# Patent

## Title
Binomial and multinomial-based slot machine

## Abstract

A gaming apparatus performs a gaming method with a symbol display system for a wagering game, a processor controlling the symbol display system and software executed by the processor, has software perform electronic functions of:

    
    
        a) providing a method of value crediting and debiting system;
        b) providing a game control component that determines rules of play of a game;
        c) providing activation of selection from virtual spinners that have individual game determinant outcomes or individual symbol determinant outcomes mathematically distributed within the virtual outcome determinant space of the virtual spinner;
        d) providing a file of images available for display on the symbol display system;
        e) the software randomly accessing the predetermined weighted portions of the outcome determinant space to select individual symbols, sets of symbols or collective symbols for use in the game;
        f) determining game outcomes; and
        g) resolving all value placed at risk in the play of the game.

## Background
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to the field of gaming, especially to electronic gaming in processor based apparatus, and in particular to video gaming apparatus in which outcomes are based on random generation of symbols into fields and the attainment of predetermined orders or sets or collections of symbols to identify winning events.
2. Background of the Art
Electronic casino games, whether video poker or slot games, have grown exponentially in numbers in the last twenty years, as have the revenues generated by such machine games. It is estimated that more than three fourths of any casino's revenue is now provided by machine games as opposed to table games.
The casino patron usually gravitates to either table games or machine games due to the very nature of each genre. The table player can be drawn by the camaraderie of group interaction and the typically lower house advantage games with less dramatic win/loss swings. Odds of approximately 1-to-1(within 1-6%) are common in casino table games, and can provide the player with more frequent wins and a slower depreciation of assets. By way of contrast, the machine player is more likely to enjoy solitary play. The solitary player also is motivated to play games that may have larger house advantages but which can provide huge payouts, albeit with a higher degree of volatility. This higher volatility is due to the fact that to provide large or jackpot wins, the game would have many more results which are either a complete loss, a push or a win of less than the total wager. The machine player can become disheartened with a streak of these losing results. Additionally, in games that feature a multiple step game play, the initial spin or deal may appear to be both a losing event and a poor start, which can compound the player's frustration and lead to less time on the machine. There is often the perception that the machine game is “rigged” to provide an inordinate amount of these bad starts, especially after a player has had some initial winning results. Prior art has sought to address these issues, but there is still a need for new inventive game play that gives the player more positive expectations and a feeling that even poor starts can be turned into a win.
U.S. Pat. No. 6,855,054 (White) describes methods of playing games of chance and gaming devices and systems comprising a display of a plurality of symbols where at least one symbol may be interchanged (two way exchange) with another symbol of the plurality of symbols. After a combination of symbols initially is randomly generated and the initial results are displayed to a player, the player may have the opportunity to interchange at least one displayed symbol with another symbol in order to configure a more advantageous symbol arrangement.
U.S. Pat. Nos. 6,641,477 and 5,704,835 (Dietz, II) describe an electronic slot machine and method of use which allows a player to completely replace up to all of the initial symbols displayed after the first draw in order to create, improve or even lose a winning combination. If a suitable winning combination is not formed with the initial symbols, the player is given opportunities to select up to all of the symbol display boxes for replacement.
US Patent Publication No. 20060183532 (Jackson) discloses a display on which symbols may be provided for use in a slot-type wagering game. Symbols are displayed on sectioned geometrical shapes such as ovals, squares, circles, polygons, etc. Specific symbol combinations, particularly comprised of one symbol appearing on one section of each sectioned geometric shape or all symbols appearing on all sections of one sectioned geometric shape, may constitute a winning combination according to a predetermined pay table. Preferably the invention incorporates three 3-section circular reels, providing 30 different pay lines and an additional pay line incorporating all nine sections of the reels.
Disclosed herein is a family of pure-luck slot machines based on mechanized playout of simple one and two player games, using a method of calculating pay tables for two or more spinner devices based on the game. The machines are simple enough to be implemented with physical hardware is random number generators for players who are suspicious of a computer controlling the random element. Computers would still be used to scan the result of the physical events, calculate payout, and operate the payment mechanisms, whether coin, magnetic, printed, wireless, or other future payment methods. The same games could be implemented in existing slot machine platforms, pure software for computers and video game consoles, mobile gaming platforms, pocket computers, cellular phones capable of running game programs, and so forth.
U.S. Pat. Nos. 7,470,182 (Martinek et al.); 6,159,096 (Yoseloff); and 6,117,009 (Yoseloff) disclose novel mapping systems in which all possible final outcomes (e.g., all of the displays available on a three-reel slot) are defined as templates, and each template is assigned a specific probability. A random number generator then selects an individual template to be displayed based on the probability of the specific template.
The present technology advances gaming systems and games as described herein. All references cited in this disclosure are incorporated herein by reference in their entirety to provide background on technical enablement for apparatus, components and methods.
SUMMARY OF THE INVENTION
A gaming apparatus includes a symbol display system for a wagering game, a processor controlling the symbol display system and software executed by the processor, wherein the software comprises executable steps to perform electronic functions of:
a) providing a method of value crediting and value debiting system that identifies value risked in the play of the wagering game and credits awards won in the play of the wagering game;
b) providing a game control component that determines rules of play of a game played on the gaming apparatus;
c) providing activation of symbol and/or event outcome selection by the processor from virtual spinners that have individual game determinant outcomes or individual symbol determinant outcomes mathematically distributed within a virtual outcome determinant space of the virtual spinner;
d) providing a file of images available for display on the symbol display system, the specific display of individual symbols, sets of symbols or collective symbols being determined by predetermined weighted portions of the outcome determinant space;
e) the software responding to user commands to initiate a game by randomly accessing the predetermined weighted portions of the outcome determinant space to select individual symbols, sets of symbols or collective symbols for use in the game;
f) determining whether the randomly accessed predetermined weighted portions of the outcome determinant space has provided individual symbols, sets of symbols or collective symbols that constitute a win according to the game; and
g) resolving all wagers on all value placed at risk in the play of the game.
The “virtual spinners” are distributions of probabilities of outcomes (e.g., specific portions of the virtual spinner or mathematically defined regions of probability) that totals effectively 100% from all of its regions of probability. Specific regions (which can be equated to specific symbol outcomes or event outcomes) of the virtual spinner are determined to have weighted probabilities of being selected, and each region (outcome) will have associated with it a predetermined symbol display outcome (when individual symbols or less than complete subsets of symbols are displayed) or predetermined complete symbol display outcome (event outcome) that is selected. These outcomes or regions may be final outcomes (end of game outcomes with all steps completed for game play) or may be an intermediate event determination (e.g., a first move of the markers in a Nannon® virtual board game, with subsequent outcomes indicating subsequent steps or moves by random weighted selection of die or dice outcomes or a bonus triggering event in combination with any of the preceding steps.) Another example may be final outcomes of a blackjack hand, with intermediate events being the sequential deal of cards to the hand. This is more complex, as there are options that may be exercised by players that could differentiate play of blackjack hands to conclusion.
The game may be a game in which outcomes are determined by one or more displays of symbols selected from group consisting of playing cards, specialty cards, dice and spinners and wherein the file of images stored in memory and accessible by the processor for display may include virtual dice and virtual token positions on a virtual game board. The game outcome may be determined by repeated random selection of predetermined weighted portions to make repeated moves of the virtual tokens on the virtual game board. The virtual game board may be a truncated backgammon board, e.g., wherein the virtual game board has only six available positions on the virtual game board for positioning of virtual tokens. The symbols may be selected from the group consisting of symbols to be randomly displayed, symbols or markers (location markers, pegs in cribbage, etc.) to fill preexisting spaces in a game board, playing cards, dice and coins. Each symbol or a set of symbols may be determined by the software according to the random selection of the predetermined weighted portions of the outcome determinant space. The gaming apparatus may have the predetermined weighted portions of the outcome determinant space (virtual space or mathematical space) selected so that on a long-term probability basis, for example, so that between 92 and 99% of total wagers placed by players (or whatever total is designed into the game) will be returned to players in winning or pushing events. These spaces may remain constant through repeated games or vary from game to game in a further random manner, with different spinners randomly selected for each game.
A method of playing a game on the gaming apparatus described above would have a payout system wherein none of, portions of or the total of player credits or winnings are returned to players at player direction by player input to the gaming apparatus either as coins, credits, tokens or printed credit slip. The random selection of predetermined weighted portions of the outcome determinant space may determine discrete (e.g., intermediate, partial, single step, etc.) outcomes in a board game or card game. For example, outcomes from the virtual spinner are selected from the group consisting of a distinguished LOSE state, and a set of winning states each determined by a weighted probability, wherein each weighted probability is used to calculate binomial or multinomial coefficients which may be used to determine the payout levels.



BRIEF DESCRIPTION OF THE FIGURES
FIG. 1 is a graph representing change in level of wagering based on statistical changes in fairness of spinner values and distributions.
FIG. 2 is a graphic representation of when a player one rolls first the outcomes form a weighted coin, shown by this spinner.
FIG. 3 is a graphic representation of a distribution of random play of Tic Tac Toe, from Player 2's perspective.
FIG. 4 is a graphic representation that increasing returns can be set up on 3, 6, 9, and 12 boards of random TicTacToe.
FIG. 5 is a graphic representation of spinner outcomes from a fair, six-sided die.
FIG. 6 shows a graphic representation of Nannon® game endings as a 7-sided spinner.
FIG. 7 shows a graphic representation of Clustered outcomes of 4×4 Othello game
FIG. 8 is a graphic representation of a Spinner derived from 2-card poker hands.
FIG. 9: is a graph of a Spinner derived from 5-card poker hands.
FIG. 10 is a graphic representation of a Spinner derived from 3-card poker.
FIG. 11 is a graphic representation of a Spinner derived from Blackjack



DETAILED DESCRIPTION OF THE INVENTION
One aspect of the present invention is to turn traditional recognizable games using coins, dice, spinners, cards, checkers, and so forth, into slot machine concepts which are easy to recognize, to understand and play, while providing the house with flexibility at setting the return and reinforcement. A gaming apparatus comprising a symbol display system for a wagering game, a processor controlling the symbol display system and software executed by the processor. The software has the ability to perform electronic functions enabling play of a wagering game. The functions a) provide a method of value crediting and debiting system that identifies value risked in the play of the wagering game and awards won in the play of the wagering game; b) provide a game control component that determines rules of play of a game played on the gaming apparatus; c) provide activation of selection from virtual spinners that have individual game determinant outcomes or individual symbol determinant outcomes mathematically distributed within the virtual outcome determinant space of the virtual spinner; d) provide a file of images available for display on the symbol display system, the specific display of individual symbols, sets of symbols or collective symbols being determined by predetermined weighted portions of the outcome determinant space; e) the software responds to user commands to initiate a game by randomly accessing the predetermined weighted portions of the outcome determinant space to select individual symbols, sets of symbols or collective symbols for use in the game; f) determines whether the randomly accessed predetermined weighted portions of the outcome determinant space has provided individual symbols, sets of symbols or collective symbols that constitute a win according to the game; and g) resolves all value placed at risk in the play of the game.
The present technology may be incorporated into gaming events using either real (physical) spinners or electronic spinners. In using physical or mechanical spinners, the physical spinners may be used in real time, or a table established for continual use in a game or multiple spinners used contemporaneously to establish the probabilities or outcomes. For example, a spinner (e.g., two dice) may be cast, observed by image capturing systems (e.g., analog or digital cameras), and the spinner outcome analyzed and used upon electronic entry into a gaming processor system, to determine symbol outcome or event outcome (based on an existing look-up table for event outcomes). In this practice, it is to be understood that the roll of the dice is not itself the event outcome, but is a spinner determining separate symbol or event outcomes. Distal image capture of actual gaming events and use of those distal outcomes in standard wagering formats (e.g., Rapid Roulette® systems) is known in the art. Non-limiting Examples of physical play that can be used in the practice of the present technology includes, but is not limited to, flipping a predetermined number of coins (e.g., 5, 8, 10, 12 or 15 physical coins, using computer vision to count the number of heads that come up, then paying out from the paytable), or randomly ordering 9 numbered marbles into a permutation, reading the order with computer analysis of the outcome, and using the outcome as the RNG for a software tictactoe game; rolling a sequence of dice which are read by computer vision and used in moving Nannon® game pieces on a virtual board; and dealing 5 cards from a new shuffle and using vision/barcode and the game algorithm to sequentially reveal a blackjack hand from 2 to 5 cards according to the rules of blackjack.
An alternative method is to use virtual spinners in the determination of symbol outcomes (i.e., individual symbol occurrence during play of a game) or event (including partial event) outcome (e.g., an initial hand dealt in 5-card draw poker, or a complete 5-card hand in stud poker, or any other final game event outcome). In using a virtual spinner, a look up table is provided with the distribution of probabilities already established (by mathematic or actual event outcome performance over a statistically significant number of events, as is required in the gaming industry for compliance) and that look-up table is accessed by use of a random number generator selecting a specific outcome in the table, and that outcome being already associated with specific symbol outcomes or event outcomes is used to determine the symbol or event occurring in the play of the game.
An important element in an appreciation of the advance of the present technology is the definition of the term “virtual spinner.” A statistical or probabilistic distribution is created based on real-life events having determinable probabilities. Existing event series (consecutive coin flips, consecutive selections from among equally weighted selections, etc.), games (poker games, blackjack games, baccarat games, Tic-Tac-Toe games, etc.) or defined physical events (die roll, dice roll, card cutting, coin flipping, candy wheel spinning, etc,). The actual probability distribution of the real-life event is then mathematically distributed as segments within a region that is the basis of selection by a random number generator. The random number generator then randomly selects among the statistical regions provided by the real-life event. The symbol outcomes or event outcomes are associated with each of these regions so that the random number generator's selection of any region determines a symbol outcome (in a specific or general location) or an event outcome. Once the probabilities of the regions of the real-life event have been determined, those regions may be artificially weighted in association with specific symbol outcomes, symbol locations and/or event outcomes. The weighting of the regions offers a core basis of probabilities based on real-life events that can be adjusted to create designed returns from wagering games on automated wagering systems. The automated wagering systems may be in the form of slot type machines (either reel-type or video type), poker-type machines (single game, multi-line, stud, draw, 2-card, 3-card, 4-card, 5-card, 6-card, 7-card, hi-lo, etc.), video blackjack, bonus games and the like.
In these new machines a random element we call a SPINNER is replicated more than once at the choice of the player. The SPINNERS are operated quickly and in parallel called a THROW (a single game play or game event). Each spinner has a finite set of OUTCOMES of non-increasing probabilities which sum to 100%. The first outcome with the largest probability is considered the ZERO state, and the other outcomes may be labeled 1, 2, 3 . . . and so on. The spinner may be exemplified or displayed as simple coins, dice, or spinners or mechanical contrivances which appear in a known game such as tic-tac-toe, checkers, chess, Othello, or backgammon and the like. A spinner can be a solitaire or two player games where robots or other automated systems shuffle, deal or roll randomly, using checkers, markers, marbles, or playing cards. The SPINNER can be implemented physically or purely in software, with or without display to the player. Once the final outcome of each SPINNER is determined, the SUM of the outcomes is used to resolve a wager against a payout table based on the size of the bet and the player is paid according to that resolution.
Allowing the player to choose how many SPINNERS to bet on, and calculating the reward based on the binomial or multinomial coefficients leads to a new class of simple slot machines based on known games.
As will be disclosed below, the bet, the size of the jackpot, the player return (house edge), and the win/lose ratio (the reinforcement) are all adjustable to achieve the values required by profitability, legal framework, and player psychology.
From several examples, the novelty of this new kind of slot machine will be clear to those experienced in the art. Even though many video poker games exist, including ones which allow 5, 50 or 100 “hands” to be played in parallel, each payout event only leads to an independent payoff summed for each hand, such as $3 for each flush or $10 for each full house. In the present invention when applied to poker, the total payout in a single round of play will be exponentially increased as each independent deal of hands played contemporaneously results in a good hand.
Machines Using Binomial Distribution.
The new board game of NANNON® game, by this inventor, is a simplified family of games based on the ancient game of Backgammon. It is a two-player dice/race/hitting/blocking game, but uses a shorter board, fewer checkers, and employs adjacency rather than stacking for creating blockades which can cause an opponent to lose their turn. This family game is cyclical and enjoys a lot of turnabout in expectations, yet has no draw or stalemate and inevitably ends. When a computer strategy plays against itself, each player will win 50% of the time, just like flipping a coin It was through diligent design of a slot machine based on NANNON® game that the present invention emerged.
Consider a machine which used a fair random binary element, such as a coin with two landing states “heads” and “tails”. There are two outcomes with non-increasing probability distribution [0.5 and 0.5]. Tails would be considered a ZERO, a worse outcome then heads (1). Consider a machine which flips multiple fair coins and guarantees flat landings and no interference between the coins. A computer sensor would count the resultant number of heads and calculate the sum (which is counting the “heads”). The sum would indicate a line in a payout table to return to the player. A virtual coin-flip can also be done with any software random number generator (RNG) and a threshold. We conceptualize the flipping coin as a SPINNER as a pie-chart in FIG. 1 where a spinning arrow would land according to the distribution.
The present system may be implemented by various combinations of processors, RAM, EPROM, video displays, interconnected through I/O ports and USB ports. A central server or controller communicates the generated or selected game outcome to the initiated gaming device. The gaming device receives the generated or selected game outcome and provides the game outcome to the player. In an alternative embodiment, how the generated or selected game outcome is to be presented or displayed to the player, such as a reel symbol combination of a slot machine or a hand of cards dealt in a card game, may also be determined by the central server or controller and communicated to the initiated gaming device to be presented or displayed to the player. Central production or control can assist a gaming establishment or other entity in maintaining appropriate records, controlling gaming, reducing and preventing cheating or electronic or other errors, controlling, altering, reducing or eliminating win-loss volatility and the like.
There are hundreds of available computer languages that may be used to implement embodiments of the invention, among the more common being Ada; Algol; APL; awk; Basic; C; C++; Cobol; Delphi; Eiffel; Euphoria; Forth; Fortran; HTML; Icon; Java; Javascript; Lisp; Logo; Mathematica; MatLab; Miranda; Modula-2; Oberon; Pascal; Perl; PUI; Prolog; Python; Rexx; SAS; Scheme; sed; Simula; Smalltalk; Snobol; SQL; Visual Basic; Visual C++; and XML.
Any commercial processor may be used to implement the embodiments of the invention either as a single processor, serial or parallel set of processors in the system. Examples of commercial processors include, but are not limited to MercedTM, PentiumTM, Pentium IITM, XeonTM, CeleronTM, Pentium PrOTM, EfficeonTM, AthlonTM, AMD and the like. Display screens may be segment display screen, analogue display screens, digital display screens, CRTs, LED screens, Plasma screens, liquid crystal diode screens, and the like.
It will be understood that this implementation is merely illustrative. For example, the there could be more or less reels with scatter symbols. The reels selected for the example are purely illustrative. Embodiment of the present invention can be readily added to existing games with modifications as required.
The term reels should be understood in include games in which symbols are arranged in different geometric patterns, with specific groups of symbols which move in a coordinated way being considered as reels. It will be appreciated that the present invention is of broad application, and can be implemented in a variety of ways. Variations and additions are possible within the general scope of the present invention.
One further basis of appreciating the scope of the present technology is to consider flipping 10 coins. The probability that all coins would come up “heads” is just 1/1024. A machine can take a $1 bet, and pay $1000 just in the case of ALL HEADS. The RETURN of this slot machine is 1000/1024 or 97.66% but this is not a fun machine.
When n coins are flipped, the probability that the sum of “heads” will be k (for k from 0 to n) is given by




   
  
    (
    
      
        
          n
        
      
      
        
          k
        
      
    
    )
  




which can be calculated as





  
    
      n
      !
    
    
      
        k
        !
      
      ⁢
      
        
          (
          
            n
            -
            k
          
          )
        
        !
      
    
  
  .




The Binomial coefficients are popularly known as “Pascal's Triangle” in the table below, where each entry is the sum of the two elements above it and above it to the left. Each row show the binomial coefficients for 1 through n coins, the columns represent k for 0 through n heads. Each row sums to 2n accounting for all possible events.














heads




















coins
0
1
2
3
4
5
6
7
8
9
10
Total Even






















1
1
1









2


2
1
2
1








4


3
1
3
3
1







8


4
1
4
6
4
1






16


5
1
5
10
10
5
1





32


6
1
6
15
20
15
6
1




64


7
1
7
21
35
35
21
7
1



128


8
1
8
28
56
70
56
28
8
1


256


9
1
9
36
84
126
126
84
36
9
1

512


10
1
10
45
120
210
252
210
120
45
10
1
1024









When each element in a row is divided by the sum, we get in each column the probability of the sum of coins adding up to k. The total probability distribution sums to 100%. This is shown in the table below.
























0
1
2
3
4
5
6
7
8
9
10
total



























1
0.5
0.5
0
0
0
0
0
0
0
0
0
100%


2
0.25
0.5
0.25
0
0
0
0
0
0
0
0
100%


3
0.125
0.375
0.375
0.125
0
0
0
0
0
0
0
100%


4
0.0625
0.25
0.375
0.25
0.0625
0
0
0
0
0
0
100%


5
0.0313
0.1563
0.3125
0.3125
0.1563
0.0313
0
0
0
0
0
100%


6
0.0156
0.0938
0.2344
0.3125
0.2344
0.0938
0.0156
0
0
0
0
100%


7
0.0078
0.0547
0.1641
0.2734
0.2734
0.1641
0.0547
0.0078
0
0
0
100%


8
0.0039
0.0313
0.1094
0.2188
0.2734
0.2188
0.1094
0.0313
0.0039
0
0
100%


9
0.002
0.0176
0.0703
0.1641
0.2461
0.2461
0.1641
0.0703
0.0176
0.002
0
100%


10
0.001
0.0098
0.0439
0.1172
0.2051
0.2461
0.2051
0.1172
0.0439
0.0098
0.00098
100%









In order to make flipping 10 coins “fun” we establish a minimum pay event which is more likely than “all heads” and derive a set of payoffs. The table below shows for the 10-coin problem, the number of heads, the binomial coefficients, which sum to 1024, the probability distribution, and 100% “fair” return calculations for minimums from 10 (all heads) to 5 coins (half must be heads). With a “Pay on 5 heads or more” policy, instead of winning 1000× once every 1024 plays, the player gets “positive feedback” 62% of the time with a maximum 200× jackpot. Pay on 6 heads gets reinforcement 38% of the time. For this invention, the choice of reinforcement level is discrete, linked mathematically to the number of paylines chosen.



















HEADS
10
distribution
10
9
8
7
6
5























0
1
0.000976563
0
0
0
0
0
0


1
10
0.009765625
0
0
0
0
0
0


2
45
0.043945313
0
0
0
0
0
0


3
120
0.1171875
0
0
0
0
0
0


4
210
0.205078125
0
0
0
0
0
0


5
252
0.24609375
0
0
0
0
0
0.67725


6
210
0.205078125
0
0
0
0
0.97524
0.8127


7
120
0.1171875
0
0
0
2.13333
1.70667
1.42222


8
45
0.043945313
0
0
 7.58519
5.68889
4.55111
3.79259


9
10
0.009765625
0
51.2
34.1333
25.6
20.48
17.0667


10 
1
0.000976563
1024
512
341.333
256
204.8
170.667


total
1024
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%




Feedback
0.10%
1.07%
5.47%
17.19%
37.70%
62.30%









Even though modern slot machines are electronic and can calculate fractions of coins, integer returns are still expected. On a single coin bet, a “0.67” return would mean 2⁄3rds of a coin. In order to get these payoffs integer multiples of the bet, the machine can work with multiple coin bets. The table below shows a simple “rounding” of the “minimum 5 heads” payoff for multiple coin bets of 1, 2, 3, 5, 10, and 100 coins.












COINS
















HEADS
10
distribution
5
1
2
3
5
10
100



















0
1
0.000976563
0








1
10
0.009765625
0








2
45
0.043945313
0








3
120
0.1171875
0








4
210
0.205078125
0








5
252
0.24609375
0.67725
1
1
2
3
7
68


6
210
0.205078125
0.8127
1
2
2
4
8
81


7
120
0.1171875
1.42222
1
3
4
7
14
142


8
45
0.043945313
3.79259
4
8
11
19
38
379


9
10
0.009765625
17.0667
17
34
51
85
171
1707


10 
1
0.000976563
170.667
171
341
512
853
1707
17067


total
1024
100.00%
100.00%
107.71%
101.22%
95.08%
97.54%
100.11%
99.98%









Thus, with multiple coins which allow fractional payoffs to be given as integers, it is now clear that this arrangement of binomial payoffs, with minor adjustments, can return 93%-100% to the player, providing a normal house profit, and that the player wins something 62% of the time, and there is a “jackpot”, in this case of 175-250 times the bet. The table below shows several manually adjusted integer pay-tables for 10 coins in and flipping 10 coins with a “5 heads minimum”



















HEADS
10
distribution
5
10
10
10
10
10























0
1
0.000976563
0







1
10
0.009765625
0


2
45
0.043945313
0


3
120
0.1171875
0


4
210
0.205078125
0


5
252
0.24609375
0.67725
6.77249
6
5
6
7


6
210
0.205078125
0.8127
8.12698
8
10
8
8


7
120
0.1171875
1.42222
14.2222
15
15
14
14


8
45
0.043945313
3.79259
37.9259
35
25
35
35


9
10
0.009765625
17.0667
170.667
150
100
175
150


10 
1
0.000976563
170.667
1706.67
1500
2500
1750
2000


total
1024
100.00%
100.00%
100.00%
93.43%
95.56%
97.14%
99.60%









A 200× payoff is not enough to provoke dreams of instant retirement. In order to achieve a big enough jackpot, 15 or more coins must be flipped. It is an object of this invention that the player can choose how many SPINNERS to bet on, and thus which paytable they want. One simple way is to set the number of spinners by the number of coins bet, e.g. bet 7 coins on 7 spinners, 10 coins on 10 spinners. Alternatively, a multiple of spinners may be triggered by each coin, e.g. 3 spinners for each coin so 5 coins trigger 15 binary spinners. Each change in the number of spinners brings up a different pay table, and the house may adjust these paytables with a slightly increasing return to encourage the player to make larger bets.

Below are sequences of paytables for 2 through 16 fair coin-flips. The first column indicates the number of Heads to show; the second column is the binomial coefficient; the third column is the probability of that many heads showing. The 4th column only shows the paying lines for a single bet, indicated by the number at the top of the column. The 5th column multiplies the paying lines by the number of coins bet to get 100% return, while the 6th column rounds the pay lines to integers.















HEADS/COINS
2 pay more than:
1 multibet
















0
1
0.25
0
0



1
2
0.5
1
2
1


2
1
0.25
2
4
5


total
4
100.00%
100.00%
100.00%
87.50%













HEADS/COINS
3 pay more than:
2 multibet
















0
1
0.125
0
0



1
3
0.375
0
0


2
3
0.375
1.333333333
4
3


3
1
0.125
4
12
13


total
8
100.00%
100.00%
100.00%
91.67%













HEADS/COINS
4 pay more than:
2 multibet
















0
1
0.0625
0
0



1
4
0.25
0
0


2
6
0.375
0.888888889
3.5555556
3


3
4
0.25
1.333333333
5.3333333
5


4
1
0.0625
5.333333333
21.333333
21


total
16
100.00%
100.00%
100.00%
92.19%













HEADS/COINS
5 pay more than:
3 multibet
















0
1
0.03125
0
0



1
5
0.15625
0
0


2
10
0.3125
0
0


3
10
0.3125
1.066666667
5.3333333
5


4
5
0.15625
2.133333333
10.666667
10


5
1
0.03125
10.66666667
53.333333
50


total
32
100.00%
100.00%
100.00%
93.75%













HEADS/COINS
6 pay more than:
3 multibet
















0
1
0.015625
0
0



1
6
0.09375
0
0


2
15
0.234375
0
0


3
20
0.3125
0.8
4.8
4


4
15
0.234375
1.066666667
6.4
6


5
6
0.09375
2.666666667
16
16


6
1
0.015625
16
96
100


total
64
100.00%
100.00%
100.00%
95.31%













HEADS/COINS
7 pay more than:
4 multibet
















0
1
0.0078125
0
0



1
7
0.0546875
0
0


2
21
0.1640625
0
0


3
35
0.2734375
0
0


4
35
0.2734375
0.914285714
6.4
6


5
21
0.1640625
1.523809524
10.666667
10


6
7
0.0546875
4.571428571
32
30


7
1
0.0078125
32
224
225


total
128
100.00%
100.00%
100.00%
95.42%













HEADS/COINS
8 pay more than:
4 multibet
















0
1
0.00390625
0
0



1
8
0.03125
0
0


2
28
0.109375
0
0


3
56
0.21875
0
0


4
70
0.2734375
0.731428571
5.8514286
5


5
56
0.21875
0.914285714
7.3142857
7


6
28
0.109375
1.828571429
14.628571
15


7
8
0.03125
6.4
51.2
50


8
1
0.00390625
51.2
409.6
400


total
256
100.00%
100.00%
100.00%
95.80%













HEADS/COINS
9 pay more than:
5 multibet
















0
1
0.001953125
0
0



1
9
0.017578125
0
0


2
36
0.0703125
0
0


3
84
0.1640625
0
0


4
126
0.24609375
0
0


5
126
0.24609375
0.812698413
7.3142857
7


6
84
0.1640625
1.219047619
10.971429
10


7
36
0.0703125
2.844444444
25.6
25


8
9
0.017578125
11.37777778
102.4
100


9
1
0.001953125
102.4
921.6
900


total
512
100.00%
100.00%
100.00%
95.96%













HEADS/COINS
10 pay more than:
5 multibet
















0
1
0.000976563
0
0



1
10
0.009765625
0
0


2
45
0.043945313
0
0


3
120
0.1171875
0
0


4
210
0.205078125
0
0


5
252
0.24609375
0.677248677
6.7724868
6


6
210
0.205078125
0.812698413
8.1269841
8


7
120
0.1171875
1.422222222
14.222222
14


8
45
0.043945313
3.792592593
37.925926
40


9
10
0.009765625
17.06666667
170.66667
150


10 
1
0.000976563
170.6666667
1706.6667
1750


total
1024
100.00%
100.00%
100.00%
96.89%













HEADS/COINS
11 pay more than:
6 multibet
















0
1
0.000488281
0
0



1
11
0.005371094
0
0


2
55
0.026855469
0
0


3
165
0.080566406
0
0


4
330
0.161132813
0
0


5
462
0.225585938
0
0


6
462
0.225585938
0.738816739
8.1269841
8


7
330
0.161132813
1.034343434
11.377778
11


8
165
0.080566406
2.068686869
22.755556
22


9
55
0.026855469
6.206060606
68.266667
68


10 
11
0.005371094
31.03030303
341.33333
320


11 
1
0.000488281
341.3333333
3754.6667
3700


total
2048
100.00%
100.00%
100.00%
97.28%













HEADS/COINS
12 pay more than:
6 multibet
















0
1
0.000244141
0
0



1
12
0.002929688
0
0


2
66
0.016113281
0
0


3
220
0.053710938
0
0


4
495
0.120849609
0
0


5
792
0.193359375
0
0


6
924
0.225585938
0.63327149
7.5992579
7


7
792
0.193359375
0.738816739
8.8658009
8


8
495
0.120849609
1.182106782
14.185281
14


9
220
0.053710938
2.65974026
31.916883
31


10 
66
0.016113281
8.865800866
106.38961
106


11 
12
0.002929688
48.76190476
585.14286
600


12 
1
0.000244141
585.1428571
7021.7143
7150


total
4096
100.00%
100.00%
100.00%
97.45%













HEADS/COINS
13 pay more than:
7 multibet
















0
1
0.00012207
0
0



1
13
0.001586914
0
0


2
78
0.009521484
0
0


3
286
0.034912109
0
0


4
715
0.087280273
0
0


5
1287
0.157104492
0
0


6
1716
0.209472656
0
0


7
1716
0.209472656
0.681984682
8.8658009
8


8
1287
0.157104492
0.909312909
11.821068
11


9
715
0.087280273
1.636763237
21.277922
22


10 
286
0.034912109
4.091908092
53.194805
50


11 
78
0.009521484
15.003663
195.04762
200


12 
13
0.001586914
90.02197802
1170.2857
1200


13 
1
0.00012207
1170.285714
15213.714
15000


total
8192
100.00%
100.00%
100.00%
97.76%













HEADS/COINS
14 pay more than:
7 multibet
















0
1
6.10352E−05
0
0



1
14
0.000854492
0
0


2
91
0.005554199
0
0


3
364
0.022216797
0
0


4
1001
0.061096191
0
0


5
2002
0.122192383
0
0


6
3003
0.183288574
0
0


7
3432
0.209472656
0.596736597
8.3543124
8


8
3003
0.183288574
0.681984682
9.5477855
9


9
2002
0.122192383
1.022977023
14.321678
14


10 
1001
0.061096191
2.045954046
28.643357
28


11 
364
0.022216797
5.626373626
78.769231
80


12 
91
0.005554199
22.50549451
315.07692
300


13 
14
0.000854492
146.2857143
2048
2000


14 
1
6.10352E−05
2048
28672
30000


total
16384
100.00%
100.00%
100.00%
98.07%













HEADS/COINS
15 pay more than:
8 multibet
















0
1
3.05176E−05
0
0



1
15
0.000457764
0
0


2
105
0.003204346
0
0


3
455
0.013885498
0
0


4
1365
0.041656494
0
0


5
3003
0.091644287
0
0


6
5005
0.152740479
0
0


7
6435
0.196380615
0
0


8
6435
0.196380615
0.636519037
9.5477855
9


9
5005
0.152740479
0.818381618
12.275724
12


10 
3003
0.091644287
1.363969364
20.45954
20


11 
1365
0.041656494
3.000732601
45.010989
45


12 
455
0.013885498
9.002197802
135.03297
135


13 
105
0.003204346
39.00952381
585.14286
600


14 
15
0.000457764
273.0666667
4096
4000


15 
1
3.05176E−05
4096
61440
60000


total
32768
100.00%
100.00%
100.00%
98.45%













HEADS/COINS
16 pay more than:
8 multibet
rounded















0
1
1.52588E−05
0
0



1
16
0.000244141
0
0


2
120
0.001831055
0
0


3
560
0.008544922
0
0


4
1820
0.027770996
0
0


5
4368
0.066650391
0
0


6
8008
0.122192383
0
0


7
11440
0.174560547
0
0


8
12870
0.196380615
0.565794699
9.0527152
9


9
11440
0.174560547
0.636519037
10.184305
10


10
8008
0.122192383
0.909312909
14.549007
14


11
4368
0.066650391
1.667073667
26.673179
25


12
1820
0.027770996
4.000976801
64.015629
65


13
560
0.008544922
13.0031746
208.05079
200


14
120
0.001831055
60.68148148
970.9037
1000


15
16
0.000244141
455.1111111
7281.7778
7000


16
1
1.52588E−05
7281.777778
116508.44
120000


total
65536
100.00%
100.00%
100.00%
98.59%









With 16 fair spinners, the jackpot can be 7000× the bet. The increased return to the player as they increase their bet size, shown in the graph of FIG. 2, may encourage them to make larger bets.

Greater Detail on NANNON® Slot Game.

Nannon® game is an invented game which is a simplification of backgammon. It is played in turns with dice rolls, and involves cyclical dynamics. In theory a game may last forever, but in practice games always end. The starting roll of a Nannon® game is that both players roll dice (or a die), and the player with the higher roll moves the calculated distance numerically indicated by the difference between the value on the dice (or between the separate die for each player). When the same computer strategy, whether random or expert, is used to play both sides of the game, the outcome is always 50-50, a fair coin. It is clear that using the NANNON® game instead of flipping coins provides a differently animated game with the same paytables as coin-flipping. Here we will demonstrate that several other interpretations of the game-ending of Nannon® game which provide a variety of payout structures.
Nannon® Game with UNFAIR COINS
When one player moves first with a regular die in a Nannon® game involving one checker each on a 6-point board, that player wins 68.11% of the time. This can be considered as a coin which lands on tails 68% of the time, or a spinner as shown in FIG. 3.
With this configuration, fewer coins can be flipped to achieve the goals of a “jackpot.” In the case of a slot machine based on “second mover in Nannon® game,” the following Pay tables can be determined.
















Probability
Payout
Multibet
Rounded




















one game






lose
0.6811


win
0.3189
3.135779
3
3


total

100.00%
95.67%
95.67%


two


0
0.463846


1
0.434433
1.150926
2.301851
3


2
0.101721
4.915393
9.830786
6




100.00%
100.00%
95.68%


three


0
0.315908


1
0.443814
0.751066
2.253197
2


2
0.207836
1.603832
4.811496
5


3
0.032443
10.27451
30.82353
30


total

100.00%
100.00%
96.67%


four


0
0.215153


1
0.40302


2
0.283098
1.177449
4.709796
5


3
0.088382
3.771502
15.08601
16


4
0.010347
32.21479
128.8591
100


total

100.00%
100.00%
96.61%


five


0
0.146533


1
0.343102


2
0.321346
0.777979
3.889894
4


3
0.150484
1.661303
8.306513
8


4
0.035235
7.095119
35.4756
30


5
0.0033
75.75489
378.7744
400


Total

100.00%
100.00%
97.33%


Six


0
0.099798


1
0.280409


2
0.328284


3
0.204978
1.219641
7.317844
7


4
0.071993
3.472575
20.83545
20


5
0.013486
18.53841
111.2305
110


6
0.001053
237.5225
1425.135
1425


Total

100.00%
100.00%
97.63%


Seven


0
0.067969


1
0.222805


2
0.313015


3
0.244305
0.818648
5.730533
6


4
0.114407
1.748147
12.23703
12


5
0.032146
6.22168
43.55176
40


6
0.005018
39.85749
279.0024
275


7
0.000336
595.7841
4170.489
4000


Total

100.00%
100.00%
97.82%


Eight


0
0.046291


1
0.173422


2
0.284244


3
0.266219


4
0.155836
1.283397
10.26718
10


5
0.058382
3.425722
27.40578
30


6
0.01367
14.63063
117.0451
100


7
0.001829
109.3483
874.7865
850


8
0.000107
1868.026
14944.21
15000


Total

100.00%
100.00%
97.97%


Nine


0
0.031527


1
0.132875


2
0.248898


3
0.271968


4
0.191042
0.87241
7.851688
8


5
0.089464
1.862951
16.76656
17


6
0.02793
5.967243
53.70519
50


7
0.005606
29.7325
267.5925
250


8
0.000656
253.9642
2285.677
2500


9
3.41E−05
4880.855
43927.7
40000


Total

100.00%
100.00%
98.37%


Ten


0
0.021472


1
0.100551


2
0.211894


3
0.26461


4
0.216852


5
0.121861
1.367681
13.67681
14


6
0.047556
3.504668
35.04668
35


7
0.012726
13.09682
130.9682
125


8
0.002235
74.57884
745.7884
750


9
0.000233
716.6532
7166.532
7000


10 
1.09E−05
15303.47
153034.7
150000


Total

100.00%
100.00%
98.99%









Thus using 10 Nannon® games with a “second player” model, a jackpot of 15,000 times the bet is obtained. The return to the player may be adjusted to encourage larger bets.

Tic Tac Toe and the Trinomial Distribution

Consider the simple game of tic tac toe. When both players choose moves randomly, tic tac toe is turned into a spinner with 3 segments, where player 1 wins, there is a draw, or player 2 wins. Of the 9! Or 362880 possible permutations of the 9 positions, of Player 1 wins 212256 or 59%, there is a draw 12% and Player 2 wins 29% of the time. In FIG. 4 we show that random tic-tac-toe can be viewed as a spinner and thus our invention allows the construction of a tic-tac-toe slot by letting the human play as player 2. Beyond a pure software model, a physical machine could play such a random game of tic-tac-toe by simply permuting 9 numbered marbles into a sequence like a bingo machine, and then considering them the alternating moves of the two players.
The trinomial distribution is much like the binomial one. A table can be constructed by adding up items instead of two. Here in column form are the trinomial coefficients:




























1









1
9








1
8
45







1
7
36
156






1
6
28
112
414





1
5
21
77
266
882




1
4
15
50
161
504
1554



1
3
10
30
90
266
784
2304


1
2
6
16
45
126
357
1016
2907


1
3
7
19
51
141
393
1107
3139


1
2
6
16
45
126
357
1016
2907



1
3
10
30
90
266
784
2304




1
4
15
50
161
504
1554





1
5
21
77
266
882






1
6
28
112
414







1
7
36
156








1
8
45









1
9










1









However, instead of simply dividing each by 3^n to get the probabilities of each sum, because the distribution is not fair, we need to sum the odds across all the possible polynomials in the expansion. The following code in a commercial language called Matlab calculates the multinomials for any game covered by this patent. Given a vector for a spinner (a discrete probability distribution which sums to 1, considered to be numbered events 0, 1, 2 . . . ) and the number of spinners desired, it calculates both the multinomial coefficient as well as the probability of attaining a sum of output events. The RADIX subroutine is used to convert numbers to different base arithmetic.















function z=spintest(probs,numdice)



n=length(probs);



z=zeros((n−1)*numdice+1,3); %col 1 count col 2 prob



for i=0:(n{circumflex over ( )}numdice)−1



 vec=radix(i,n,numdice);



 s=sum(vec)+1;



 z(s,2)=z(s,2)+1;



 z(s,3)=z(s,3)+prod(probs(vec+1));



end



for i=1:size(z,1)



 z(i,1)=i−1;



end



function z=radix(n,base,digits,v)



if nargin < 3 digits=floor(1+log(n)/log(base));end



if nargin < 4 v=base.{circumflex over ( )}(digits−(1:digits));end



z=zeros(1,digits);



indx=first(find(n>=v));



if indx*n



 powr=v(indx);



 digit=divide(n,powr);



 z(indx)=digit;



 z=z+radix(n−powr*digit,base,digits);



end










With this random Tic-Tac-Toe game as the SPINNER, the following tables provide an incrementally increasing player return with exponential possibilities. Using multiple boards per coin in, the player bets 1 through 4 coins to choose how many games (3, 6, 9, or 12) to start, and the machine automatically plays that many random games of tic-tac-toe in parallel. Software judges whether each outcome is a lose, draw, or win for player 2, and the pay table is consulted and the player is rewarded. In this game, we can establish a LOSE=0, Draw=1 and Win=2, and that to be paid, the games return a minimum sum of the number of coins in. With 12 games, a jackpot of 250,000 times the bet of 4 coins can be achieved.

















three


raw
multibet
adjusted





0
1
0.200120153911


0


1
3
0.130336056821


0


2
6
0.323995409363


0


3
7
0.130438357589
1.916614136
1.916614136
1


4
6
0.159579828492
1.566614041
1.566614041
3


5
3
0.031618615700
7.906734513
7.906734513
5


6
1
0.023911578123
10.45518613
10.45518613
8




0.345548379905
100.00%
100.00%
95.86%





six


raw
two coins
adjusted





0
1
0.040048076001


0


1
6
0.052165743502


0


2
21
0.146663510084


0


3
50
0.136663256562


0


4
90
0.202844947339


0


5
126
0.138775913796
0.900732675
1.801465349
1


6
141
0.138232897621
0.904270996
1.808541992
2


7
126
0.068352315750
1.828760279
3.657520557
4


8
90
0.049208765349
2.540197851
5.080395702
6


9
50
0.016329360497
7.654923169
15.30984634
15


10
21
0.008631347931
14.48209492
28.96418984
30


11
6
0.001512101999
82.66638103
165.3327621
150


12
1
0.000571763568
218.6218341
437.2436683
400




0.421614466511
100.00%
100.00%
97.17%





nine


games


raw
three coins
adjusted





0
1
0.008014427133


0


1
9
0.015659124928


0


2
45
0.049124794299


0


3
156
0.068589882200


0


4
414
0.119119095047


0


5
882
0.127209548065


0


6
1554
0.155309215692


0


7
2304
0.130810288980


0


8
2907
0.121842984401
0.75
2.238350235
2


9
3139
0.081685566320
1.11
3.338744958
3


10
2907
0.060012216198
1.51
4.544529264
5


11
2304
0.031733661105
2.86
8.59425806
8


12
1554
0.018557293564
4.90
14.69650042
15


13
882
0.007486455689
12.14
36.42942456
35


14
414
0.003452844906
26.33
78.98625051
75


15
156
0.000979252931
92.84
278.5054444
300


16
45
0.000345441655
263.17
789.5031452
750


17
9
0.000054235118
1676.20
5028.610331
5000


18
1
0.000013671769
6649.40
19948.20627
20000




0.326163623656
100.00%
100.00%
97.98%





Twelve


raw
4 coins
adjusted





0
1
0.001603848391


0


1
12
0.004178275321


0


2
78
0.014468447592


0


3
352
0.026247823067


0


4
1221
0.052015565701


0


5
3432
0.072365557487


0


6
8074
0.103727370280


0


7
16236
0.116046428660


0


8
28314
0.130697384725


0


9
43252
0.120574225461


0


10
58278
0.110850208779


0


11
69576
0.085358174965
0.84
3.35
3


12
73789
0.065241753619
1.09
4.38
4


13
69576
0.042042086177
1.70
6.80
7


14
58278
0.026891485266
2.66
10.62
10


15
43252
0.014406944805
4.96
19.83
20


16
28314
0.007691719641
9.29
37.15
35


17
16236
0.003363779081
21.23
84.94
85


18
8074
0.001480908380
48.23
192.93
200


19
3432
0.000508868857
140.37
561.47
500


20
1221
0.000180155043
396.48
1,585.94
1500


21
352
0.000044776024
1,595.24
6,380.97
7000


22
78
0.000012156633
5,875.69
23,502.75
25000


23
12
0.000001729130
41,308.97
165,235.89
150000


24
1
0.000000326914
218,493.74
873,974.97
1000000




0.247224864536
100.00%
100.00%
98.70%









These four paytables for random TicTacToe show an increasing player return as more coins are bet and are represented in the graph of FIG. 5.

Multinomial Games

Thus any game played with a random element which has a finite set of outcomes can be turned into a SPINNER, and this spinner can be turned into a slot machine using the method of this patent. We will demonstrate for fair six sided dice, 6-outcome Nannon® game, and then for Poker and Blackjack, for which despite a century of art, this invention leads to new family of slot machines.
The probability of a fair dice coming up each of 6 sides is 1⁄6th each. Portrayed as a spinner it is shown in FIG. 6.
When two dice are rolled, the multinomial coefficients which count up and down by 1 are familiar to players of craps and backgammon. The multinomial theorem, using a Pascal's triangle adding up 6 previous entries gives the multinomial coefficients, and dividing each by 6^n (for n dice) provides the probabilities of each total coming up. From these calculations, we establish a minimum total for payout of (max−min)/2, and we can calculate the raw 100% payback for those paylines. Again, assuming the player bets multiple coins we can round to integer paybacks. Here we can multiply the theoretical payback by the number of coins bet which is also the number of dice thrown, and adjust the paybacks to integer numbers. There is enough flexibility to manage the reinforcement as well as make the return to the player increase with increased bet.

















Events
Probability
Raw Pay
Multibet
Adjusted





















one







1
1
0.166666666667


2
1
0.166666666667


3
1
0.166666666667


4
1
0.166666666667
1
1
1


5
1
0.166666666667
2
2
2


6
1
0.166666666667
3
3
3





100.00%
100.00%
100.00%


two


2
1
0.027777777778


3
2
0.055555555556


4
3
0.083333333333


5
4
0.111111111111


6
5
0.138888888889


7
6
0.166666666667
1
2.00
2.00


8
5
0.138888888889
1.2
2.40
2.00


9
4
0.111111111111
1.5
3.00
3.00


10
3
0.083333333333
2
4.00
4.00


11
2
0.055555555556
3
6.00
6.00


12
1
0.027777777778
6
12.00
11.00



36

100.00%
100.00%
95.83%


three


3
1
0.004629629630


4
3
0.013888888889


5
6
0.027777777778


6
10
0.046296296296


7
15
0.069444444444


8
21
0.097222222222


9
25
0.115740740741


10
27
0.125000000000


11
27
0.125000000000
1
3.00
3.00


12
25
0.115740740741
1.08
3.24
3.00


13
21
0.097222222222
1.285714
3.86
4.00


14
15
0.069444444444
1.8
5.40
5.00


15
10
0.046296296296
2.7
8.10
8.00


16
6
0.027777777778
4.5
13.50
13.00


17
3
0.013888888889
9
27.00
25.00


18
1
0.004629629630
27
81.00
80.00



216

100.00%
100.00%
96.91%


four


4
1
0.000771604938


5
4
0.003086419753


6
10
0.007716049383


7
20
0.015432098765


8
35
0.027006172840


9
56
0.043209876543


10
80
0.061728395062


11
104
0.080246913580


12
125
0.096450617284


13
140
0.108024691358


14
146
0.112654320988
0.806974
3.23
3.00


15
140
0.108024691358
0.841558
3.37
3.00


16
125
0.096450617284
0.942545
3.77
4.00


17
104
0.080246913580
1.132867
4.53
5.00


18
80
0.061728395062
1.472727
5.89
6.00


19
56
0.043209876543
2.103896
8.42
8.00


20
35
0.027006172840
3.366234
13.46
12.00


21
20
0.015432098765
5.890909
23.56
21.00


22
10
0.007716049383
11.78182
47.13
50.00


23
4
0.003086419753
29.45455
117.82
100.00


24
1
0.000771604938
117.8182
471.27
500.00



1296

100.00%
100.00%
97.34%


five


5
1
0.000128600823


6
5
0.000643004115


7
15
0.001929012346


8
35
0.004501028807


9
70
0.009002057613


10
126
0.016203703704


11
205
0.026363168724


12
305
0.039223251029


13
420
0.054012345679


14
540
0.069444444444


15
651
0.083719135802


16
735
0.094521604938


17
780
0.100308641975


18
780
0.100308641975
0.766864
3.83
3


19
735
0.094521604938
0.813815
4.07
4


20
651
0.083719135802
0.918823
4.59
5


21
540
0.069444444444
1.107692
5.54
6


22
420
0.054012345679
1.424176
7.12
7


23
305
0.039223251029
1.96116
9.81
10


24
205
0.026363168724
2.917824
14.59
15


25
126
0.016203703704
4.747253
23.74
20


26
70
0.009002057613
8.545055
42.73
40


27
35
0.004501028807
17.09011
85.45
80


28
15
0.001929012346
39.87692
199.38
200


29
5
0.000643004115
119.6308
598.15
600


30
1
0.000128600823
598.1538
2,990.77
3000



7776

100.00%
100.00%
97.63%


six


6
1
0.000021433471


7
6
0.000128600823


8
21
0.000450102881


9
56
0.001200274348


10
126
0.002700617284


11
252
0.005401234568


12
456
0.009773662551


13
756
0.016203703704


14
1161
0.024884259259


15
1666
0.035708161866


16
2247
0.048161008230


17
2856
0.061213991770


18
3431
0.073538237311


19
3906
0.083719135802


20
4221
0.090470679012


21
4332
0.092849794239
0.67313
4.04
3


22
4221
0.090470679012
0.690832
4.14
4


23
3906
0.083719135802
0.746544
4.48
5


24
3431
0.073538237311
0.849898
5.10
6


25
2856
0.061213991770
1.021008
6.13
7


26
2247
0.048161008230
1.29773
7.79
8


27
1666
0.035708161866
1.7503
10.50
10


28
1161
0.024884259259
2.511628
15.07
15


29
756
0.016203703704
3.857143
23.14
20


30
456
0.009773662551
6.394737
38.37
35


31
252
0.005401234568
11.57143
69.43
70


32
126
0.002700617284
23.14286
138.86
130


33
56
0.001200274348
52.07143
312.43
300


34
21
0.000450102881
138.8571
833.14
800


35
6
0.000128600823
486
2,916.00
3000


36
1
0.000021433471
2916
17,496.00
15000



46656

100.00%
100.00%
97.79%


seven


7
1
0.000003572245


8
7
0.000025005716


9
28
0.000100022862


10
84
0.000300068587


11
210
0.000750171468


12
462
0.001650377229


13
917
0.003275748743


14
1667
0.005954932556


15
2807
0.010027291952


16
4417
0.015778606539


17
6538
0.023355338363


18
9142
0.032657464563


19
12117
0.043284893690


20
15267
0.054537465706


21
18327
0.065468535665


22
20993
0.074992141061


23
22967
0.082043752858


24
24017
0.085794610197


25
24017
0.085794610197
0.647541
4.53
3


26
22967
0.082043752858
0.677145
4.74
4


27
20993
0.074992141061
0.740818
5.19
5


28
18327
0.065468535665
0.848584
5.94
6


29
15267
0.054537465706
1.018668
7.13
7


30
12117
0.043284893690
1.283486
8.98
9


31
9142
0.032657464563
1.701159
11.91
12


32
6538
0.023355338363
2.378709
16.65
17


33
4417
0.015778606539
3.520942
24.65
25


34
2807
0.010027291952
5.540435
38.78
40


35
1667
0.005954932556
9.329334
65.31
70


36
917
0.003275748743
16.95965
118.72
120


37
462
0.001650377229
33.66234
235.64
250


38
210
0.000750171468
74.05714
518.40
500


39
84
0.000300068587
185.1429
1,296.00
1400


40
28
0.000100022862
555.4286
3,888.00
4000


41
7
0.000025005716
2221.714
15,552.00
15000


42
1
0.000003572245
15552
108,864.00
100000



279936

100.00%
100.00%
97.99%


eight


8
1
0.000000595374


9
8
0.000004762993


10
36
0.000021433471


11
120
0.000071444902


12
330
0.000196473480


13
792
0.000471536351


14
1708
0.001016899101


15
3368
0.002005220241


16
6147
0.003659765089


17
10480
0.006239521414


18
16808
0.010007049230


19
25488
0.015174897119


20
36688
0.021843087944


21
50288
0.029940176802


22
65808
0.039180384088


23
82384
0.049049306508


24
98813
0.058830708924


25
113688
0.067686899863


26
125588
0.074771852614


27
133288
0.079356233806


28
135954
0.080943501371
0.5883
4.71
3


29
133288
0.079356233806
0.600067
4.80
4


30
125588
0.074771852614
0.636858
5.09
5


31
113688
0.067686899863
0.703519
5.63
6


32
98813
0.058830708924
0.809425
6.48
7


33
82384
0.049049306508
0.97084
7.77
8


34
65808
0.039180384088
1.21538
9.72
10


35
50288
0.029940176802
1.590473
12.72
13


36
36688
0.021843087944
2.180051
17.44
17


37
25488
0.015174897119
3.138015
25.10
25


38
16808
0.010007049230
4.75855
38.07
35


39
10480
0.006239521414
7.631843
61.05
60


40
6147
0.003659765089
13.0115
104.09
100


41
3368
0.002005220241
23.74754
189.98
200


42
1708
0.001016899101
46.8277
374.62
400


43
792
0.000471536351
100.987
807.90
800


44
330
0.000196473480
242.3688
1,938.95
2,000


45
120
0.000071444902
666.5143
5,332.11
5,000


46
36
0.000021433471
2221.714
17,773.71
20,000


47
8
0.000004762993
9997.714
79,981.71
80,000


48
1
0.000000595374
79981.71
639,853.71
600,000



1679616

100.00%
100.00%
98.36%


nine


9
1
0.000000099229


10
9
0.000000893061


11
45
0.000004465306


12
165
0.000016372790


13
495
0.000049118370


14
1287
0.000127707762


15
2994
0.000297091716


16
6354
0.000630501257


17
12465
0.001236889861


18
22825
0.002264902613


19
39303
0.003899998571


20
63999
0.006350558699


21
98979
0.009821590173


22
145899
0.014477416267


23
205560
0.020397519433


24
277464
0.027532483615


25
359469
0.035669760231


26
447669
0.044421760688


27
536569
0.053243221466


28
619569
0.061479230967


29
689715
0.068439750514


30
740619
0.073490905064


31
767394
0.076147762346


32
767394
0.076147762346
0.570972
5.14
4


33
740619
0.073490905064
0.591614
5.32
5


34
689715
0.068439750514
0.635278
5.72
6


35
619569
0.061479230967
0.707202
6.36
7


36
536569
0.053243221466
0.816597
7.35
8


37
447669
0.044421760688
0.97876
8.81
9


38
359469
0.035669760231
1.218911
10.97
11


39
277464
0.027532483615
1.579162
14.21
14


40
205560
0.020397519433
2.131546
19.18
19


41
145899
0.014477416267
3.003178
27.03
27


42
98979
0.009821590173
4.426805
39.84
40


43
63999
0.006350558699
6.846368
61.62
60


44
39303
0.003899998571
11.14828
100.33
100


45
22825
0.002264902613
19.19653
172.77
170


46
12465
0.001236889861
35.15128
316.36
300


47
6354
0.000630501257
68.95825
620.62
600


48
2994
0.000297091716
146.3463
1,317.12
1,300


49
1287
0.000127707762
340.4512
3,064.06
3,000


50
495
0.000049118370
885.1731
7,966.56
8,000


51
165
0.000016372790
2655.519
23,899.67
24,000


52
45
0.000004465306
9736.904
87,632.14
85,000


53
9
0.000000893061
48684.52
438,160.70
400,000


54
1
0.000000099229
438160.7
3,943,446.26
4,000,000



10077696

100.00%
100.00%
98.69%









In the case of 9 plain dice, we can return a jackpot of over 400,000 times the players bet.

Consideration of a Nannon® Game with Measured Outcomes as a Spinner

The mini-backgammon game modeled before as both a fair coin and a biased coin, can also be used as a multinomial spinner. Consider that when one player wins, the opponent is left on one of the 6 positions of the board. We thus have a 7-way non-increasing probability distribution with a 50% “zero” outcome, which can be used as a SPINNER under this invention. Because Nannon® game is cyclic it is difficult to solve directly like tic tac toe, dice, or poker. Using Monte Carlo methods, we use a computer to play millions of games to arrive at the spinner probabilities. Advanced robotic automation could be used to roll the dice and move the pieces to make a physical random number generator, but this is most likely implemented in software or firmware.
Following earlier derivations, we establish a minimum sum of outcomes, which is one greater than the number of boards, and extract the raw 100% payback for those paylines, multiply it by the multiple bet, and then adjust the values to integers to get the following tables for up to 6 Nannon® games, achieving a nearly 2,000,000 times jackpot potential.
Using these tables, the house edge and player's return increases from 94% to 97% and the reinforcement varies between 35 and 50% of the time when the player receives any payback.

















count
prob
raw
Multibet
adjusted





















one







0

0.5000000000000000


1

0.1332420000000000


2

0.1097420000000000
1.82
1.82
1


3

0.0875080000000000
2.29
2.29
2


4

0.0681380000000000
2.94
2.94
3


5

0.0545040000000000
3.67
3.67
4


6

0.0468660000000000
4.27
4.27
5




  100%
  100%
  100%
94.15%


two


0
1
0.2500000000


1
2
0.1322000000


2
3
0.1286768400


3
4
0.1164012800
0.86
1.72
1


4
5
0.1030682400
0.97
1.94
2


5
6
0.0914486800
1.09
2.19
3


6
7
0.0846559600
1.18
2.36
4


7
6
0.0364457600
2.74
5.49
5


8
5
0.0246225700
4.06
8.12
7


9
4
0.0156384800
6.39
12.79
10


10
3
0.0093962200
10.64
21.29
18


11
2
0.0051706800
19.34
38.68
25


12
1
0.0022752900
43.95
87.90
70




100.00%
100.00%
100.00%
95.21%


Three


games


0
1
0.1250000000


1
3
0.0991500000


2
6
0.1096152600


3
10
0.1116623582


4
15
0.1096576338
0.61
1.82
1


5
21
0.1059886087
0.63
1.89
2


6
28
0.1038072572
0.64
1.93
2


7
33
0.0697110194
0.96
2.87
3


8
36
0.0539237469
1.24
3.71
4


9
37
0.0398218961
1.67
5.02
5


10
36
0.0282784100
2.36
7.07
7


11
33
0.0191265159
3.49
10.46
10


12
28
0.0119070138
5.60
16.80
15


13
21
0.0058249752
11.44
34.33
35


14
15
0.0033610877
19.83
59.50
60


15
10
0.0018032359
36.97
110.91
100


16
6
0.0008824877
75.54
226.63
200


17
3
0.0003699622
180.20
540.60
500


18
1
0.0001085313
614.26
1,842.79
2,000




100.00%
100.00%
100.00%
96.18%


four


0
1
0.0625000000


1
4
0.0661000000


2
10
0.0818152600


3
20
0.0922227965


4
35
0.0988683476


5
56
0.1029318804
0.49
1.94
2


6
84
0.1065812598
0.47
1.88
2


7
116
0.0881351203
0.57
2.27
2


8
149
0.0756466157
0.66
2.64
3


9
180
0.0622679312
0.80
3.21
3


10
206
0.0494678420
1.01
4.04
4


11
224
0.0378226705
1.32
5.29
5


12
231
0.0274716775
1.82
7.28
7


13
224
0.0180175101
2.78
11.10
11


14
206
0.0120836302
4.14
16.55
16


15
180
0.0077566606
6.45
25.78
25


16
149
0.0047517996
10.52
42.09
40


17
116
0.0027466262
18.20
72.82
70


18
84
0.0014694127
34.03
136.11
125


19
56
0.0007143654
69.99
279.97
300


20
35
0.0003620591
138.10
552.40
500


21
20
0.0001683338
297.03
1,188.12
1,000


22
10
0.0000694942
719.48
2,877.94
3,000


23
4
0.0000235296
2,124.98
8,499.93
8,000


24
1
0.0000051769
9,658.21
38,632.83
35,000




100.00%
100.00%
100.00%
97.03%


five


hands


0
1
0.0312500000


1
5
0.0413125000


2
15
0.0565960500


3
35
0.0697151956


4
70
0.0807058344


5
126
0.0897719084


6
210
0.0980185312
0.41
2.04
2


7
325
0.0920360144
0.43
2.17
2


8
470
0.0858761310
0.47
2.33
2


9
640
0.0769338233
0.52
2.60
3


10
826
0.0665558800
0.60
3.00
3


11
1,015
0.0556097361
0.72
3.60
4


12
1,190
0.0446362553
0.90
4.48
5


13
1,330
0.0336697248
1.19
5.94
6


14
1,420
0.0251013912
1.59
7.97
8


15
1,451
0.0180813014
2.21
11.06
11


16
1,420
0.0125819908
3.18
15.90
16


17
1,330
0.0084282092
4.75
23.73
20


18
1,190
0.0054060497
7.40
37.00
37


19
1,015
0.0033097644
12.09
60.43
60


20
826
0.0019963208
20.04
100.18
100


21
640
0.0011527943
34.70
173.49
150


22
470
0.0006344188
63.05
315.25
300


23
325
0.0003301886
121.14
605.71
450


24
210
0.0001613929
247.84
1,239.21
1,000


25
126
0.0000744419
537.33
2,686.66
2,000


26
70
0.0000337214
1,186.19
5,930.95
6,000


27
35
0.0000138395
2,890.29
14,451.43
14,000


28
15
0.0000049407
8,096.09
40,480.46
40,000


29
5
0.0000014030
28,511.31
142,556.55
150,000


30
1
0.0000002469
161,982.50
809,912.50
1,000,000




100.00%
100.00%
100.00%
97.42%


six


0
1
0.0156250000


1
6
0.0247875000


2
21
0.0372345375


3
56
0.0496522956


4
126
0.0615725593


5
252
0.0727220578


6
462
0.0834781378


7
786
0.0857379317
0.39
2.33
2


8
1,251
0.0857570235
0.39
2.33
2


9
1,876
0.0823589824
0.40
2.43
2


10
2,667
0.0763563108
0.44
2.62
3


11
3,612
0.0684553825
0.49
2.92
3


12
4,676
0.0592416413
0.56
3.38
4


13
5,796
0.0489629636
0.68
4.08
4


14
6,891
0.0395753696
0.84
5.05
5


15
7,872
0.0310283334
1.07
6.45
6


16
8,652
0.0236125218
1.41
8.47
9


17
9,156
0.0174232154
1.91
11.48
12


18
9,331
0.0124428374
2.68
16.07
15


19
9,156
0.0085920574
3.88
23.28
20


20
8,652
0.0057992619
5.75
34.49
35


21
7,872
0.0037936946
8.79
52.72
50


22
6,891
0.0024025086
13.87
83.25
80


23
5,796
0.0014699396
22.68
136.06
125


24
4,676
0.0008675966
38.42
230.52
200


25
3,612
0.0004945890
67.40
404.38
400


26
2,667
0.0002740313
121.64
729.84
700


27
1,876
0.0001454244
229.21
1,375.29
1,400


28
1,251
0.0000736105
452.83
2,717.00
2,500


29
786
0.0000353645
942.56
5,655.38
5,500


30
462
0.0000160785
2,073.16
12,438.97
12,000


31
252
0.0000069340
4,807.23
28,843.37
30,000


32
126
0.0000028426
11,726.32
70,357.89
75,000


33
56
0.0000010444
31,916.58
191,499.48
200,000


34
21
0.0000003284
101,493.85
608,963.10
800,000


35
6
0.0000000803
415,084.31
2,490,505.83
2,500,000


36
1
0.0000000118
2,829,882.94
16,979,297.64
10,000,000




1.0000000000
100.00%
100.00%
97.62%









Multinomial Random Play Othello® Game

Similar to the construction of a TicTacToe slot, Othello® game, or Reversi® game is a well loved board game. There have been attempts to convert it to a slot machine, e.g. (http://www.ledgaming.com/Othello/html/). Under our invention, the slots depend on how the games end up, so we ran 1,200,000 random games on a 4 by 4 board and collected the following statistics of outcomes, where a −16 means player 2 captured the entire board and +16 means Player 1 captured the entire board, and 0 means a tie. In Othello-4, player 2 has a great advantage winning 55% of the time, while player 1 wins only 35% of the time with 9% being a draw. In a mode exactly like our tic-tac-toe, we can use lose, draw, and win as outcomes 0 1 and 2, and derive a multinomial. However, there are many more paylines available.














−16
35685
0.029738


−15
3575
0.002979


−14
32601
0.027168


−13
1631
0.001359


−12
34907
0.029089


−11
8292
0.00691


−10
72193
0.060161


−9
16131
0.013443


−8
96486
0.080405


−7
3642
0.003035


−6
90111
0.075093


−5
5919
0.004933


−4
121885
0.101571


−3
9085
0.007571


−2
124529
0.103774


−1
5413
0.004511


0
116600
0.097167


1
8328
0.00694


2
104138
0.086782


3
9643
0.008036


4
68290
0.056908


5
11002
0.009168


6
39365
0.032804


7
5630
0.004692


8
46065
0.038388


9
11750
0.009792


10
38273
0.031894


11
6439
0.005366


12
16405
0.013671


13
10646
0.008872


14
26086
0.021738


15
8979
0.007483


16
10276
0.008563



1200000
100.00%









There are 33 different outcomes, or 3 different outcomes, so by recombining we get a reduced set of outcomes for this game. Odd number outcomes are much rarer than even numbered outcomes. Therefore we sort the Player-1 wins by decreasing likelihood and we find that it is more common to win by low even numbers, high even numbers then by odd numbers, and that winning by 1, 7, and 15, are the rarest forms of win for player 1. Using the decreased sorted table, with the “16” line moved up to the evens, provides a reduced outcomes tables as follows and a gives a pie chart with 35% reinforcement, as shown in FIG. 8.
















2
104138
8.68%




4
68290
5.69%


8
46065
3.84%


6
39365
3.28%
0.21488
2-4-6-8


10
38273
3.19%


14
26086
2.17%


12
16405
1.37%


16
10276
0.86%
0.07587
10-12-14-16


9
11750
0.98%


5
11002
0.92%


13
10646
0.89%


3
9643
0.80%


15
8979
0.75%
0.04335
3-5-9-13-15


1
8328
0.69%
0.00694
1


11
6439
0.54%
0.00537
11


7
5630
0.47%
0.00469
7









Using this spinner, we can derive a 4 coin multinomial slot machine for Othello-4 with random legal play to have a potential $100,000,000 jackpot if all four games are won by player 1 with a 7 point lead. As in other games shown, the player return can be slightly increased with increased bets as an incentive.

















prob
raw
multibet
adjusted




















one game






0
0.64890416667


1
0.21488166667
0.77562069
0.77562069
1


2
0.07586666667
2.196836555
2.196836555
2


3
0.04335000000
3.844675125
3.844675125
3


4
0.00694000000
24.01536984
24.01536984
20


5
0.00536583333
31.06072371
31.06072371
30


6
0.00469166667
35.52397869
35.52397869
35



100.00%
100.00%
100.00%
96.06%


two games


0
0.42107661795


1
0.27887522215


2
0.14463452870
0.628543486
1.257086973
1


3
0.08886470477
1.023005603
2.046011206
2


4
0.03339278224
2.722417385
5.44483477
5


5
0.01652401676
5.501633908
11.00326782
10


6
0.01132717733
8.025749776
16.05149955
15


7
0.00343218108
26.48726535
52.97453069
50


8
0.00122526382
74.19552378
148.3910476
150


9
0.00048124551
188.9037698
377.8075395
400


10 
0.00009391251
968.0189546
1936.037909
2000


11 
0.00005034941
1805.564272
3611.128545
3500


12 
0.00002201177
4130.022333
8260.044666
9000



100.00%
100.00%
100.00%
96.73%


Three game


0
0.27323837201


1
0.27144494059


2
0.18572480264


3
0.12815499213
0.487690717
1.463072151
1


4
0.06674856438
0.936349726
2.809049179
3


5
0.03510459579
1.780393666
5.341180999
5


6
0.02176235823
2.871931403
8.61579421
8


7
0.01006355448
6.210529305
18.63158791
20


8
0.00449541924
13.90304145
41.70912435
50


9
0.00203777078
30.67077058
92.01231174
100


10 
0.00073003638
85.61217131
256.8365139
250


11 
0.00030460198
205.1857992
615.5573975
600


12 
0.00013315272
469.3858167
1408.15745
1,500


13 
0.00003863794
1617.581197
4852.743592
4,500


14 
0.00001283517
4869.431369
14608.29411
15,000


15 
0.00000406540
15373.64592
46120.93775
45,000


16 
0.00000086353
72376.95462
217130.8639
200,000


17 
0.00000035433
176387.1484
529161.4452
500,000


18 
0.00000010327
605198.2228
1815594.668
1,500,000



100.00%
100.00%
100.00%
97.57%


Four games


—
0.17730551818


1
0.23485567075


2
0.19957582590


3
0.15550767133


4
0.09860531508
0.48
1.931703076
2


5
0.05824640270
0.82
3.270179473
3


6
0.03631193113
1.31
5.245553859
5


7
0.01992292681
2.39
9.560653024
10


8
0.01027464397
4.63
18.53847112
20


9
0.00519823601
9.16
36.64246679
40


10 
0.00234146911
20.34
81.34900846
100


11 
0.00105531760
45.12
180.4918167
200


12 
0.00048287887
98.61
394.4595708
400


13 
0.00019391927
245.56
982.2447588
1000


14 
0.00007702954
618.19
2472.768131
2500


15 
0.00002969134
1,603.80
6415.210986
6000


16 
0.00001006626
4,730.56
18922.24134
20000


17 
0.00000369203
12,897.78
51591.12765
50000


18 
0.00000130601
36,461.43
145845.7339
150000


19 
0.00000036487
130,509.88
522039.5014
500000


20 
0.00000011122
428,148.54
1712594.171
2000000


21 
0.00000003064
1,553,993.96
6215975.824
6000000


22 
0.00000000667
7,139,904.60
28559618.39
25000000


23 
0.00000000222
21,483,321.74
85933286.96
75000000


24 
0.00000000048
98,281,296.25
393125185
100000000



1.00000002800
100.00%
100.00%
98.03%









Multinomial Poker

A poker hand, drawn from a full 52 card deck has a stable distribution of hands which may be used as a SPINNER for this invention. Many varieties of card shuffling machines exist, and single card shufflers can be employed to each mix a deck of cards, and then deal out a hand, which can be read by a computer sensor using vision or a bar code scanner.
Here we show 3 new, simple, pure-luck poker machines, for 5-card, 3-card and 2-card varieties of poker. The history of poker machines leading to the modern “video poker” slots is interesting and never arrived upon our invention. Initially 10 cards were placed on 5 reels, using up 50 of the 52 cards. The reels were spun and a cam mechanism inside “read” the hand and paid out. Stud poker 5 reel machines were improved to allow “draw” poker by holding reels and re-spinning, and these evolved into the modern video poker machine, which involve a hold cycle. Multi-hand video poker games of up to 100 hands drawn from different remainder decks are more complex than our machine below, which require no “draw” or strategy.

First consider drawing 2 cards from a deck of 52. Out of 1326 hands, the following table is derived:


















nothing
792
0.597285



flush
264
0.199095



straight
144
0.108597



pair
78
0.058824



sflush
48
0.036199



total
1326










This may be viewed as a 5-way spinner with non-increasing probabilities for our invention as shown in FIG. 9.

Following earlier constructions, we calculate the multinomial probabilities for sums of multiple spinners, and arrive at a set of pay tables as below for the basis of a slot machine. The machine would deal out between 1 and 5 hands of “two-card poker” and then pay the player an exponentially increasing amount as the sum of the hands increases. As can be seen, betting 5 coins can trigger a payment of 5 million coins back, a 1 million times return on the bet.


















One Hand
Count
Probability
T-payoff

adjusted





0-nothing
792
0.597285

0


1-flush
264
0.199095
1.255682
1
1


2-straight
144
0.108597
2.302083
2
2


3-pair
78
0.058824
4.25
4
5


4-sflush
48
0.036199
6.90625
9
8


total
1326

100.00%
97.74%
100.00%














two hands
Probability
t-payoff
multibet
adjusted





0
0.356749


0


1
0.237833


0


2
0.169366
0.843482
1.686965
2


3
0.113511
1.258529
2.517058
3


4
0.078459
1.820795
3.64159
4


5
0.02719
5.25398
10.50796
9


6
0.011322
12.61715
25.23431
20


7
0.004259
33.54464
67.08929
55


8
0.00131
109.0201
218.0402
200




100.00%
100.00%
98.03%





three hands
Probability
t-payoff
multibet
adjusted





0
0.213081


0


1
0.213081


0


2
0.187253


0


3
0.148332
0.674165
2.022494
2


4
0.114759
0.871395
2.614184
3


5
0.06276
1.593369
4.780106
5


6
0.033505
2.984665
8.953994
9


7
0.016475
6.069832
18.2095
20


8
0.0073
13.69918
41.09755
40


9
0.002374
42.12896
126.3869
100


10
0.000803
124.5829
373.7487
400


11
0.000231
432.4464
1297.339
1000


12
4.74E−05
2108.176
6324.528
6000




100.00%
100.00%
98.40%





Four hands
Probability
t-payoff
multibet
adjusted





0
0.12727


0


1
0.169694


0


2
0.177407


0


3
0.161552


0


4
0.138658
0.554767
2.219067
2


5
0.09517
0.808269
3.233074
3


6
0.060473
1.272018
5.08807
5


7
0.035446
2.170125
8.6805
9


8
0.019125
4.022223
16.08889
18


9
0.008903
8.640189
34.56076
35


10
0.003927
19.5898
78.35921
75


11
0.001581
48.64017
194.5607
200


12
0.000565
136.0484
544.1935
500


13
0.000168
458.7024
1834.81
1450


14
4.78E−05
1608.933
6435.734
6000


15
1.12E−05
6892.114
27568.46
30000


16
1.72E−06
44798.74
179195
200000




100.00%
100.00%
98.55%





bet 5
Probability
t-payoff
multibet
adjusted





0
0.076017


0


1
0.126694


0


2
0.153569


0


3
0.157728


0


4
0.148838


0


5
0.118572
0.527104
2.635519
3


6
0.086051
0.726317
3.631583
4


7
0.057551
1.08599
5.429952
5


8
0.035665
1.752429
8.762147
9


9
0.019977
3.128614
15.64307
15


10
0.010469
5.970056
29.85028
30


11
0.005101
12.25176
61.25881
60


12
0.002295
27.23318
136.1659
125


13
0.000938
66.64982
333.2491
300


14
0.000359
174.3296
871.6482
900


15
0.000125
500.3853
2501.926
2500


16
3.88E−05
1612.005
8060.026
7000


17
1.04E−05
5988.451
29942.25
30000


18
2.57E−06
24284.3
121421.5
120000


19
5.05E−07
123756.5
618782.6
600000


20
6.22E−08
1005522
5027609
5000000




100.00%
100.00%
98.77%









These 5 games, which allow the player to choose how many decks to play on, can be arranged to encourage larger bets.

Three-Card Poker

Three-Card poker has 22,100 different hands, in which three-of-a-kind is a rarer hand than the straight or flush. Counting the hands results in the following table and the spinner shown in
















0 - nothing
16500



1 - pair
3744



2 - flush
1100



3 - straight
660



4 - three-kind
52



5 - str-flush
44



Total Hand
22100










We can build a slot machine which uses 3 hands of 3-card poker to generate a large jackpot as follows.




















0 - nothing
16500
74.66%





1 - pair
3744
16.94%
1.18
1



2 - flush
1100
4.98%
4.02
4



3 - straight
660
2.99%
6.70
7



4 - three-kind
52
0.24%
85.00
80



5 - str-flush
44
0.20%
100.45
100



Total Hand
22100
100.00%
100.00%
96.49%















two games
probability
t-payoff
multibet
adjusted





0
0.557421


1
0.252968
0.395307238
0.790614477
1


2
0.103023
0.970655632
1.941311264
2


3
0.061458
1.627122156
3.254244313
3


4
0.01611
6.207486584
12.41497317
12


5
0.006743
14.83007194
29.66014388
30


6
0.001801
55.53446058
111.0689212
100


7
0.000339
295.2188397
590.4376794
600


8
0.000124
803.5174757
1607.034951
1200


9
9.37E−06
10673.29747
21346.59494
18000


10 
3.96E−06
25227.79499
50455.58998
50000




100.00%
100.00%
96.92%














3 games
t-payoff
multibet
adjusted
bet 3





0
0.416174


1
0.283301
0.24
0.71
1


2
0.147518
0.45
1.36
2


3
0.092577
0.72
2.16
3


4
0.036433
1.83
5.49
6


5
0.015604
4.27
12.82
14


6
0.00587
11.36
34.07
40


7
0.001724
38.66
115.98
120


8
0.000602
110.82
332.46
333


9
0.000147
454.58
1,363.75
1200


10 
3.85E−05
1,730.74
5,192.22
5000


11 
9.24E−06
7,217.63
21,652.89
20000


12 
1.44E−06
46,157.54
138,472.62
120000


13 
3.88E−07
171,731.55
515,194.65
500000


14 
2.80E−08
2,382,625.26
7,147,875.78
1000000


15 
7.89E−09
8,447,489.89
25,342,469.67
5000000




100.00%
100.00%
97.45%









We note that the probabilities in this spinner are so small, that getting 3 3-card straight flushes invokes a $25 m payoff.

Using 5-Card Poker as a Spinner

We consider the natural probability of the hands in 5-card stud poker drawn from a full deck, which are well known. We can reduce from 11 outcomes to 9 by combining the Straight Flush and Royal Flush and ignoring the jacks-or-better pair distinction leading to a spinner as shown in figure.
















Royal flush
4



Straight Flush
36



4-kind
624



Full House
3744



Flush
5108



Straight
10200



3-Kind
54912



Two-pair
123552



Jack-Ace Pair
337920



2-10 Pair
760320



Busted
1302540



Total hands
2598960










Following our earlier derivations, we replicate the spinner, calculate the multinomial expansion, then choose the minimum sum (1 pair) to pay on, providing a number of theoretical paylines 1/probability/number-of-lines to get a raw 100% return with fractional values. Consider two hands of 5-card poker below, where at least one pair must be received. According to this, with a $2 Bet, one pair might return 30 c, while two straight flushes could pay 1⁄2 a Billion dollars!



















0
1
0.251178780291249000000000
t-payoff



1
2
0.423564088115623000000000
0.15



2
3
0.226215543009151000000000
0.28



3
4
0.061355235602413500000000
1.02



4
5
0.024050304807631700000000
2.60



5
6
0.007295749557770070000000
8.57



6
7
0.003924563268838790000000
15.93



7
8
0.001810857134431940000000
34.51



8
9
0.000453763098823709000000
137.74



9
8
0.000112136488882404000000
557.36



10
7
0.000026779348187937500000
2,333.89



11
6
0.000008197572130708410000
7,624.21



12
5
0.000003139836810861780000
19,905.49



13
4
0.000000752251384889702000
83,083.93



14
3
0.000000101989267403332000
612,809.58



15
2
0.000000007390526623429870
8,456,772.19



16
1
0.000000000236875853315060
263,851,292.25






100.00%










We can further reduce from 9 to 7 outcomes by combining the fullhouse, 4-of-a-kind, straight-flush and royal flush into a single top category (called royalty). Here are the derived multinomial pay tables for that condensed game with only 7 outcomes per spinner.
















raw
multibet
adjusted





















one hand







Busted
1
0.501177394034537000


Pair
1
0.422569027611044000
0.39
0.39
1


2Pair
1
0.047539015606242500
3.51
3.51
2


3Kind
1
0.021128451380552200
7.89
7.89
5


Straigh
1
0.003924646781789640
42.47
42.47
20


Flush
1
0.001965401545233480
84.80
84.80
50


Royalty
1
0.001696063040600860
98.27
98.27
100










(Full house, 4k, or strflush)
100.00%
100.00%
96.97%


two hands












 0
1
0.251178780291249000





 1
2
0.423564088115623000


 2
3
0.226215543009151000
0.40
0.80
1


 3
4
0.061355235602413500
1.48
2.96
2


 4
5
0.024050304807631700
3.78
7.56
8


 5
6
0.007295749557770070
12.46
24.92
25


 6
7
0.004180651696239700
21.75
43.49
40


 7
6
0.001786117346560010
50.90
101.80
100


 8
5
0.000259712969057858
350.04
700.07
700


 9
4
0.000087097384682223
1,043.76
2,087.53
2000


10
3
0.000017175699942019
5,292.89
10,585.78
10000


11
2
0.000006666889841621
13,635.91
27,271.81
25000


12
1
0.000002876629837692
31,602.64
63,205.28
65000





100.00%
100.00%
97.55%


three hands


 0
1
0.125885126543142000


 1
3
0.318421118832604000


 2
6
0.304299973137634000


 3
10
0.151784377571857000
0.41
1.24
1


 4
15
0.058669396800093000
1.07
3.20
3


 5
21
0.023671736870242800
2.64
7.92
8


 6
28
0.009764179508745600
6.40
19.20
20


 7
33
0.004920547408068560
12.70
38.11
40


 8
36
0.001836464543345060
34.03
102.10
100


 9
37
0.000506602947298249
123.37
370.11
400


10
36
0.000167034613875549
374.17
1,122.52
1250


11
33
0.000047827660247848
1,306.78
3,920.33
4000


12
28
0.000018536040017160
3,371.81
10,115.43
10000


13
21
0.000005777042353564
10,818.68
32,456.05
30000


14
15
0.000000956692666552
65,329.23
195,987.70
200000


15
10
0.000000268423723726
232,840.82
698,522.46
700000


16
6
0.000000053523941500
1,167,701.75
3,503,105.24
3000000


17
3
0.000000016961198184
3,684,881.18
11,054,643.54
10000000


18
1
0.000000004878945549
12,810,145.01
38,430,435.04
35000000





1.00
1.00
97.82%









Under this invention 3 hands of 5 card stud poker with the 3 rarest hands combined can be used to provide over a million times return on a 3 coin bet offering a $35 m jackpot on a $3 bet.

Multinomial Blackjack

While blackjack is the most popular table-game, it has not translated to video format very well. It is too slow, and the payoffs are not high enough. The value of the table game is often in the camaraderie of the table, not the mechanics of playing.
Our invention of a multinomial pure-luck way of converting outcomes into spinners, and spinners into slots provides a fun version of multi-hand blackjack. It is different from poker in that each hand can range from 2 cards to 5 cards. In order to remove the skill element, the player automatically stands on 17. To create the spinner, we ran a program over all exhaustive hands of 5 ORDERED cards under the hit/stand rule and collected the outcomes classified by total of cards, and whether all 5 were needed. Then, to smooth out the SPINNER, we first combined all regular hands from 17-20, and separated the 20's into “royal weddings” of 2 picture cards, a new category similar to but more frequent than blackjack. Finally, a rarest hand is using all 5 cards without getting busted. The derived spinner is shown in the following table and in FIG. 11
Using only 4 games under our multinomial construction, we can achieve a high payback of 2 million coins on a 4 coin bet, as shown by the tables below.

















Exact number of hands
Probility
t-payoff
adjusted





0 - busted
87826656
0.281608


1 - 17-20
166444800
0.533690
0.374749106
1


2 - twentyone
21988992
0.070506
2.836648447
1


3 - royal wedding
15523200
0.049774
4.018181818
2


4 - blackjack
15052800
0.048265
4.14375
3


5 - Five-Charlie
5038752
0.016156
12.37906529
7



311875200
100.00%
100.00%
96.16%
















T-payoff
Multibet
adjusted





Two hands


0
0.079303255


1
0.30058333


2
0.324535451
0.342369719
0.684739438
1


3
0.103289883
1.075721146
2.151442291
2


4
0.085282522
1.302859111
2.605718221
3


5
0.067635799
1.642785526
3.285571052
4


6
0.026528344
4.188392208
8.376784416
9


7
0.007082931
15.68716454
31.37432908
21


8
0.003937875
28.21600799
56.43201597
50


9
0.001559583
71.24409973
142.4881995
100


10 
0.000261026
425.6701598
851.3403195
500



1.0000000
100.00%
100.00%
96.42%


Three Hands


0
0.022332458


1
0.126970157


2
0.25740166


3
0.227428818
0.338229244
1.014687731
1


4
0.12081147
0.636719983
1.91015995
2


5
0.103786187
0.741168738
2.223506214
3


6
0.075241382
1.022350665
3.067051994
4


7
0.035394707
2.173293199
6.519879598
7


8
0.015910903
4.834614275
14.50384282
10


9
0.009002924
8.544232545
25.63269764
25


10 
0.003909176
19.67756648
59.03269945
50


11 
0.001215731
63.27312169
189.8193651
150


12 
0.000400528
192.0543982
576.1631947
500


13 
0.000151888
506.4467439
1519.340232
1500


14 
3.78E−05
2035.235451
6105.706354
5000


15 
4.22E−06
18240.22627
54720.67881
50000



1.0000000
100.00%
100.00%
97.30%


Four hands


0
0.006289006


1
0.047674473


2
0.141823773


3
0.211482341


4
0.180944056
0.325092356
1.300369423
1


5
0.129038944
0.455858731
1.823434923
2


6
0.110891159
0.530461851
2.121847405
3


7
0.078589428
0.748491639
2.993966554
4


8
0.043346667
1.357048489
5.428193956
5


9
0.024228526
2.427862478
9.711449911
10


10 
0.01409753
4.172612507
16.69045003
15


11 
0.006779318
8.676909838
34.70763935
30


12 
0.002825139
20.82146181
83.28584723
80


13 
0.001228413
47.88580442
191.5432177
200


14 
0.000514588
114.3119354
457.2477415
500


15 
0.000173839
338.3790268
1353.516107
1200


16 
5.14E−05
1143.339032
4573.356127
5000


17 
1.60E−05
3680.944943
14723.77977
15000


18 
4.49E−06
13106.61883
52426.47533
50000


19 
8.14E−07
72248.39625
288993.585
200000


20 
6.81E−08
863341.2869
3453365.148
2000000



1.0000000
100.00%
100.00%
97.95%









The foregoing relates to a preferred set of embodiments for the invention of multinomial based slot machines using traditional game models like backgammon and tic tac toe, coin-flipping, dice rolling and variants of poker, blackjack, other card games, as well as random play of board games such as chess, checkers, Othello, and Go. These other embodiments are possible and within the spirit and scope of the invention the latter being defined by the appended claims.

## Claims

1. A gaming apparatus comprising a symbol display system for a wagering game, a processor controlling the symbol display system and software executed by the processor, wherein the software is executed by the processor to perform electronic functions of:
a) providing a method of value crediting and debiting that identifies value risked in the play of the wagering game and awards won in the play of the wagering game;
b) providing a game control component that determines rules of play of a first game and a wagering game played on the gaming apparatus;
c) providing activation of selection from virtual spinners consisting of first games that provide first game outcomes that have individual first game determinant outcomes or individual symbol determinant outcomes mathematically distributed within the virtual outcome determinant space of the virtual spinner;
d) providing a file of individual symbols, values and/or images available for display on the symbol display system or retention in memory by the processor, the specific display of individual symbols, values and/or images, sets of symbols or collective symbols being determined by predetermined weighted portions of the virtual outcome determinant space selected by the virtual spinners in the first game;
e) the software responding to user commands to initiate the first game by randomly accessing the predetermined weighted portions of the outcome determinant space to select individual symbols, values and/or images, sets of symbols or collective symbols for use in the wagering game;
f) determining whether the randomly accessed predetermined weighted portions of the outcome determinant space has provided a final set of individual symbols, values and/or images, sets of symbols or collective symbols that constitute a win according to the wagering game; and
g) resolving all value placed at risk in the play of the wagering game according to the determination in f).

2. The gaming apparatus of claim 1 wherein the first game comprises a game in which outcomes are determined by one or more displays of symbols selected from group consisting of playing cards, specialty cards, dice and spinners, with each possible outcome for the one or more displays having a probability assigned thereto in the game control component.

3. A method of playing a game on the gaming apparatus of claim 2 wherein portions or totals of player credits are returned to players at player direction by player input to the gaming apparatus either as coins, tokens or printed credit slip.

4. The gaming apparatus of claim 2 wherein the individual symbols, values and/or images consist of numerical values, and the numerical values provided by multiple rounds of play of the first game are summed to form a total value, and the processor is configured to execute code to determine if the total value is a winning outcome.

5. The gaming apparatus of claim 4 wherein first game outcomes are determined by the processor executing code to effect repeated virtual play of the first game producing repeated random selection of predetermined weighted portions of a virtual spinner selected from the group consisting of dice, playing cards and spinners, the processor configured so that each first game outcome makes a limited number of defined moves of the virtual tokens on a virtual backgammon board.

6. The gaming apparatus of claim 5 wherein the virtual game board comprises a truncated backgammon board of at least four positions on a single player side of a backgammon board.

7. The gaming apparatus of claim 6 wherein the processor executes code to perform a random selection of predetermined weighted portions of the outcome determinant space determine discrete outcomes in a board game or card game as the first game.

8. The gaming apparatus of claim 6 wherein the virtual game board has only Six available positions on the virtual game board for positioning of virtual tokens.

9. A method of playing a game on the gaming apparatus of claim 8 wherein portions or totals of player credits are returned to players at player direction by player input to the gaming apparatus either as coins, tokens or printed credit slip.

10. The gaming apparatus of claim 1 wherein the selective symbols are selected from the group consisting of symbols to be randomly displayed, markers to fill preexisting spaces in a game board, playing cards, dice and coins.

11. The gaming apparatus of claim 10 wherein the processor is configured to execute code such that each symbol or a set of symbols in the wagering game is determined by the software according to the random selection of the predetermined weighted portions of the outcome determinant space in the first game.

12. The gaming apparatus of claim 1 wherein outcomes from the virtual spinner are selected from the group consisting of a distinguished LOSE state, and a set of winning states each determined by a weighted probability of outcomes in the first game, wherein each weighted probability is used to calculate binomial or multinomial coefficients that are summed and which determine the payout levels.

13. The apparatus of claim 1 wherein the processor executes code such that predetermined weighted portions of the outcome determinant space is constructed based on real-life events in the first game having determinable probabilities, wherein an actual probability distribution of the real-life event is mathematically distributed as segments within a region that is the basis of selection by a random number generator, further wherein the random number generator randomly selects among the statistical regions provided by the real-life event for the first game and symbol outcomes or event outcomes are associated with each of these regions so that selection of any region determines a symbol outcome or an event outcome in the wagering game.

14. The method of claim 1 wherein the processor is configured to execute code such that predetermined weighted portions of the outcome determinant space are constructed based on real-life events of the first game having determinable probabilities, wherein an actual probability distribution of the real-life event is mathematically distributed as segments within a virtual region that is the basis of selection by a random number generator, further wherein the random number generator randomly selects among the statistical regions provided by the real-life event of the first game and symbol outcomes for the wagering game or event outcomes for the wagering game are associated with each of these regions so that selection of any region in the first game determines a symbol outcome or an event outcome for the wagering game.

15. A gaming apparatus comprising a symbol display system for a wagering game, a processor executing code to control images displayed on the symbol display system and software executed by the processor, wherein the software present in memory and executable by the processor enables the ability to perform electronic functions of:
a) providing a method of value crediting and debiting system that identifies value risked in the play of the wagering game and awards won in the play of the wagering game;
b) providing a game control component that determines rules of play of a first game and a wagering game played on the gaming apparatus;
c) providing activation of selection from virtual spinners that have individual first game determinant outcomes mathematically distributed within a virtual outcome determinant space of the virtual spinner;
d) providing a file of images available for display on the symbol display system, the specific display of individual symbols, sets of symbols or collective symbols useful in the wagering game being determined by the processor executing coder to play the first game, and the first game results, the processor containing code that when executed provides from the predetermined weighted portions of the outcome determinant space a symbol, number or first game outcome;
e) the processor executing code on the software in response to user commands to initiate a wagering game by first randomly accessing the predetermined weighted portions of the outcome determinant space in the first game and then executing code to select individual symbols, sets of symbols, values that are summed by the processor or collective symbols for use in the wagering game;
f) determining whether the randomly accessed predetermined weighted portions of the outcome determinant space has provided individual symbols, sets of symbols, sums of values or collective symbols that constitute a win according to the game; and
g) resolving all value placed at risk in the play of the game according to the determination in f),
wherein the virtual spinner used in the first game consists of a processor executed code that simulates a virtually played game of cards selected from the group consisting of blackjack, poker and baccarat and an ending of the game of cards determine a separate symbol or event outcome for use in the wagering game from a look-up table.

16. The gaming apparatus of claim 15 wherein the first game comprises a first game in which the first game outcomes are determined by one or more displays of symbols selected from group consisting of playing cards, specialty cards, dice and spinners, with each possible outcome for the one or more displays having a probability assigned thereto in the first game control component and the physical spinner provides different symbols than are used in the wagering game outcome.

17. The gaming apparatus of claim 16 wherein the file of images stored in memory and accessible by the processor for display in the wagering game include virtual dice and virtual token positions on a virtual game board, the virtual token positions representing at least three positions on a virtual backgammon board.