Abstract:
A method and apparatus for playing casino-style roulette that offers players the option of placing specific wagers that limit the winning and losing outcomes to a subset of all possible outcomes for any given spin of the roulette wheel. Using a system of tokens to indicate the unique betting scenarios, the invention provides players with additional betting opportunities that increase the dynamics and enjoyment of the game while providing casino operators with the opportunity for incremental revenue.

Description:
RELATED APPLICATIONS  
       [0001]     This application claims priority to U.S. Provisional Patent Application No. 60/648,804, entitled “Method and Apparatus for Playing an Enhanced Casino Roulette Game” filed Feb. 1, 2005, which is hereby incorporated by reference. 
     
    
     FIELD OF THE INVENTION  
       [0002]     This application relates generally to a method and apparatus for playing a game of chance; more particularly to a method and apparatus for playing a modified version of roulette (e.g., a game where players wager on the particular slot of a rotating disk in which a small ball will come to rest) in which working wagers may be made that are active until a specific losing event occurs; and more particularly still to a method and apparatus for playing roulette in which a player may make working wagers which remain active until either a losing event occurs or the player requests that the working wager be removed/discontinued.  
       BACKGROUND OF THE INVENTION  
       [0003]     Roulette is one of the easiest casino games to play because there is no strategy to learn that can increase the likelihood of winning, such as in blackjack. Therefore, players usually bet on what they feel are trends of occurrence or lucky numbers. It also offers players a slower-paced game that can increase the gambling time with a given bankroll. This combination of attributes account, at least partly, for the game&#39;s popularity.  
         [0004]     Roulette offers casino operators one of the highest advantages of all casino games, with a 5.26% house edge on virtually all bets for a standard American roulette table with 38 numbers (e.g., eighteen black numbers, eighteen red numbers, a green zero, and a green double zero). The game features a variety of bets which are made by players placing one or more chips having a color designated specifically for that player on either a specific number, a combination of numbers (by placing a chip on the intersecting lines for those numbers), and/or a group of numbers designated by a special box. For any spin of a roulette wheel, players are generally free to make as many of these described bets as are allowed by minimum and maximum bets for that table. The chart set forth below in Table 1 summarizes the available bets in a standard game of American roulette:  
                                             TABLE 1                       Betting Scenario   Probability   Payout   House Edge (%)                                Any Red   47.37%   1:1   5.26       Any Black   47.37%   1:1   5.26       Any Even   47.37%   1:1   5.26       Any Odd   47.37%   1:1   5.26        1 to 18   47.37%   1:1   5.26       19 to 36   47.37%   1:1   5.26        1 to 12   31.58%   2:1   5.26       13 to 24   31.58%   2:1   5.26       25 to 36   31.58%   2:1   5.26       Any one number   2.63%   35:1    5.26       Either of two   5.26%   17:1    5.26       numbers       Any of three   7.89%   11:1    5.26       numbers       Any of four   10.53%   8:1   5.26       numbers       Any of six numbers   15.79%   5:1   5.26       0, 00, 1, 2 or 3   13.16%   6:1   7.89                  
 
         [0005]     The betting options set forth in Table 1 have remained essentially the same since the game was introduced. It will be appreciated that the common factors being that virtually every bet has the same house advantage and every bet either wins or loses based on a single spin of the roulette wheel. Numerous variations of roulette have been proposed that combine the game with other casino games, as well as variations of the game that include additional wheels or bonus bets and jackpots. However, these games have not been successfully marketed due to the appeal of the original game&#39;s inherent simplicity.  
       SUMMARY OF THE INVENTION  
       [0006]     The present invention introduces the concept of “working bets” to the game of roulette. A “working bet” is defined as a bet that is “live” or active until either a specific losing event occurs or the player asks that his bet be discontinued. By introducing the concept of working bets to roulette, the present invention offers roulette players betting options that have the potential to pay off numerous times and ensures that the player does not lose the bet unless a specific losing event occurs. Much like the “place number” bets in the game of craps, where a player can bet that certain dice values will occur before the dice value of seven occurs and vice versa, the present invention offers roulette players the same style of action by letting them bet that specific outcomes from the spin of the roulette wheel will occur before other specific outcomes from the spin of the roulette wheel. The working bets of the present invention are preferably implemented by using a system of tokens to help identify the specific working bets made by the player.  
         [0007]     By applying the concept working wagers to certain combinations of numbers—based on their numeric value, color orientation and/or arrangement on the roulette table—the invention enables a number of new wagering options. One feature of these working wagers is that they may be working for more than one spin of the wheel. Another feature is that the working wagers allow players to limit the risk of a bet to a select group of possible outcomes and the wagers may be provided with different odds than those typically offered in roulette. Yet another feature is that the working wagers provide the casino with an opportunity for incremental income without requiring additional space, tables or game surveillance. Further, implementation of the invention does not alter the playing surface, playing procedure or fundamental mechanics of the game.  
         [0008]     The present invention can easily be adapted to various gaming formats and media platforms in addition to standard roulette tables, including single-player and multi-player video slot machines, electronic games and devices, lottery terminals, lottery scratch-off cards, home-entertainment software, online gaming applications and online non-gaming applications.  
         [0009]     Therefore, according to one aspect of the invention, there is provided a method of playing roulette, comprising: placing wagers prior to a first spin of the roulette wheel that at least one of a first set of designated outcomes from the first spin of the roulette wheel will occur before one of a second set of designated outcomes from the first spin of the roulette wheel, wherein the sum of the probabilities for all designated winning and losing outcomes on any given spin of the roulette wheel is less than the total probability for all possible outcomes on any given spin of the roulette wheel; spinning the roulette wheel to determine the result of the first spin; and determining if the wager is satisfied, wherein: the placed wager is won only when at least one of the first set of designated outcomes occurs; the placed wager is lost only when at least one of the second set of designated outcomes occurs; and the placed wager is neither won or lost if neither of the first or second sets of designated outcomes occurs during the first spin of the roulette wheel.  
         [0010]     According to another aspect of the invention, there is provided a method of playing roulette in accordance with the preceding paragraph, and further comprising: allowing the placed wager to be removed or maintained prior to a second spin of the roulette wheel; and spinning the roulette wheel and determining if the placed wager is satisfied in accordance with the preceding paragraph if the placed wager has been maintained.  
         [0011]     According to yet another aspect of the invention, there is provided a method for playing roulette according to the preceding paragraphs, and further comprising allowing placement of additional wagers prior to a second spin of the roulette wheel and determining if the placed wager and placed additional wager is satisfied.  
         [0012]     While the invention will be described with respect to preferred embodiment configurations, particular groups of working wagers, and recommended payouts, it will be understood that the invention is not to be construed as limited in any manner by either such configuration, working wagers or recommended payouts described herein. Also, while the particular roulette game table layouts, tokens and equipment are described herein, it will be understood that such particular table layouts, tokens and equipment are not to be construed in a limiting manner. Instead, the principles of this invention extend to a roulette game in which players may make active working wagers. These and other variations of the invention will become apparent to those skilled in the art upon a more detailed description of the invention.  
         [0013]     The advantages and features which characterize the invention are pointed out with particularity in the claims annexed hereto and forming a part hereof. For a better understanding of the invention, however, reference should be had to the drawings which form a part hereof and to the accompanying descriptive matter, in which there is illustrated and described a preferred embodiment of the invention. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0014]     The present invention will be described in detail with reference to the following drawings. It is to be understood that the invention is not limited to the configurations represented by the following drawings and that the various embodiments of the invention can be configured in other manners in view of the present disclosure.  
         [0015]      FIG. 1  is an illustration of a token used to indicate a wager in accordance with the principles of the invention.  
         [0016]      FIG. 2  illustrates a first working wager example according to an embodiment of the present invention.  
         [0017]      FIG. 3  illustrates a second working wager example according to an embodiment of the present invention.  
         [0018]      FIG. 4  illustrates a third working wager example according to an embodiment of the present invention.  
         [0019]      FIG. 5  illustrates a fourth working wager example according to an embodiment of the present invention.  
         [0020]      FIG. 6  illustrates a fifth working wager example according to an embodiment of the present invention.  
         [0021]      FIG. 7  illustrates a sixth working wager example according to an embodiment of the present invention.  
         [0022]      FIG. 8  is an example of a video gaming machine apparatus of the type on which the various embodiments of the present invention may be played.  
         [0023]      FIG. 9  is an example of a personal computer apparatus of the type on which the various embodiments of the present invention may be played. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0024]     An underlying principle of the invention is based on the mathematics governing the probability of one of one or more given events occurring prior to one of one or more different events within a larger set of possible events. This principle allows the construction of new wagering options for the game of roulette without changing the fundamental method of play, the game equipment and/or the playing layout. Accordingly, a preferred embodiment of the invention utilizes standard casino roulette table.  
         [0025]     In view of this embodiment, the figures and description of the invention will be in reference to a standard game of casino roulette, which is well known to those skilled in the art. It will be understood that while the preferred embodiment of the invention pertains to roulette tables and wheels that have 38 numbers (e.g., those with both a “0” and “00”), the underlying mathematics of the invention and vast majority of betting scenarios specific to the invention can also be applied to tables and wheels with 37 numbers (e.g., those with only a single “0”).  
         [0026]     Turning now to  FIG. 1 , the invention can be implemented using a system of tokens  10  that are inscribed with relevant information to help the dealer track the bets specific to the invention, including the identification of the bet as a working bet  12 , specific winning and losing parameters for the bet  14  and the payout consequent to a winning event  16 .  
         [0027]     As noted above, the game is played following the normal sequence of events and rules of roulette. While such sequence of events and rules are well known to those skilled in the art, a brief description will be provided herein.  
         [0028]     First, at the beginning of play, players place their chips on numbers or combination of numbers or a color to indicate the bets they wish to make. At many tables, there is a minimum wager amount for each wager and a maximum wager. Further, a player may generally make as many wagers as he or she wishes (e.g., a player is not constrained to make a single bet. Instead, by way of example, wagers might be made on the color red, the number 12, and the sequence of numbers 1, 2, and 3 prior to a single spin of the roulette wheel).  
         [0029]     In the prior art, a winning or losing event for a wager is always determined by a single spin of the roulette wheel. The present invention, however, introduces roulette wagers where a winning or losing event is not necessarily determined by a single spin of the roulette wheel. In the preferred embodiment, this is implemented by asking the roulette dealer to mark such a working wager with a specially marked token, indicating specific winning and losing parameters for the working wager. That wager then remains “working” (e.g., live) until either a losing event occurs or the player indicates that the bet should be discontinued. If a winning outcome occurs on any given spin of the roulette wheel, the player wins and the dealer pays off the bet according to the predetermined payout indicated for the bet. The player can then opt to maintain the same bet for the next spin of the roulette wheel or indicate that the bet should be discontinued. If a losing outcome occurs on any given spin of the roulette wheel, then the player loses and the dealer collects the bet. If none of the winning or losing outcomes occur on any given spin of the roulette wheel, the bet remains untouched and the player can choose to leave the bet working for the next spin of the roulette wheel or can indicate that the bet be discontinued. If the player indicates that the bet is discontinued, then the dealer removes the token signifying the working bet and the player can then retrieve the chips that made up the wager. This procedure may be utilized for any of the representative working wagers described herein.  
       Sample Embodiments of the Invention  
       [0030]      FIGS. 2-7  illustrate various examples of a method and apparatus according to an embodiment of the present invention that utilizes a traditional roulette table layout  20  with thirty-eight numbers (1-36, 0 and 00) identified individually by color and groups, a traditional roulette wheel  22  having thirty-eight numbers (1-36, 0 and 00), an example of a game wagering chip  24 , a token that indicates a bet specific to the invention  26 , a group of wagering chips with a token specific to the invention placed on top of the chips  28  and the placement of a wager specific to the invention on the roulette layout  30 . It will be appreciated that the hatched shaded numbers represent black numbers and the unshaded numbers represent red numbers in the FIGURES. Both the numbers 0 and 00, are also unshaded, but are typically colored green on prior art roulette wheels and tables.  
         [0031]      FIG. 2  represents an example of a method and apparatus according to an embodiment of the present invention that limits the winning and losing outcomes for a bet to any one number occurring before either of two other numbers. In this particular example, the player is betting five chips that the number 12 will occur before either one of the numbers 0 or 00. Five chips are placed on the number 12 on the roulette table and the appropriate token is placed on top of the chips to indicate that the wager is limited to a losing outcome of 0 or 00. The specific working wager then remains live until either a 0 or 00 occurs or the player indicates a desire to discontinue the bet. If the number 12 occurs before 0 or 00, the player wins and the dealer pays off the bet according to the predetermined payout indicated for the bet. Once either a 0 or 00 occurs, the player loses and the dealer collects the bet. If none of the winning or losing outcomes occur on any given spin of the roulette wheel, the bet remains untouched and the player can choose to leave the bet working for the next spin of the roulette wheel or he can retrieve the bet by indicating that he wishes to discontinue the working wager.  
         [0032]     Note that the initial working wager may optionally be left on the number 12 after a winning roulette spin and the player is paid off—such that the bet remains live, and if the next spin is also a 12, the player would be paid off again according to the predetermined payout indicated for the bet. This option may be used in connection with some or all of the working wagers described herein.  
         [0033]      FIG. 3  represents an example of a method and apparatus according to an embodiment of the present invention that limits the winning and losing outcomes for a bet to either of two numbers occurring before any of twelve other numbers. In this particular example, the player wagers five chips that either of the numbers 0 or 00 will occur before any of the twelve numbers 25 through 36. In this example, five chips are placed on the line between 0 and 00 on the roulette table and the appropriate token is placed on top of the chips to indicate that the wager is limited to a losing outcome of any of the numbers 25 through 36. The working wager then remains live until any of the numbers 25 through 36 occurs or the player indicates a desire to discontinue the bet. If either of the numbers 0 or 00 occurs before any of the numbers 25 through 36, the player wins and the dealer pays off the bet according to the predetermined payout indicated for the bet. If any of the numbers 25 through 36 occurs before either of the numbers 0 or 00, the player loses and the dealer collects the bet. If none of the winning or losing outcomes occur on any given spin of the roulette wheel, the bet remains untouched and the player can choose to leave the bet working for the next spin of the roulette wheel or the player can retrieve the bet by indicating a desire to discontinue the working wager.  
         [0034]      FIG. 4  represents an example of a method and apparatus according to an embodiment of the present invention that limits the winning and losing outcomes for a bet to any one number occurring before any of eleven other numbers. In this example, the player wagers five chips that the number 8 will occur before any other number within the first twelve numbers. In this example, five chips are placed on the number 8 on the roulette table and the appropriate token is placed on top of the chips to indicate that the wager is limited to a losing outcome of any of the numbers 1 through 7 or 9 through 12. The working wager then remains live until any of the numbers 1 through 7 or 9 through 12 occurs or the player indicates a desire to discontinue the working wager. If the number 8 occurs before any of the numbers 1 through 7 or 9 through 12, the player wins and the dealer pays off the bet according to the predetermined payout indicated for the bet. If any of the numbers 1 through 7 or 9 through 12 occurs before the number 8, the player loses and the dealer collects the bet. If none of the winning or losing outcomes occur on any given spin of the roulette wheel, the working wager remains untouched and the player can choose to leave the bet working for the next spin of the roulette wheel or can retrieve the bet by indicating a desire to discontinue the working wager.  
         [0035]      FIG. 5  represents an example of a method and apparatus according to an embodiment of the present invention that limits the winning and losing outcomes for a bet to any of twelve numbers occurring before any of twelve other numbers. In this example, the player wagers five chips that any of the twelve numbers 25 through 36 will occur before any of the twelve numbers 1 through 12. Here, five chips are placed in the box on the roulette table that covers all of numbers 25 through 36 and the appropriate token is placed on top of the chips to indicate that the wager is limited to a losing outcome of any of the numbers 1 through 12. The specific bet then remains live until any of the numbers 1 through 12 occurs or the player indicates a desire to discontinue the working wager. If any of the numbers 25 through 36 occurs before any of the numbers 1 through 12, then the player wins and the dealer pays off the bet according to the predetermined payout indicated for the bet. If any of the numbers 1 through 12 occurs before any of the numbers 25 through 36, then the player loses and the dealer collects the bet. If none of the winning or losing outcomes occur on any given spin of the roulette wheel, the bet remains untouched and the player can choose to leave the bet working for the next spin of the roulette wheel or he can retrieve the bet by indicating a desire to discontinue the working wager.  
         [0036]      FIG. 6  represents an example of a method and apparatus according to an embodiment of the present invention that limits the winning and losing outcomes for a bet to any of six numbers occurring before any of twelve other numbers. In this example, the player wagers five chips that any of the six numbers 4 through 9 will occur before any of the twelve numbers 13 through 24. In this example, five chips are placed on the line between the two rows on the roulette table that cover all six numbers 4 through 9 and the appropriate token is placed on top of the chips to indicate that the wager is limited to a losing outcome of any of the numbers 13 through 24. The specific bet then remains live until any of the numbers 13 through 24 occurs or the player indicates a desire to discontinue the working wager. If any of the numbers 4 through 9 occurs before any of the numbers 13 through 24, the player wins and the dealer pays off the bet according to the predetermined payout indicated for the bet. If any of the numbers 13 through 24 occurs before any of the numbers 4 through 9, the player loses and the dealer collects the bet. If none of the winning or losing outcomes occur on any given spin of the roulette wheel, the bet remains untouched and the player can choose to leave the bet working for the next spin of the roulette wheel or the player can retrieve the bet by indicating a desire to discontinue the working wager.  
         [0037]      FIG. 7  represents an example of a method and apparatus according to an embodiment of the present invention that limits the winning and losing outcomes for a bet to any of eighteen numbers occurring before any of ten other numbers. In this example, the player wagers ten chips that any of the eighteen red numbers will occur before any of the ten even black numbers. Here, ten chips are placed in the box on the roulette table that covers all 18 red numbers and the appropriate token is placed on top of the chips to indicate that the wager is limited to a losing outcome of any even black number, of which there are 10. The specific bet then remains live until an even black number occurs or the player indicates a desire to discontinue the bet. If any odd or even red number occurs before any even black number, the player wins and the dealer pays off the bet according to the predetermined payout indicated for the bet. If any even black number occurs before any odd or even red number, the player loses and the dealer collects the bet. If none of the winning or losing outcomes occur on any given spin of the roulette wheel, the bet remains untouched and the player can choose to leave the bet working for the next spin of the roulette wheel or the player can retrieve the bet by indicating a desire to discontinue the working wager.  
         [0038]     It is to be understood that depending upon the payout of any given bet specific to the invention (as well as the table rules for minimum bets specific to the invention), the player may have to place a minimum amount on any given bet specific to the invention. It is also to be understood that the invention can utilize any design for a token or need not be limited to tokens as an identifying device for the bets specific to the invention.  
         [0039]     Alternatively, the various embodiments of the invention may be played in a virtual environment.  
         [0040]      FIG. 8  is an illustration of a casino-style video gaming machine apparatus on which the various embodiments of the invention can be played and typically includes a physical housing  32 , an electronic display monitor  34 , a computer processor and computer program designed to conduct the game through random number generation technology  36 , a player interface console  38  that includes operative buttons (and/or other player input devices) by which the player can input decisions for the game  40 , an input device for depositing funds via coins, paper currency, bank cards or casino debit cards  42  and an output device for returning sums to the player  44 .  
         [0041]      FIG. 9  is an illustration of a personal computer apparatus on which the various embodiments of the invention can be played off-line or online over the internet at virtual casinos and typically includes a physical housing  46 , an electronic display monitor  48 , a computer processor and computer program designed to conduct the game through random number generation technology or via the internet at an online virtual casino  50 , and a computer keyboard  52  and mouse device  54  to assist in play of the game.  
       Underlying Mathematical Principles of the Invention  
       [0042]     The various bets and examples of wagers provided herein in accordance with the present invention are based on the mathematical principles governing the probability of one of one or more given events occurring before one of one or more different given events within a greater finite set of events. The methodologies used to calculate the true odds payouts, recommended payouts and house edges for any betting scenario specific to the invention, when played on a standard American roulette wheel with 38 numbers, each having a 2.63% probability of occurrence on any given spin of the roulette wheel, are next described.  
         [0043]     To calculate the true odds payout (T) for any betting scenario, the sum of the probabilities for the winning events (W) is divided into the sum of the probabilities for the losing events (L). 
 
 T=L/W  
 
         [0044]     As an example, to calculate the true odds payout for a wager that the number 12 will occur before either of the numbers 0 or 00, one divides the probability of a 12 occurring on any given spin of the roulette wheel (2.63%) into the probability of either a 0 or 00 occurring on any given spin of the roulette wheel (5.26%) to reach a result of 2:1 
 
 T= 5.26/2.63=2:1 
 
         [0045]     By offering slightly less than the true odds payout, the house retains a given edge for any given betting scenario. To calculate the house edge based on a recommended payout (P) that is less than the true odds payout (T), one must first consider that there are really two house edge calculations to be performed—the house edge based on a single spin of the wheel (S) and the total house edge (E) based on the average number of spins of the wheel needed to ensure a resolution for the betting scenario (A).  
         [0046]     To calculate the single-spin house edge (S) for any given betting scenario, one multiplies the sum of the probabilities for the winning events by the recommended payout (P) for the betting scenario and subtracts that number from the sum of the probabilities for the losing events. 
 
 S=L −( W×P ) 
 
         [0047]     Using the same example for a wager that the number 12 will occur before either a 0 or 00, with a recommended payout of 9:5, one calculates the single-spin house edge by multiplying the probability of a 12 occurring on any given spin of the roulette wheel (2.63%) by the recommended payout (9:5) and subtracting that number from the probability of either a 0 or 00 occurring on any given spin of the roulette wheel (5.26%) to achieve a single-spin house edge of approximately 0.526%. 
 
 S= 5.26−(9/5×2.63) 
 
 S= 5.26−4.734 
 
S≈0.526% 
 
         [0048]     To determine the total house edge (E) for any given betting scenario, one calculates how many spins of the roulette wheel on average must be played for one of the winning or losing events in the betting scenario to occur (A) and then multiplies that number by the single-spin house edge (S). The average number of spins of the roulette wheel that must be played for one of the winning or losing events in the betting scenario to occur (A) is defined as the sum of the probabilities for all possible outcomes on any given spin of the roulette wheel divided by the sum of the probabilities for all winning and losing outcomes. 
 
 A= 100/( W+L ) 
 
 E=A×S  
 
         [0049]     Using the same betting scenario of the number 12 occurring before either of the numbers 0 or 00 with a recommended payout of 9:5, one calculates the total house edge for the betting scenario as follows: 
 
 A= 100/(2.63+5.26) 
 
 A= 100/7.89 
 
A=12.674271 spins 
 
 E= 12.674271×0.526 
 
E=6.67% 
 
         [0050]     Alternatively, the following shortcut probability equation can also be used to determine the total house edge (E) for any given betting scenario where the true odds (T) and proposed payout (P) are known. 
 
 E =( T−P )/( T+ 1) 
 
         [0051]     Using the same example, the total house edge (E) for a 12 occurring before either a 0 or 00, where the true odds (T) is 2:1 and the payout (P) is 9:5, would be calculated as follows: 
 
 E =(2−9/5)/(2+1) 
 
 E =(2−1.8)/3 
 
 E= 0.2/3 
 
E=0.066666 
 
E=6.67% 
 
         [0052]     The underlying mathematical principles of the invention can be applied to any betting scenario where the sum of the probabilities for all winning and losing outcomes is less than the total probability for all possible outcomes.  
       Working Wager Examples and Payout Schedules  
       [0053]     In addition to the working wager embodiments and examples set forth above, the true odds, payout options that may be utilized, and associated house edges for various betting scenarios specific to the invention are set forth in TABLE 2 below. It will be appreciated that the working wager examples presented in the following table can be applied to roulette wheels with 38 numbers (those with the numbers 1-36 plus both a 0 and 00), where each number has a 2.63% probability of occurrence on any given spin of the roulette wheel, as well as roulette wheels with 37 numbers (those with only a single 0), where each number has a 2.70% probability of occurrence. It should also be understood that while these betting scenarios cover many of the desirable and feasible betting scenarios for the invention, they should not be construed as limiting or as the only betting scenarios for the invention. Further, alternate payouts may be substituted for any of the recommended payouts.  
                                             TABLE 2                           True Odds   Recommended   House       Betting Scenario   Payout   Payout(s)   Edge (%)                                Any one number occurring   1:1    9:10   5.0       before any one other        4:5   10.0       number       Any one number occurring   2:1    19:10   3.3       before either of two other        9:5   6.7       numbers       Any one number occurring   7:1   33:5   5.0       before any of seven other       32:5   7.5       numbers       Any one number occurring   8:1   38:5   4.4       before any of eight other       37:5   6.7       numbers       Any one number occurring   9:1   43:5   4.0       before any of nine other       42.5   6.0       numbers       Any one number occurring   10:1    48:5   3.6       before any of ten other       47:5   5.5       numbers        9:1   9.1       Any one number occurring   11:1    53:5   3.3       before any of eleven other       51:5   6.7       numbers       10:1   8.3       Any one number occurring   12:1    58:5   3.1       before any of twelve other       56:5   6.2       numbers       11:1   7.7       Any one number occurring   17:1    82:5   3.3       before any of seventeen       16:1   5.6       other numbers       Any one number occurring   18:1    87:5   3.2       before any of eighteen other       17:1   5.3       numbers       Either of two numbers   .5:1     2:5   6.7       occurring before any one       other number       Either of two numbers   1:1    9:10   5.0       occurring before either of        4:5   10.0       two other numbers       Either of two numbers   4:1   19:5   4.0       occurring before any of       18:5   8.0       eight other numbers       Either of two numbers   5:1   24:5   3.3       occurring before any of ten       23:5   6.7       other numbers       Either of two numbers   6:1   29:5   2.9       occurring before any of       28:5   5.7       twelve other numbers       Either of two numbers   9:1   43:5   4.0       occurring before any of       42:5   6.0       eighteen other numbers       Any of three numbers   .3:1     1:5   10.0       occurring before any one       other number       Any of three numbers   .7:1     5:10   10.0       occurring before either of       two other numbers       Any of three numbers   3:1   14:5   5.0       occurring before any of nine       13:5   10.0       other numbers       Any of three numbers   4:1   19:5   4.0       occurring before any of       18:5   8.0       twelve other numbers       Any of three numbers   5:1   24:5   3.3       occurring before any of       23:5   6.7       fifteen other numbers       Any of three numbers   6:1   29:5   2.9       occurring before any of       28:5   5.7       eighteen other numbers       Any of four numbers   .25:1      1:5   4.0       occurring before any one       other number       Any of four numbers   .5:1     2:5   6.7       occurring before either of       two other numbers       Any of four numbers   2:1    19:10   3.3       occurring before any of        9:5   6.7       eight other numbers       Any of four numbers   3:1   14:5   5.0       occurring before any of       13:5   10.0       twelve other numbers       Any of four numbers   3.5:1     16:5   6.7       occurring before any of       fourteen other numbers       Any of four numbers   4.5:1     21:5   5.5       occurring before any of       eighteen other numbers       Any of five numbers   1.8:1      8:5   7.1       occurring before any of nine       other numbers       Any of five numbers   2.4:1     11:5   5.9       occurring before any of       twelve other numbers       Any of five numbers   3:1   13:5   10.0       occurring before any of       fifteen other numbers       Any of five numbers   3.6:1     16:5   8.7       occurring before any of       eighteen other numbers       Any of six numbers   .17:1      1:10   5.7       occurring before any one       other number       Any of six numbers   .33:1      2:10   10.0       occurring before either of       two other numbers       Any of six numbers   1:1    9:10   5.0       occurring before any of six        4:5   10.0       other numbers       Any of six numbers   2:1    19:10   3.3       occurring before any of        9:5   6.7       twelve other numbers       Any of six numbers   3:1   14:5   5.0       occurring before any of       13.5   10.0       eighteen other numbers       Any of eight numbers   .25:1      1:10   12.0       occurring before either of       two other numbers       Any of eight numbers   1:1    9:10   5.0       occurring before any of        4:5   10.0       eight other numbers       Any of eight numbers   1.25:1      11:10   6.7       occurring before any of ten       other numbers       Any of eight numbers   2.25:1      2:1   7.7       occurring before any of       eighteen other numbers       Any of ten numbers   .2:1     1:10   8.3       occurring before either of       two other numbers       Any of ten numbers   .8:1     7:10   5.6       occurring before any of       eight other numbers       Any of ten numbers   1:1    9:10   5.0       occurring before any of ten        4:5   10.0       other numbers       Any of ten numbers   1.8:1      17:10   3.6       occurring before any of        8:5   7.1       eighteen other numbers       Any of twelve numbers   .17:1      1:10   5.7       occurring before either of       two other numbers       Any of twelve numbers   .5:1     2:5   6.7       occurring before any of six       other numbers       Any of twelve numbers   1:1    9:10   5.0       occurring before any of        4:5   10.0       twelve other numbers       Any of eighteen numbers   .44:1      2:5   3.1       occurring before any of        3:10   10.0       eight other numbers       Any of eighteen numbers   .56:1      5:10   3.6       occurring before any of ten        4:10   10.0       other numbers                  
 
         [0054]     It is to be understood that the embodiments shown and described herein are merely illustrative of the principles of this invention and that various modifications may be implemented by those skilled in the art without departing from the scope and spirit of the invention.