy for the district populations). 
     Players take turns beginning with the player to the left of the player who took the final turn during phase 1. During a player&#39;s turn, he/she makes a move by changing the district to which one sector is assigned. This is done by removing the (home base or expansion) marker that occupies one sector tile and replacing it with an expansion marker of a different color. The net result is that one district loses a sector and one district gains a sector. The other seven districts remain unchanged. The player then updates the scoreboard to reflect the move that has been made. 
     Every move made during this phase of the game must be of type 3A:
         3A Reassign a community from one district (say District X) to another (say District Y). This move must meet four requirements. (j) District Y must exist prior to this move. (k) District Y must not be expandable prior to this move. (1) District X must be at least 2 sectors larger than District Y prior to this move. (m) Districts X and Y must each remain connected.       

     This phase of the game concludes when no more moves of type 3A exist. 
     Phase 3: Run an Election 
     Phase 3 in this game is optional and is very similar to phase 2 in Example 11. During this phase of the game, the political status of each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice. 
     Phase 4: Identify the Winner(s) 
     In the game&#39;s final phase, the overall winner is identified. If phase 3 was played, the winner is the player whose party wins the most district elections. If more than one party ties for winning the most district elections, these parties together win the game and the result is a tie. 
     If phase 3 was not played, the winner is determined by identifying the party that controls each district, i.e. the party with the most voters in each district. Each party receives 6 points for each district that it solely controls; 3 points for each district that it jointly controls with one other party; and 2 points for each district that it jointly controls with two other parties. The winner is the player whose party has more points than any other player&#39;s party. If parties represented by two or more players tie for having the most points, those players jointly win. 
     Symmetric Games 
     A symmetric version of this game may be played if there is a desire to eliminate bias in the initial sector arrangement. A symmetric game has three additional rules compared to a regular game. Rule 1 guarantees an initial sector arrangement that is symmetric, whereas rules 2 and 3 minimize the possibility of a symmetric position during play. The three rules are as follows.
         1. During the game setup, the sector tile arrangement must be rotationally symmetric. In a rotationally symmetric sector arrangement the number of (Blue, Red, Green) Party supporters in every sector is the same as the number of (Red, Green, Blue) Party supporters in the sector that is a 120 degree clockwise rotation from it. Such an arrangement guarantees that the initial map is unbiased, favoring no party.
             FIG. 53  shows a rotationally symmetric sector arrangement. Bold lines distinguish three diamond-shaped portions of the board: the upper-right diamond  1220 , lower-right diamond  1222 , and the left diamond  1224 . Note that the number of (Blue, Red, Green) Party supporters in every sector is the same as the number of (Red, Green, Blue) Party supporters in the sector that is a 120 degree clockwise rotation from it.   
           2. The second, third, and fourth moves made during phase 1 (i.e. the 1 st  move made by the player who goes second, the 1 st  move made by the player who goes third, and the 2 nd  move made by the player who goes first) may not involve the placement of a marker on a sector that is either a 120 degree clockwise or 120 degree counterclockwise rotation from a sector on which a marker has been placed.   3. During phase 1, no move may expand an existing district so that it has size six and it coincides with the small hexagon at the center of the region.       

     Example 22 
     Strategy games of this example encompass several games that are played with more sectors than Example 21 but are otherwise very similar to Example 21. 
     The taxonomic codes for three possible games included in this example are listed below. All games are analog, multi-player games with triangular sectors and a political focus in which three types of elements—namely three political parties—are present, the game proceeds according to alternating turn-based play, there are 2-3 players, and moves in categories “E,” “X,” and “R” are allowed.
         A/G/T2/P/3/12/U/2-3/EXR   A/G/T3/P/3/15/U/2-3/EXR   A/G/T4/P/3/27/U/2-3/EXR       

     The three games above get progressively larger with 96, 150, and 216 sectors respectively. 
     Example 23 
     Strategy games of Example 23 combine the simultaneous independent play of Example 19 with the three-party environment in Example 21. These games have taxonomic code A/G/T1/P/3/9/I. They are analog, multi-player games with 54 triangular sectors and a political focus in which three types of elements—three political parties—are present and nine districts are formed. The game paradigm is simultaneous independent play, so any number of players may participate. 
     Components 
     In one example, the game components are similar to the components used in Example 21. A digital or mechanical timer is needed to play this game. In addition, each player should have a copy of the same game set which contains (a) 54 sectors having the same markings as in Example 21, (b) dozens of markers, (c) a scoreboard, and (d) a game board. 
     Playing the Game 
     Play may include the following four phases. Phases 2-4 are optional.
         1. Build political districts that maximize the Red Party&#39;s advantage (time limit 10 min)   2. Build political districts that maximize the Green Party&#39;s advantage (time limit 10 min)   3. Build political districts that maximize the Blue Party&#39;s advantage (time limit 10 min)   4. Build political districts that equalize the advantage of all parties (time limit 10 min)       

     Each phase 1-3 proceeds like a phase in Example 19. Phase 4 proceeds as in Example 20. 
     At the end of each phase of the game, each player tracks his/her score with respect to the goal at hand by placing markers on the appropriate squares on his/her game board. 
     Identifying the Winner(s) 
     The winner is the player who does the overall best job of achieving the goals that were pursued during the different phases of the game. 
     Example 24 
     Strategy games of Example 24 are played with hexagonal sectors. Some examples of these games have taxonomic code A/G/H1/P/2/9/U/2/EXR. They are analog, multi-player games with 37 hexagonal sectors and a political focus in which two types of elements—two political parties—are present and nine districts are formed. Taxonomic codes for five additional games included in this example are listed below. All five games are analog, multi-player games with hexagonal sectors and a political focus in which two types of elements—two political parties—are present, the game proceeds according to alternating turn-based play, and there are 2 players.
         A/G/H2/P/2/15/U/2/EXR   A/G/H3/P/2/15/U/2/EXR   A/G/H4/P/2/21/U/2/EXR   A/G/H5/P/2/21/U/2/EXR   A/G/H6/P/2/27/U/2/EXR       

     The five games above get progressively larger with 61, 91, 127, 169, and 217 sectors respectively. The shape of the region in each game is essentially a regular hexagon. 
     The strategy games of this example proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed. 
       FIG. 54  shows a region  1302  having 37 sector placeholders  1304 . Sector tiles  1306 , two examples of which shown in  FIG. 55 , may be placed on the region, one on each sector placeholder  1304 , to form the sectors. Each sector has a set of elements  1308 . During play, nine districts (whose average size is roughly 4) are formed from the sectors. It should be understood that regions of these games can be any size, and contain any number of sectors. The number of districts to be formed may vary depending upon the number of sectors in the region. 
     Rules that can be used for strategy games of this type are generally similar or identical to those of previous examples. 
     Example 25 
     Strategy games of Example 25 have taxonomic code A/G/H1/P/2/9/I. They are analog, multi-player game with 37 hexagonal sectors and a political focus in which two types of elements—two political parties—are present and nine districts are formed. These games proceed according to simultaneous independent play, so any number of players may participate. 
     Rules that can be used for strategy games of this type are generally similar or identical to those of previous examples with two parties and simultaneous independent play. 
     Example 26 
     Strategy games of Example 26 have taxonomic code A/G/H1/P/6/9/U/2-6/EXR. They are analog, multi-player games with 37 hexagonal sectors and a political focus in which six types of elements—six political parties—are present and nine districts are formed. These games proceed according to alternating, turn-based play; there are 2-6 players; and moves in categories “E,” “X,” and “R” are allowed. 
     Rules that can be used for strategy games of this type are generally similar or identical to those in Examples 11 and 21. 
     Example 27 
     Strategy games of Example 27 have taxonomic code A/G/H1/P/6/9/I. They are analog, multi-player games with 37 hexagonal sectors and a political focus in which six types of elements—six political parties—are present and nine districts are formed. These games proceed according to simultaneous independent play, and a variety of pre-defined goals can be pursued in each phase of these games. 
     Rules that can be used for strategy games of this type are generally similar or identical to those in Examples 13-14 and 23. 
     Example 28 
     Strategy games of Example 28 have a real-world focus in which U.S. congressional districts are formed in a real U.S. state, namely Wisconsin. These games have taxonomic code A/G/C82/P/3/8/U/2/EXR. They are analog, multi-player games with 82 complex sectors and a political focus in which three types of elements—population, Red Party supporters, and Blue Party supporters—are considered and eight districts are formed. These games proceed according to alternating, turn-based play; there are 2 players; and moves in categories “E,” “X,” and “R” are allowed. 
     Game Summary 
       FIGS. 56-70  illustrate aspects of a strategy game  1400 . In this game, two players representing opposing political parties (Red and Blue) vie for political control of the region  1402  (i.e., the state of Wisconsin) by competitively creating eight U.S. congressional districts out of 82 sectors  1404  that largely coincide with the counties currently existing in Wisconsin. Wisconsin has a two-party political system in which one person is elected to the U.S. House of Representatives to represent each congressional district. As shown in  FIG. 56 , at the outset, the districts are formless and the players know the shape, location, population, and political composition of each sector (i.e. which party its citizens favor and by how much) based on the set of elements  1408  depicted in each sector  1404 . During the first phase of the game, players gradually build political districts by assigning sectors to political districts one sector at a time. They may also reassign sectors from more populated districts to less populated districts in order to better equalize the district populations. During the (optional) second phase of the game, the political margin in each district is converted into a probability of each party winning the district, and an election is simulated by rolling dice (such as dice  1622  of  FIG. 44 ). The winner of the game is the player whose party controls more districts than his/her opponent. If both players control an equal number of districts, the result is a tie. 
     Components 
     The components of the game are listed below:
         Region  1402  (Wisconsin map) having 82 sectors  1404 , each sector having a set of elements  1408     8 marker subsets (one for each of the eight districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, and Gray), each containing a plurality of markers  1418 , which may include, for example, 8 home base markers  1422  (one for each district) and 280 expansion markers  1424  (35 for each of the eight districts)   64 scoring tokens (eight for each of the eight districts above)   2 ten-sided dice (one black, one white) each showing values 0-9   Two-part scoreboard having a first part  1410  ( FIG. 57 ) and a second part  1412  ( FIG. 58 ).       

     Setup 
     The players decide (yes or no) if phase 2 of the game will be played. The players then decide who plays Red and who plays Blue, and who will take the first turn. 
     Sectors 
     As can be seen in  FIG. 56 , each sector on the map contains two numbers. In accordance with the key in Table 1, the first number  1414  (in a rectangle) denotes the sector&#39;s voting population (in thousands). The second number  1416  indicates the sector&#39;s voter margin (i.e. voting tendency, political margin). In the second number  1416 , a black number in a white circle indicates that the sector tends to vote for the Red Party; a white number in a black circle indicates that the sector tends to vote for the Blue Party. The number itself, in this example, is the voting margin (in thousands of votes) by which the sector supported one party over the other in the 2016 U.S. presidential election. For example, a black 4 in a white circle indicates that the sector supported the Red Party by a margin of 4000 votes in the 2016 election. A white 0 in a black circle indicates that voters in the sector were evenly divided—after rounding off to the nearest thousand voters—among the two parties in the 2016 U.S. presidential election. 
     In this game, the 82 sectors  1404  are used as building blocks to form eight non-overlapping political districts which together exhaust the land area of the state. The eight districts are identified by color: Brown, Red, Orange, Yellow, Green, Blue, Purple, and Gray. Initially, the political districts are formless and no sector belongs to any district. During the course of the game, players use markers  1418  to gradually assign these 82 sectors to political districts. Each sector eventually belongs to exactly one political district. 
     A close inspection of the sectors  1404  as shown in  FIG. 56  will reveal that the sum of the black numbers in white circles exceeds the sum of the white numbers in black circles by 23. In other words, there are 23,000 more Red Party supporters than Blue Party supporters in the region  1402 . This value agrees with the results of the 2016 U.S. presidential election. Based on this information, we say that the overall political margin in the state is “+23 for the Red Party” or “+23 Red.” Although the Red Party has a slight advantage statewide, it is highly unlikely that the Red Party will be able to maintain an advantage within every district after the region  1402  is divided into eight districts. Also note that the sum of the numbers in the black rectangles is 2793. In other words, the state&#39;s voter population is 2,793,000. Dividing this value by 8, we see that, at the game&#39;s end, the average district&#39;s voter population will be 349.125, which is most closely represented by the value  349 . However, it is unlikely that any districts will have a population of exactly 349 at the game&#39;s end; most or all districts will have a population strictly greater than or less than 349. 
     Scoreboard 
     Scoring tokens are used to display the (voter) population and political margin of every district on scoreboard ( FIGS. 57-58 ) at all times. At the start of the game, these scoring tokens may be placed to show that the population of each district is zero and that no party has an advantage in any district. 
     The scoreboard should be updated after every player takes a turn. For example, consider a moment in the game when exactly three sectors with populations  40 ,  71 , and  48  and voting tendencies “+7 Red,” “+8 Blue,” and “+10 Red” respectively have been assigned to the Green District. In this case, the Green District&#39;s population—159—should be indicated by three green scoring tokens placed at positions “Population x100=1,” “Population x10=5,” and “Population x1=9” in the Green District&#39;s portion of the scoreboard. The Green District&#39;s current political marging—“+9 Red”—equals the difference between the sum of the numbers in the sectors that support the Red Party (e.g. 17) and the sum of the numbers in the sectors that support the Blue Party (e.g. 8) that belong to the district. The political margin favors the Red Party if there are more Red Party than Blue Party supporters in the district; it favors the Blue Party if the opposite is true. A district&#39;s political margin is indicated by four scoring tokens that show (1) which party has the majority of voters in the district, (2) the hundreds digit of the political margin, (3) the tens digit of the political margin, and (4) the ones digit of the political margin. In the case above, four green scoring tokens should be placed at the positions “Current Leader=Red,” “Political Margin x100=0,” “Political Margin x10=0,” and “Political Margin x1=9” in the Green District&#39;s portion of the scoreboard. If a sector with population  72  and voting tendency “+36 Blue” is added to this district, the district&#39;s new population is 231 and its new political margin is “+27 Blue,” so the seven green scoring tokens should immediately be moved to positions “Population x100=2,” “Population x10=3,” “Population x1=1,” “Current Leader=Blue,” “Political Margin x100=0,” “Political Margin x10=2,” and “Political Margin x1=7.” 
     Playing the Game 
     Play may include the following three phases (the second phase is optional):
         1. Build political districts   2. Run an election (optional)   3. Identify the winner       

     Phase 1: Build Political Districts 
     Summary 
     This is the main phase of the game. During this phase, players take turns assigning sectors  1404  to political districts, one sector at a time, until every sector  1404  belongs to a political district. The assignment of a sector to a political district is accomplished by placing a home base marker or expansion marker on a sector. Players may also reassign sectors from more populated districts to less populated districts in order to better equalize the district populations. This is done by changing the color of the marker that occupies a sector. At the end of this phase, there will be eight non-overlapping political districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, and Gray—that together cover the state. Also, each district will be connected. 
     Each district evolves in the same general way. Initially, it is formless. At some point, the district is established when its home base marker is placed on a vacant sector. (A vacant sector is a sector with no marker on it.) The district is then expanded whenever one of its expansion markers is placed on a vacant sector that is adjacent to a sector that already belongs to the district. Later, the district may be adjusted so its population is more similar to neighboring districts. 
     The overall process of building the political districts is relatively unrestricted. In general, any player may contribute to building any district during any of his/her turns. There is no requirement for the sequence in which, or locations where, districts are constructed. Once begun, the construction of a given district may be temporarily halted while players take turns establishing, expanding, and/or adjusting other districts. There is no district population requirement. However, the rules encourage the creation of districts whose population is close to the average value of 349. 
     Details 
     Players take turns beginning with the starting player. During a player&#39;s turn, he/she (A) makes one move and then (B) records the move on the scoreboard. Forfeiting a turn is not allowed. 
     The rules of this game provide that all moves made during this phase of the game must be of type 1, 1A, 2, 2A, 3, or 3A below. Moves of type 1 and 1A establish a new district. Moves of type 2 and 2A expand an existing district. Moves of type 3 and 3A adjust two adjacent districts by transferring a sector from one district to an adjacent district. “A” stands for “alternate move.” 
     Phase 1 of the game is divided into three stages. During stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no more moves of type 1 exist and (ii) fewer than eight districts have been established. During stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 8 th  district is established. During stage 3, the next move must be of type 2 or 3A if at least one move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. This phase of the game ends when no more legal moves exist. 
     The six types of legal moves are as follows. Explanations for the asterisked terms are provided at the end of these descriptions.
         1 Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (i) The sector must be at least three steps away from all previously placed home base markers. (ii) There must be space to grow this district into a connected* district with a population of at least 349. (In  FIG. 59 , sectors  11  and  34  are three steps away from each other. Sectors  25  and  44  are two steps away from each other. Also, sectors  57  and  72  are two steps away from each other.)   1A Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (i) The sector must be in the most populated open space on the board. (An open space is a connected* group of vacant sectors.) (ii) Among the sectors satisfying the criterion above, the sector must be the farthest (in number of steps) from a previously placed home base marker. Ties may be broken arbitrarily.   2 Expand a district by placing one of its expansion markers on a vacant sector. This move must meet four requirements. (i) The district must remain connected.* (ii) The district&#39;s initial population before the move must be 348 or less. (iii) The district&#39;s new population—including the new sector and any sectors that are captured**—must be 398 or less. (iv) No district may be trapped.***   2A Expand the least populous expandable district by placing one of its expansion markers on a vacant sector. The district must remain connected* and no sectors may be captured.** (A district is expandable if there is at least one vacant sector adjacent to it.)   3 Reassign a sector from one district (e.g. District X) to another (e.g. District Y) by removing the District X marker from a sector and replacing it with a District Y expansion marker. This move must meet four requirements. (i) Districts X and Y must remain connected.* (ii) The populations of Districts X and Y must become strictly more balanced. In other words, before this move is made, the population of X must exceed the population of Y by more than the population of the reassigned sector. (iii) District Y must exist and be confined, i.e. it must not be expandable, prior to this move. In other words, there must be no vacant sectors adjacent to District Y prior to this move. (iv) The District X marker that is removed must be an expansion marker; it may not be a home base marker.   3A Same as move type 3 but without requirement (iv) for move type 3.       * See subsection entitled “Connectedness” below   * See subsection entitled “Captured sectors” below   ** See subsection entitled “Trapped districts” below   

     Connectedness 
     Two sectors are adjacent—and connected—if and only if they share a common edge. For example, in the map shown in  FIG. 59  (in which each of the 82 sectors  1404  has been given a reference number  1420  for ease of reference), sectors  1  and  2  are adjacent; sectors  35  and  45  are adjacent; and sectors  69  and  74  are adjacent. However, sectors  1  and  8  are not adjacent; sectors  44  and  56  are not adjacent; and sectors  30  and  41  are not adjacent. By analogy the territory consisting of sectors  25 ,  33 ,  44 , and  54  is connected, and the territory consisting of sectors  57 ,  64 ,  68 , and  72  is connected. The territory consisting of sectors  59  and  68 - 70  is not connected. 
     In this game, every political district must be connected at all times. That is, at all times and for any two sectors that belong to a given district (say District X), there must be a path within District X (i.e. a sequence of adjacent sectors that all belong to District X) connecting those two sectors. 
     Captured Sectors 
     A set of connected, vacant sectors is captured if it is (i) surrounded by the edge of the board on one side and a single district on the other side or (ii) entirely surrounded by a single district. In  FIG. 59 , sectors  60  and  76  are captured by the Green District (consisting of sectors  54 ,  61 - 62 , and  77 ); sector  18  is captured by the Blue District (consisting of sectors  17 ,  19 , and  24 ); sector  41  is captured by the Gray District (consisting of sector  40 ); sector  29  is captured by the Yellow District (consisting of sectors  23 ,  28 , and  30 ); and sectors  73 - 75  are captured by the Red District (consisting of sectors  69 - 70 ,  72 , and  81 ). 
     A move of type 2 which captures one or more sectors is allowed if the district&#39;s initial population before the move is 348 or less and the district&#39;s new population—including the sector where the marker is placed and any sectors that are captured—is 398 or less. All other moves that capture sectors are forbidden. For example, in  FIG. 59 , it is permissible to add sector  54  to a district consisting of sectors  61 - 62  and  77 . In this case, the district immediately grows to include sectors  54 ,  60 - 62 , and  76 - 77  after the move is made, and the district&#39;s new population is 68 which is well below the 398 threshold ( FIG. 56  shows the sector populations). However, it is not permissible to add sector  72  to a district consisting of sectors  69 - 70  and  81  because the expanded district—which after capturing three sectors would consist of sectors  69 - 70 ,  72 - 75 , and  81 —would have a population of 569 which is well above the 398 threshold. 
     Any sectors that are captured during a legal move of type 2 are immediately assigned to the district that has captured them. Expansion markers are immediately placed on these sectors. 
     Trapped Districts 
     A district is trapped if (i) it (and the open spaces beside it) is surrounded either by the edge of the board on one side and a single district on the other side or by a single district on all sides and (ii) its population plus the populations of all open spaces beside it is 348 or less. In  FIG. 59 , the Orange District (consisting of sector  11 ) is trapped by the Brown District (consisting of sectors  8 ,  10 ,  12 ,  20 ,  22 ,  26 - 27 , and  34 ) because the combined population of sectors  11  and  21  (=16) is 348 or less ( FIG. 56  shows the sector populations). 
     A move of type 2 which traps a district is forbidden. For example, if the Brown District consists of sectors  10 ,  12 ,  20 ,  22 ,  26 - 27 , and  34  and the Orange District consists of sector  11 , then an expansion of the Brown District to sector  8  is forbidden. 
     Phase 2: Run an Election 
     In this phase, the political margin in each district is converted into a numerical likelihood of each party winning the district, and an election in each district is simulated by rolling the two 10-sided dice. 
     Each district is considered one at a time. First, using the table below, the political margin for the party with more voters in the district is converted into a numerical likelihood of that party winning an election in the district. For example, a “+4 Blue” political margin in the Yellow District converts to a 66% chance for the Blue Party to win an election in the Yellow District. 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                 Political 
                 Winning 
               
               
                   
                 Margin 
                 Likelihood 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                   
                 0 
                 50% 
               
               
                   
                 +1 
                 54% 
               
               
                   
                 +2 
                 58% 
               
               
                   
                 +3 
                 62% 
               
               
                   
                 +4 
                 66% 
               
               
                   
                 +5 
                 70% 
               
               
                   
                 +6 
                 73% 
               
               
                   
                 +7 
                 76% 
               
               
                   
                 +8 
                 79% 
               
               
                   
                 +9 
                 82% 
               
               
                   
                 +10 
                 85% 
               
               
                   
                 +11 
                 87% 
               
               
                   
                 +12 
                 89% 
               
               
                   
                 +13 
                 91% 
               
               
                   
                 +14 
                 93% 
               
               
                   
                 +15 
                 95% 
               
               
                   
                 +16 
                 96% 
               
               
                   
                 +17 
                 97% 
               
               
                   
                 +18 
                 98% 
               
               
                   
                 +19 
                 99% 
               
               
                   
                 +20 or more 
                 100%  
               
               
                   
                   
               
            
           
         
       
     
     Next, a random number from 1-100 is produced by rolling the two 10-sided dice. 
     The random number is then compared to the winning percentage (e.g. 66 for the above case). If the random number is less than or equal to the winning percentage, the party with more voters in the district wins the district election. If the random number is greater than the winning percentage, the party with fewer voters in the district wins the district election. In the above example, the Blue Party wins the Yellow District election if the random number is from 1-66, and the Red Party wins the Yellow District election if the random number is from 67-100. If both parties have a 50% chance of winning the election, the Blue Party wins if the random number is 1-50 and the Red Party wins if the random number is 51-100. After the winner of an election is identified, the unused scoring token that matches the district color is placed on the “Winner=Blue” or “Winner=Red” square in the appropriate district&#39;s portion of the scoreboard. 
     The above procedure is repeated for each political district. 
     Phase 3: Identify the Winner 
     In the game&#39;s final phase, the overall winner is identified. 
     If phase 2 is played, the winner is the player whose party wins five or more district elections. If each party wins four district elections, the result is a tie. 
     If phase 2 is not played, the scoreboard is used to identify the party that controls each district, i.e. the party with more voters in each district. The winner is the player whose party controls more districts than his/her opponent. If the two players control an equal number of districts, the result is a tie. 
     Example of Play 
     An example of play is now provided, with reference to  FIGS. 60-70 , to illustrate the rules of the game. The initial position is shown in  FIG. 56 . During play, the population and voting tendency of each sector are visible to both players. However, in  FIGS. 60-69 , the reference number  1420  of each sector  1404  is shown instead of the voting tendency for ease of discussion. 
     During stage 1, only moves of type 1, 2, and 3 are allowed. After 40 moves have been made—six of type 1 and 34 of type 2—assume the board position, with home base markers  1422  and expansion markers  1424 , is as shown in  FIG. 60 . Note that six districts—Brown, Red, Orange, Yellow, Blue, and Purple—have been established. Two districts—Green and Gray—have not (yet) been established. 
     There are 43 possibilities for the next move which must be of type 1, 2, or 3. No legal moves of type 3 exist because no district is confined. Regarding moves of type 1, it is possible to establish a new district in sector  73  or  74 . Establishing a new district in another sector is not allowed because either (i) the sector is less than three steps away from a previously placed home base marker or (ii) there is not enough space to grow the new district into a connected district with a population of at least 349. Regarding moves of type 2, it is possible to (a) expand the Brown District to sector  5 ,  13 ,  22 ,  21 ,  19 ,  56 ,  50 ,  46 ,  28 ,  29 ,  30 , or  16 ; (b) expand the Red District to sector  28 ,  46 ,  51 ,  57 ,  53 ,  40 ,  38 , or  37 ; (c) expand the Orange District to sector  19 ,  24 ,  31 ,  32 ,  42 ,  43 ,  54 ,  60 ,  76 ,  65 ,  63 , or  56 ; (d) expand the Yellow District to sector  57 ,  67 ,  70 ,  72 , or  59 ; or (e) expand the Blue District to sector  4 ,  9 ,  21 , or  12 . The Purple District may not be expanded during the next move because its population—352—is already at least 349. 
     Several of the above moves of type 2 capture one or more sectors including the expansion of the (a) Brown District to sector  30  (which captures sector  16 ); (b) Red District to sector  40  (which captures sector  41 ); (c) Red District to sector  38  (which captures sectors  40  and  41 ); and (d) Orange District to sector  24 ,  31 ,  32 ,  42 ,  54 , or  60 . Note that an expansion of the Brown District to sector  9  is not allowed because the Blue District would be trapped (by the Brown District). Also, an expansion of the Yellow District to sector  53 ,  73 , or  74  is not allowed because one or more sectors would be captured and added to the Yellow District, putting its population over the limit of 398. 
     The next move played is the establishment of the Green District in sector  74 . This results in the game position shown in  FIG. 61 . In this position, (i) there is no way to make a move of type 1 that satisfies its criteria and (ii) fewer than eight districts have been established. Thus, stage 2 of play begins. 
     There are 52 possibilities for the next move which must be of type 1A, 2, or 3. No legal moves of type 3 exist because no district is confined. The feasible moves of type 2 include the 41 moves of type 2 mentioned above and (f) expanding the Green District to sector  70 ,  75 , or  73 . 
     A move of type 1A requires that a new district be established in a sector that is within the most populous connected open space on the board. Among the sectors satisfying this criterion, a sector that ties for being the most steps away from a previously placed home base marker must be selected. In the current board position, the most populous open space has population  511  and consists of sectors  46 ,  50 - 51 ,  56 - 57 ,  63 - 65 ,  67 ,  70 , and  75 . Among these sectors, eight tie for being two steps away from a previously placed home base marker— 46 ,  50 - 51 ,  56 ,  63 - 65 , and  67 —and none is three or more steps away from all previously placed home base markers. Thus, there are eight possible moves of type 1A: establish the Gray District in sector  46 ,  50 - 51 ,  56 ,  63 - 65 , or  67 . 
     The next five moves in this example game are as follows (move type in parentheses):
         1. (2) Expansion of Red District to sector  40  (one sector captured)   2. (2) Expansion of Blue District to sector  12     3. (2) Expansion of Brown District to sector  30  (one sector captured)   4. (2) Expansion of Orange District to sector  24  (seven sectors captured)   5. (1A) Establishment of Gray District in sector  56         

     The establishment of the 8 th  district during move #5 above ushers in stage 3 of play. During stage 3, the next move must be of type 2 or 3A if at least one type 2 move exists. Otherwise the next move must be of type 2A or 3A. The new board position is shown in  FIG. 62 . 
     There are 40 possibilities for the next move. Regarding moves of type 2, it is possible to (a) expand the Brown District to sector  5 ,  13 ,  22 ,  21 ,  19 ,  50 ,  46 ,  28 ,  29 , or  38  (but not  9 ); (b) expand the Red District to sector  28 ,  46 ,  51 ,  57 ,  53 ,  38 , or  37 ; (c) expand the Orange District to sector  18 ,  65 ,  63 , or  19 ; (d) expand the Yellow District to sector  57 ,  67 ,  70 ,  72 , or  59  (but not  53  or  73 ); (e) expand the Green District to sector  70 ,  75 , or  73 ; (f) expand the Blue District to sector  4 ,  9 ,  21 ,  22 ,  13 , or  5 ; or (g) expand the Gray District to sector  50 ,  63 ,  64 ,  57 , or  51 . The Purple District may not be expanded during the next move because its population—352—is already at least 349. There are no legal moves of type 3A because no district is confined. 
     After the next 23 moves in the game, the exemplary board position is shown in  FIG. 63 . The preceding 23 moves were all of type 2, and these moves involved the expansion of seven districts: Brown, Red, Orange, Yellow, Green, Blue, and Gray. 
     The table below shows the current population of each district. Note that no feasible move of type 2 exists. This is because all districts already have a population of at least 349 and/or are confined. In particular, the only two districts with a population of 348 or less—Blue and Gray—are confined. Thus, the next move must be of type 2A or 3A. 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                 District 
                 Population 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                   
                 Brown 
                 366 
               
               
                   
                 Red 
                 359 
               
               
                   
                 Orange 
                 360 
               
               
                   
                 Yellow 
                 355 
               
               
                   
                 Green 
                 352 
               
               
                   
                 Blue 
                 243 
               
               
                   
                 Purple 
                 352 
               
               
                   
                 Gray 
                 141 
               
               
                   
                   
               
            
           
         
       
     
     A move of type 3A is possible—i.e. a sector may be reassigned from District X to District Y—if and only if (i) Districts X and Y remain connected after the reassignment, (ii) before the reassignment the population of X exceeds the population of Y by more than the population of the reassigned sector, and (iii) District Y is confined before the reassignment. Note that accurate district population information (as shown in the table above) is needed in order to make a correct assessment regarding item (ii) above. 
     To search for moves of type 3A, we consider each district one at a time and ask if that district can “steal” a sector from another district.
         The Brown District may not steal a sector from another district for two reasons: (A) it is the most populous district and (B) it is not confined.   The Red District is not confined, so it may not steal a sector from another district.   The Orange District is confined and its population is 6 less than the Brown District, so the Orange District could theoretically steal a sector with population 5 or less from the Brown District. However, the Brown District has no such sector that lies along the Orange-Brown border, so there is no legal way for the Orange District to steal a sector from another district.   The Yellow District is not confined, so it may not steal a sector from another district.   The Green District is not confined, so it may not steal a sector from another district.   There are five legal moves of type 3A in which the Blue District—whose population is much less than the other districts except the Gray District—steals a sector from a neighboring district. These include (a) reassigning sector  10 ,  14 , or  23  from the Brown District to the Blue District and (b) reassigning sector  24  or  25  from the Orange District to the Blue District. Note that reassigning sector  20 ,  26 , or  27  from the Brown District to the Blue District destroys the connectedness of the Brown district, so these moves are not allowed.   The Purple District&#39;s population is slightly less than that of the Brown, Red, Orange, and Yellow Districts, so the Purple District could theoretically steal a small sector from one of these districts. However, the Purple District does not share a border with the Brown, Red, or Yellow District. In addition, the Orange District does not have a small sector that lies along the Purple-Orange border. Thus, there is no legal way for the Purple District to steal a sector from another district.   There are eight legal moves of type 3A in which the Gray District—the least populous district—steals a sector from another district. These include (a) reassigning sector  51  from the Brown District to the Gray District; (b) reassigning sector  55 ,  62 , or  65  from the Orange District to the Gray District; (c) reassigning sector  66  from the Purple District to the Gray District; (d) reassigning sector  64  from the Green District to the Gray District; (e) reassigning sector  58  from the Yellow District to the Gray District; and (f) reassigning sector  52  from the Red District to the Gray District. The following moves are not allowed because they destroy the connectedness of a district: (g) reassigning sector  45  or  50  from the Brown District to the Gray District; (h) reassigning sector  67  from the Green District to the Gray District; and (i) reassigning sector  68  from the Yellow District to the Gray District.       

     To search for moves of type 2A, note that the Green District—with a population of 352—is the least populous expandable district. Only one move of type 2A is available: (a) expand the Green District to sector  73 . Overall, a total of 14 legal moves exist. 
     The next five moves are as follows (move type in parentheses):
         1. (2A) Expansion of Green District to sector  73     2. (3A) Reassignment of sector  23  from Brown District to Blue District   3. (3A) Reassignment of sector  10  from Brown District to Blue District   4. (3A) Reassignment of sector  24  from Orange District to Blue District   5. (3A) Reassignment of sector  51  from Brown District to Gray District       

     The board position is now as shown in  FIG. 64 . The current populations of the districts are shown in the table below. Districts with a population of 348 or less are asterisked. Note that two districts with a population of 348 or less—Brown and Gray—are not confined. (The Gray District—the least populated district—became unconfined during move #5 above. The Brown District—which was never confined—became underpopulated—with a population of 348 or less—during move #3 above.) Thus, a move of type 2 is available. The next move must therefore be of type 2 or 3A. 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                 District 
                 Population 
               
               
                   
                   
               
             
            
               
                   
                 Brown 
                  325* 
               
               
                   
                 Red 
                 359 
               
               
                   
                 Orange 
                 356 
               
               
                   
                 Yellow 
                 355 
               
               
                   
                 Green 
                 435 
               
               
                   
                 Blue 
                  279* 
               
               
                   
                 Purple 
                 352 
               
               
                   
                 Gray 
                  150* 
               
               
                   
                   
               
            
           
         
       
     
     Two moves of type 2 are available: (a) expand the Gray District to sector  46  and (b) expand the Brown District to sector  37 . Six moves of type 3A are also available: (c) reassign sector  14 ,  29 , or  20  from the Brown District to the Blue District; (d) reassign sector  25  or  31  from the Orange District to the Blue District; and (e) reassign sector  64  from the Green District to the Purple District. Note that the Brown, Red, Yellow, Green, and Gray Districts are not confined, so they may not steal a sector from another district. Also, the Purple District may not steal sector  75  from the Green District because the population imbalance would not be reduced. Overall, eight legal moves are available. 
     The next five moves are as follows (move type in parentheses):
         1. (2) Expansion of Brown District to sector  37     2. (2) Expansion of Gray District to sector  46     3. (3A) Reassignment of sector  20  from Brown District to Blue District   4. (3A) Reassignment of sector  31  from Orange District to Blue District   5. (3A) Reassignment of sector  26  from Brown District to Blue District       

     The new board position is shown in  FIG. 65 . The current district populations are shown in the table below. Districts with a population of 348 or less are asterisked. There are no feasible moves of type 2, so the next move must be of type 2A or 3A. 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                 District 
                 Population 
               
               
                   
                   
               
             
            
               
                   
                 Brown 
                 370 
               
               
                   
                 Red 
                 359 
               
               
                   
                 Orange 
                 349 
               
               
                   
                 Yellow 
                 355 
               
               
                   
                 Green 
                 435 
               
               
                   
                 Blue 
                  329* 
               
               
                   
                 Purple 
                 352 
               
               
                   
                 Gray 
                  161* 
               
               
                   
                   
               
            
           
         
       
     
     There are 17 possibilities for the next move. To search for moves of type 2A, note that the Yellow District—with a population of 355—is the least populous expandable district. Only one move of type 2A is available: (a) expand the Yellow District to sector  72 . Several moves of type 3A are available: (b) reassign sector  14 ,  29 , or  34  from the Brown District to the Blue District; (c) reassign sector  32  or  33  from the Orange District to the Blue District; (d) reassign sector  64  from the Green District to the Purple District; (e) reassign sector  35  or  50  from the Brown District to the Gray District; (f) reassign sector  36  or  52  from the Red District to the Gray District; (g) reassign sector  55 ,  62 , or  65  from the Orange District to the Gray District; (h) reassign sector  58  from the Yellow District to the Gray District; (i) reassign sector  64  from the Green District to the Gray District; and (j) reassign sector  66  from the Purple District to the Gray District. 
     The next four moves are as follows (move type in parentheses):
         1. (3A) Reassignment of sector  64  from Green District to Gray District   2. (3A) Reassignment of sector  34  from Brown District to Blue District   3. (3A) Reassignment of sector  62  from Orange District to Gray District   4. (2A) Expansion of Yellow District to sector  72         

     The new board position is shown in  FIG. 66 . Note that there are no vacant sectors. Thus, all future moves will be of type 3A. Play continues until no more such moves exist. The new district populations are shown in the table below. 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                 District 
                 Population 
               
               
                   
                   
               
             
            
               
                   
                 Brown 
                  334* 
               
               
                   
                 Red 
                 359 
               
               
                   
                 Orange 
                  338* 
               
               
                   
                 Yellow 
                 438 
               
               
                   
                 Green 
                 363 
               
               
                   
                 Blue 
                 365 
               
               
                   
                 Purple 
                 352 
               
               
                   
                 Gray 
                  244* 
               
               
                   
                   
               
            
           
         
       
     
     There are 21 possibilities for the next move: (a) reassign sector  5 ,  13 ,  21 ,  22 , or  23  from the Blue District to the Brown District; (b) reassign sector  58  or  59  from the Yellow District to the Red District; (c) reassign sector  19  or  31  from the Blue District to the Orange District; (d) reassign sector  68  or  69  from the Yellow District to the Green District; (e) reassign sector  50  from the Brown District to the Gray District; (f) reassign sector  36  or  52  from the Red District to the Gray District; (g) reassign sector  55 ,  61 , or  65  from the Orange District to the Gray District; (h) reassign sector  58  or  68  from the Yellow District to the Gray District; (i) reassign sector  67  from the Green District to the Gray District; or (j) reassign sector  66  from the Purple District to the Gray District. 
     The next two moves are as follows (move type in parentheses):
         1. (3A) Reassignment of sector  22  from Blue District to Brown District   2. (3A) Reassignment of sector  68  from Yellow District to Gray District       

     The new board position is shown in  FIG. 67 . Note that the new district populations—shown in the table below—are more balanced than before. Each future move will make the district populations even more balanced until, at the end of the game, no further “rebalancing” is possible. At that point, an “equilibrium” will be established, and the competition between the two players—who alternate turns and are representing the Red and Blue Parties—will end. 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                 District 
                 Population 
               
               
                   
                   
               
             
            
               
                   
                 Brown 
                  348* 
               
               
                   
                 Red 
                 359 
               
               
                   
                 Orange 
                  338* 
               
               
                   
                 Yellow 
                 364 
               
               
                   
                 Green 
                 363 
               
               
                   
                 Blue 
                 351 
               
               
                   
                 Purple 
                 352 
               
               
                   
                 Gray 
                  318* 
               
               
                   
                   
               
            
           
         
       
     
     There are five possibilities for the next move: (a) reassign sector  31  from the Blue District to the Orange District; (b) reassign sector  50  from the Brown District to the Gray District; (c) reassign sector  36  from the Red District to the Gray District; (d) reassign sector  61  from the Orange District to the Gray District; or (e) reassign sector  67  from the Green District to the Gray District. 
     The next two moves are as follows (move type in parentheses):
         1. (3A) Reassignment of sector  36  from Red District to Gray District   2. (3A) Reassignment of sector  31  from Blue District to Orange District       

     The new board position is shown in  FIG. 68  and the new district populations are in the table below. 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                 District 
                 Population 
               
               
                   
                   
               
             
            
               
                   
                 Brown 
                 348* 
               
               
                   
                 Red 
                 334* 
               
               
                   
                 Orange 
                 345* 
               
               
                   
                 Yellow 
                 364  
               
               
                   
                 Green 
                 363  
               
               
                   
                 Blue 
                 344* 
               
               
                   
                 Purple 
                 352  
               
               
                   
                 Gray 
                 343* 
               
               
                   
                   
               
            
           
         
       
     
     There is only one possibility for the next move: reassign sector  29  from the Brown District to the Blue District. This move is compulsory for the player who takes the next turn. 
     The new board position is shown in  FIG. 69  and the new district populations are in the table below. No legal moves exist in this board position, so phase 1 of play concludes. The districts that have been formed are now final. All districts are connected, but some—such as Brown, Yellow, Green, Blue, and Gray—have irregular shapes with tentacle-like protrusions. As expected, the district populations sum to 2793, and the population of the average district is 349.125=2793/8. However, no district has a population equaling 349 or 350. Three districts—Yellow, Green, and Purple—have populations exceeding 350 and five districts—Brown, Red, Orange, Blue, and Gray—have populations below 349. The district populations are not perfectly smooth, but they are relatively balanced. The population difference between the most populous district—Yellow—and least populous district—Red—is only 30. This situation is not uncommon at the end of phase 1. 
     The final board position at the end of phase 1 is shown in  FIG. 70 . The markers played and each sector&#39;s population and voting tendency—which are always displayed to both players during the game—are shown. The population is displayed in a black rectangle and the voting tendency is displayed in a circle. 
     At the end of phase 1, the scoreboard should read as shown in the box below. 
     
       
         
           
               
               
               
               
             
               
                   
                   
               
               
                   
                   
                   
                 Political 
               
               
                   
                 District 
                 Population 
                 Margin 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Brown 
                 347 
                 +61 Red  
               
               
                   
                 Red 
                 334 
                 +56 Red  
               
               
                   
                 Orange 
                 345 
                 +31 Blue 
               
               
                   
                 Yellow 
                 364 
                  +4 Blue 
               
               
                   
                 Green 
                 363 
                 +69 Blue 
               
               
                   
                 Blue 
                 345 
                 +64 Red  
               
               
                   
                 Purple 
                 352 
                 +30 Blue 
               
               
                   
                 Gray 
                 343 
                 +24 Blue 
               
               
                   
                 Overall 
                 2793 
                 +23 Red  
               
               
                   
                   
               
            
           
         
       
     
     If phase 2 is not played, the game ends, and the Blue Party wins by a score of 5 districts to 3 districts. 
     If phase 2 is played, an election is simulated. In the final scoreboard (see above), the political margin of the (Brown, Red, Orange, Yellow, Green, Blue, Purple, Gray) District translates to a winning likelihood of (100%, 100%, 100%, 66%, 100%, 100%, 100%, 100%) for the party that has the majority of voters in the district. Note that the Red Party automatically wins three districts with 100% probability and the Blue Party automatically wins four districts. 
     Dice are then thrown to determine the election results for the district in which there is not an automatic winner. The results are summarized in the table below. Despite being at a disadvantage going into the election, the Red Party “gets lucky” and wins the elections in four districts. The result is a tie. 
     
       
         
           
               
               
               
               
               
               
             
               
                   
                   
               
               
                   
                   
                 Political 
                 Winning 
                 Dice 
                 Election 
               
               
                   
                 District 
                 Margin 
                 Likelihood 
                 Roll 
                 Result 
               
               
                   
                   
               
             
            
               
                   
                 Brown 
                 +61 Red  
                 100% 
                 — 
                 Red Wins 
               
               
                   
                 Red 
                 +56 Red  
                 100% 
                 — 
                 Red Wins 
               
               
                   
                 Orange 
                 +31 Blue 
                 100% 
                 — 
                 Blue Wins 
               
               
                   
                 Yellow 
                  +4 Blue 
                  66% 
                 67 
                 Red Wins 
               
               
                   
                 Green 
                 +69 Blue 
                 100% 
                 — 
                 Blue Wins 
               
               
                   
                 Blue 
                 +64 Red  
                 100% 
                 — 
                 Red Wins 
               
               
                   
                 Purple 
                 +30 Blue 
                 100% 
                 — 
                 Blue Wins 
               
               
                   
                 Gray 
                 +24 Blue 
                 100% 
                 — 
                 Blue Wins 
               
               
                   
                   
               
            
           
         
       
     
     Example 29 
     Strategy games of this Example 29 have a real-world focus in which U.S. congressional districts are formed in a real U.S. state, namely Michigan. These games have taxonomic code A/G/C108/P/3/14/U/2/EXR. These games are analog, multi-player games with complex sectors (which may mimic actual counties of the state) and a political focus in which three types of elements—population, Red Party supporters, and Blue Party supporters—are considered and a pre-determined number of districts are formed. The game proceeds according to alternating, turn-based play; there are 2 players; and moves in categories “E,” “X,” and “R” are allowed. 
     Game Summary 
       FIGS. 71-74  illustrate aspects of a strategy game  1500 . In this game, two players representing opposing political parties (Red and Blue) vie for political control of region  1502  (e.g., Michigan) by competitively creating 14 U.S. congressional districts out of 108 sectors  1504  that largely coincide with counties that currently exist in Michigan. 
     One example of an initial position is illustrated in  FIG. 71 . At the outset, the districts are formless and the players know the shape, location, population, and political composition of each sector  1504  (i.e. which party its citizens favor and by how much) based on the set of elements  1508  depicted in each sector  1504 . During the first phase of the game, players gradually build the political districts by assigning sectors to political districts one sector at a time. They may also reassign sectors from more populated districts to less populated districts in order to better equalize the district populations. During the (optional) second phase of the game, the political margin in each district is converted into a probability of each party winning the district, and an election is simulated by rolling dice (such as dice  1622  of  FIG. 44 ). The winner of the game is the player whose party controls more districts than his/her opponent. If both players control an equal number of districts, the result is a tie. The sets of elements shown in  FIG. 71  are based on the results of the 2016 U.S. presidential election, and each includes two numbers. In accordance with the key in Table 1, the first number  1514  (in a rectangle) denotes the sector&#39;s voting population (in thousands). The second number  1516  indicates the sector&#39;s voting tendency.  FIGS. 72-73  show an example of a first part of a scoreboard  1510  and a second part of a scoreboard  1512  that may be included in a strategy game  1500 . 
     Playing the Game 
     The rules are very similar to Example 28. The main difference is that this game has different population thresholds for moves of type 1 and 2. 
     All moves made during phase 1 must be of type 1, 1A, 2, 2A, 3, or 3A below. 
     Phase 1 is divided into three stages. During stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no more moves of type 1 exist and (ii) fewer than 14 districts have been established. During stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 14 th  district is established. During stage 3, the next move must be of type 2 or 3A if at least one move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. This phase of the game ends when no more legal moves exist. 
     The six types of legal moves are as follows:
         1 Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (i) The sector must be at least three steps away from all previously placed home base markers. (ii) There must be space to grow this district into a connected district with a population of at least 320.   1A Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (i) The sector must be in the most populated open space on the board. (An open space is a connected group of vacant sectors.) (ii) Among the sectors satisfying the criterion above, the sector must be the farthest (in number of steps) from a previously placed home base marker. Ties may be broken arbitrarily.   2 Expand a district by placing one of its expansion markers on a vacant sector. This move must meet four requirements. (i) The district must remain connected. (ii) The district&#39;s initial population before the move is 320 or less. (iii) The district&#39;s new population—including the new sector and any sectors that are captured—is 340 or less. (iv) No district may be trapped. A district is trapped if (a) it, and the open spaces beside it, is surrounded either by the edge of the board on one side and a single district on the other side or by a single district on all sides and (b) its population plus the populations of all open spaces beside it is 320 or less.   2A Expand the least populous expandable district by placing one of its expansion markers on a vacant sector. The district must remain connected and no sectors may be captured. (A district is expandable if there is at least one vacant sector adjacent to it.)   3 (Same as in Example 28)   3A (Same as in Example 28)       

     Example of Play 
     We now provide an example of play. After 127 moves have been made—64 by the player representing the Red Party and 63 by the player representing the Blue Party—a final position as shown in  FIG. 74  may be reached, with home base markers  1518  and expansion markers  1520  placed in a manner that defines 14 districts within the region  1502 . No legal moves exist in this board position, so phase 1 of play concludes. As expected, the district populations sum to 4548, and the population of the average district is 324.857=4548/14. However, only one district—Purple—has a population equaling 324 or 325. 
     
       
         
           
               
               
               
               
             
               
                   
                   
               
               
                   
                   
                   
                 Political 
               
               
                   
                 District 
                 Population 
                 Margin 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Brown 
                 269 
                 +65 Blue 
               
               
                   
                 Red 
                 348 
                 +52 Red 
               
               
                   
                 Orange 
                 330 
                 +21 Red 
               
               
                   
                 Yellow 
                 331 
                 +49 Red 
               
               
                   
                 Lt. Green 
                 345 
                 +70 Red 
               
               
                   
                 Dk. Green 
                 328 
                 +14 Red 
               
               
                   
                 Lt. Blue 
                 350 
                  +92 Blue 
               
               
                   
                 Dk. Blue 
                 272 
                  +24 Blue 
               
               
                   
                 Purple 
                 324 
                  +21 Blue 
               
               
                   
                 Pink 
                 338 
                 +34 Red 
               
               
                   
                 Gray 
                 356 
                   +2 Blue 
               
               
                   
                 Black 
                 339 
                  +2 Red 
               
               
                   
                 White 
                 338 
                 +81 Red 
               
               
                   
                 Gold 
                 280 
                 +108 Blue  
               
               
                   
                 Overall 
                 4548 
                 11 
               
               
                   
                   
               
            
           
         
       
     
     If phase 2 is not played, the game ends, and the Red Party wins by a score of 8 districts to 6 districts. 
     Example 30 
     Strategy games of this example 30 have taxonomic code A/G/C?/N/3/?/U/2-3/EXRBF. They are analog, multi-player, nonpolitical games with turn-based play and complex two-dimensional sectors that are designed for 2-3 players. Each player represents a tribe. The players are provided with a playing surface illustrating a region divided into a number of sectors. The premise is that three tribes have been fighting wars against each other within the region illustrated on the board for more than a century. After significant bloodshed and no clear winner, they have decided to peacefully settle their differences by forming districts that various tribes will inhabit upon conclusion of the game. 
     During the game, the players organize the region into districts such that (i) each sector is assigned in its entirety to exactly one district and (ii) each district is a single connected piece. 
     At the outset the districts are formless and the players are informed of each sector&#39;s precise shape, location, and set of elements. The set of elements for each sector provides the intensity of each of three elements—rivers, plants, and mammals—in the sector. 
     During play, the players take alternating turns, with each player taking a single turn before any other player takes another turn. During a player&#39;s turn, the player must make one of the following moves: (E) establish a new district by assigning a first sector to it; (X) expand an already established district by assigning a new, previously unassigned sector to the district; (R) reassign a sector from one established district to another established district; (B) break up two adjacent districts by returning all sectors assigned them to unassigned status; and (F) freeze a given district so that no player may modify the district during the next 8 turns. Each player may make each move B and F at most once during the game. Each player has no limit on the number of moves E, X, and R that he/she plays. Each district must be connected at all times during play. In general, any player may use their move to contribute to the construction, destruction, or freezing of any district during any of his/her turns. Play ends when no legal moves exist. 
     The method of scoring at the end is nontrivial and relates to the suitability (i.e. habitability) of each district for each tribe. 
     Tribes A, B, and C have different habitability criteria. Members of Tribe A depend on fishing for sustenance and are allergic to plants. That is, Tribe A considers rivers as a resource and plants as a hazard, and it is indifferent to mammals. Members of Tribe B depend on plants/farming for sustenance and are allergic to mammals. That is, Tribe B considers plants as a resource and mammals as a hazard, and it is indifferent to rivers. Members of Tribe C depend on hunting for sustenance and are very poor swimmers. That is, Tribe C considers mammals as a resource and rivers as a hazard, and it is indifferent to plants. 
     Once the districts are finalized (i.e., when no legal moves exist), the intensity of each element in each district is computed by summing the intensities of the element in the sectors comprising the district. 
     At the end of the game, each tribe receives points for each district as follows.
         If the total intensity of rivers exceeds the total intensity of plants in a district, Tribe A considers the district habitable and receives [(river intensity)−(plant intensity)] points for that district. Otherwise, Tribe A receives zero points for that district.   If the total intensity of plants exceeds the total intensity of mammals in a district, Tribe B considers the district habitable and receives [(plant intensity)−(mammal intensity)] points for that district. Otherwise, Tribe B receives zero points for that district.   If the total intensity of mammals exceeds the total intensity of rivers in a district, Tribe C considers the district habitable and receives [(mammal intensity)−(river intensity)] points for that district. Otherwise, Tribe C receives zero points for that district.       

     Each tribe&#39;s point total at the end of the game equals the sum of the points it receives in all districts. The winner is the player (i.e. tribe) with the most points at the end of the game. 
     Example 31 
     Strategy games of this example 31 have taxonomic code A/G/C?/N/6/?/U/2-6/EXRBF. They are analog, multi-player, nonpolitical games with turn-based play and complex two-dimensional sectors that are designed for 2-6 players. Each player represents an interplanetary transportation company. In one example, the premise is that six interplanetary transportation companies dominate the economy of the region of the Milky Way Galaxy in the year 2388. After decades of chaos in the transportation market, the companies have decided, for their mutual benefit, to set standard transportation rates within the galaxy by dividing it into districts. After the districts are formed, direct transportation between planets will only take place (i) within districts and (ii) between adjacent districts. No other direct transportation services will be offered. 
     During the game, players organize the galaxy—which is already divided into C sectors that (i) may not be further divided, (ii) do not overlap, and (iii) together cover the entire galaxy—into a given number, D, of districts (where 2≤D&lt;C) such that (a) each sector belongs in its entirety to exactly one district and (b) each district is comprised of a set of adjacent sectors. 
     At the outset the districts are formless and the players are informed of each sector&#39;s precise shape, location, and set of elements, which in this case is a set of planets. The set of planets in a sector consists of six numbers which respectively represent the number of each planet type—agricultural, metropolitan, scholarly, industrial, medical, and ecological—in the sector. 
     Players take turns in rotating fashion. During a player&#39;s turn, the player must make one of the following moves: (E) establish a new district by assigning a first sector to it; (X) expand an already established district by assigning a new, previously unassigned sector to the district; (R) reassign a sector from one established district to another established district; (B) break up two or three adjacent districts by returning all sectors previously assigned them to unassigned status; and (F) freeze a given district so that no player may modify the district during the next 5 turns. Each player may make each move B and F at most once during the game. Each player has no limit on the number of moves E, X, and R that he/she plays. Each district must be connected at all times during play. In general, any player may use their move to contribute to the construction, destruction, or freezing of any district during any of his/her turns. Play ends when no legal moves exist. 
     The method of scoring is nontrivial; it relates to the profitability of each district and each pair of adjacent districts for each company. 
     Each company specializes in a different kind of transportation and therefore has a different perspective on profitability. Company A specializes in transporting food and food equipment between agricultural and metropolitan planets. Company B specializes in transporting students and researchers between metropolitan and scholarly planets. Company C specializes in transporting workers and engineers between scholarly and industrial planets. Company D specializes in transporting injured workers between industrial and medical planets. Company E specializes in transporting people and medicinal plants between medical and ecological planets. Company F specializes in transporting flora and fauna between ecological and agricultural planets. 
     Once the districts are finalized (i.e., once no legal moves exist), the total number of each planet type in each district is computed by summing the number of that planet type in the sectors comprising the district. Each company receives points for each district as follows.
         Company A receives 3 points for each transportation leg that it can service in the district, i.e. 3 points for each unique pair of planets—one agricultural and one metropolitan—in the district. So Company A gets [3*(no. of agricultural planets in district)*(no. of metropolitan planets in district)] points for the district.   Company B receives 3 points for each transportation leg that it can service in the district, i.e. 3 points for each unique pair of planets—one metropolitan and one scholarly—in the district. So Company B gets [3*(no. of metropolitan planets in district)*(no. of scholarly planets in district)] points for the district.   Company C receives 3 points for each transportation leg that it can service in the district, i.e. 3 points for each unique pair of planets—one scholarly and one industrial—in the district. So Company C gets [3*(no. of scholarly planets in district)*(no. of industrial planets in district)] points for the district.   Company D receives 3 points for each transportation leg that it can service in the district, i.e. 3 points for each unique pair of planets—one industrial and one medical—in the district. So Company D gets [3*(no. of industrial planets in district)*(no. of medical planets in district)] points for the district.   Company E receives 3 points for each transportation leg that it can service in the district, i.e. 3 points for each unique pair of planets—one medical and one ecological—in the district. So Company E gets [3*(no. of medical planets in district)*(no. of ecological planets in district)] points for the district.   Company F receives 3 points for each transportation leg that it can service in the district, i.e. 3 points for each unique pair of planets—one ecological and one agricultural—in the district. So Company F gets [3*(no. of ecological planets in district)*(no. of agricultural planets in district)] points for the district.       

     Each company also receives points for each pair of adjacent districts (e.g. X and Y) as follows.
         Company A gets 1 point for each transportation leg that it can service between Districts X and Y. So Company A gets [(no. of agricultural planets in X)*(no. of metropolitan planets in Y)]+[(no. of agricultural planets in Y)*(no. of metropolitan planets in X)] points for this pair of districts.   Company B gets 1 point for each transportation leg that it can service between Districts X and Y. So Company B gets [(no. of metropolitan planets in X)*(no. of scholarly planets in Y)]+[(no. of metropolitan planets in Y)*(no. of scholarly planets in X)] points for this pair of districts.   Company C gets 1 point for each transportation leg that it can service between Districts X and Y. So Company C gets [(no. of scholarly planets in X)*(no. of industrial planets in Y)]+[(no. of scholarly planets in Y)*(no. of industrial planets in X)] points for this pair of districts.   Company D gets 1 point for each transportation leg that it can service between Districts X and Y. So Company D gets [(no. of industrial planets in X)*(no. of medical planets in Y)]+[(no. of industrial planets in Y)*(no. of medical planets in X)] points for this pair of districts.   Company E gets 1 point for each transportation leg that it can service between Districts X and Y. So Company E gets [(no. of medical planets in X)*(no. of ecological planets in Y)]+[(no. of medical planets in Y)*(no. of ecological planets in X)] points for this pair of districts.   Company F gets 1 point for each transportation leg that it can service between Districts X and Y. So Company F gets [(no. of ecological planets in X)*(no. of agricultural planets in Y)]+[(no. of ecological planets in Y)*(no. of agricultural planets in X)] points for this pair of districts.       

     Each company&#39;s point total at the end of the game equals the sum of the points it receives in all districts plus the sum of the points it receives in all pairs of adjacent districts. The winner is the player (i.e. company) with the most points at the end of the game. 
     From the foregoing, it will be appreciated that although specific examples have been described herein for purposes of illustration, various modifications may be made without deviating from the spirit or scope of this disclosure. For example, even though the Examples are described as being analog, they could alternatively be digital, and any component of the games could be digitally represented. It is therefore intended that the foregoing detailed description be regarded as illustrative rather than limiting, and that it be understood that it is the following claims, including all equivalents, that are intended to particularly point out and distinctly claim the claimed subject matter.