Abstract:
A dice shaker for randomly changing the positional state of dice including a sealable cylindrical body member constructed of a rigid and transparent material and defining an inner chamber for securely storing the dice therein, a sleeve defining a tube dimensioned for receiving the cylindrical body member therein and constructed of an non-transparent material, and a padding adhered to at least a portion of the inner surface of the cylindrical body member, wherein the pad is constructed of a resilient material having sound-reducing qualities.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a continuation of U.S. patent application Ser. No. 13/250,868 filed Sep. 30, 2011, which is a continuation of U.S. patent application Ser. No. 10/917,717, filed Aug. 13, 2004, now U.S. Pat. No. 8,029,356, and also claims the benefit of U.S. Provisional Patent Application No. 61/447,810, filed Mar. 1, 2011, the disclosures of which are incorporated herein by reference. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The invention generally relates to groups of gaming elements, such as pairs of dice, customized to exhibit a non-transitive relationship, and games therefor. 
       BACKGROUND 
       [0003]    As gaming continues to enjoy widespread acceptance, casinos are increasingly in need of new games of chance to retain and attract patrons. While electronic gaming devices (e.g., slot machines) attract the most attention, many players prefer the skill requirements and personal interaction of live gaming. Thus, live gaming continues to be an integral component to the success of any casino. 
       SUMMARY 
       [0004]    The invention generally relates to a plurality of gaming elements which are configured to generate random outcomes that exhibit non-transitive relationships among each other. These non-transitive relationships generally refer to the circular nature of the randomly generated outcomes among the gaming elements. 
         [0005]    In some embodiments, non-transitive gaming elements of the invention are used in the provision or play of games including a first side and a second side, and in which wagers are received as to which of the two sides will achieve a higher or lower score or rank. 
         [0006]    In some embodiments, a player selects a first gaming element and a dealer, representing the house or gaming establishment, selects a second gaming element, wherein the gaming elements exhibit a non-transitive relationship. In these embodiments, the non-transitive relationship of the gaming elements insures that no matter which gaming element the player selects, the dealer can always select another gaming element (from the remaining gaming elements) that will, on average, outscore or otherwise “beat” the player-selected gaming element. When the gaming elements are single dice cubes, “beat” simply means that one cube, on average, rolls a higher number than another cube. For ease in describing embodiments of the invention, it will be presumed that a higher number corresponds with a higher score, however, it should be readily apparent that the gaming element may be configured so that the lower number will “beat” the higher number, if it is decided that the lower number provides the better ultimate score. When the gaming elements are dice pairs, “beat” means that one pair rolls a higher combined pair total than another pair, on average. 
         [0007]    Accordingly, a first embodiment of the present invention utilizes a group of three pairs of differently colored (e.g., red, blue and amber) non-transitive dice and a second embodiment utilizes three non-transitive decks of cards. Non-transitive simply means that there exists a circular, rather than a linear, relationship among the group of objects. So, there must be at least three objects in the group in order to have a non-transitive relationship among them. In gaming, the non-transitive “relationship” is either “beats” or “loses to”. In other words, the group of three dice pairs is non-transitive if and only if each pair of dice in the group loses to one of the other pairs of dice in the group. That is, each pair of the non-transitive dice will be outscored, on average, by one and only one of the remaining two dice pairs, neglecting any ties. Accordingly, with a first embodiment of the present invention, a player first selects or designates which pair of non-transitive dice will be used for play against the bank/house. Then, the banker/dealer/house selects from the remaining two pairs of non-transitive dice. Since the dealer is educated regarding the non-transitive dice, he or she selects that particular pair of the remaining two non-transitive dice pairs, which he or she knows has the advantage over the player-selected non-transitive dice pair. 
         [0008]    To facilitate the first embodiment of the present invention, a Blackjack type table layout depicts multiple wagering areas. The wagering areas include a player or banker/dealer wager (these two wagers are mutually exclusive), a push wager, a triple-player-win wager, a double-tie wager and a triple-tie wager. In the first embodiment, a “game” comprises three rolls of two pairs of the dice; a player rolls one pair and another pair is rolled by the banker/dealer. As used herein the terms “dealer” and “banker” are used synonymously. The two wagers on the player or banker/dealer are dependent upon whether the player or the banker/dealer will obtain a higher score on at least two out of the three rolls and so are obviously mutually exclusive. The winning player or banker/dealer wagers both pay 1 to 1 or even money. A push occurs when neither the player nor banker/dealer wins two of the three rolls. The push wager pays 5 to 1. The player and banker/dealer wagers both result in no play on a push outcome. That is, the player retains the original wager but does not win anything on the player or banker/dealer wager. However, in order to maintain a house edge on the banker/dealer wager, a banker/dealer wager will lose one half of their bet if the game results in a push outcome and there is one player win and one banker win and the player win occurs before the banker win. Clearly, this rule could alternatively require that the banker win occur before the player win and the offering casino can decide which rule to use. Other wagers include a double-tie wager, which pays 30 to 1, a triple-tie wager, which pays 750 to 1, and a triple-player-win wager, which pays 10 to 1. To track and record game play, the table layout also depicts player, banker and tie indicators for each of the three roll outcomes. Based on the above noted features, the embodiments of the present invention provide a very fast-paced game since there are no player decisions once the two non-transitive gaming elements in play are selected and wagers have been placed. The game has a house edge on the even-money wagers, which is comparable to baccarat and attractive to players and acceptable to the house or casino. 
         [0009]    In order to make the game fast-paced, only one player seated at the table plays against the house during a game. All players seated at the table may place wagers on either “P” for player or “B” for the bank (as well as the other wagers discussed above). This is exactly the same betting style of Baccarat. Whether the player actually rolls the “player-dice” or only designates (by pointing to) the “player dice” container, which the dealer then subsequently rolls for the player, is not critical for the operation of the game. It is quite likely, however, that players will want to actively participate in the game by actually shaking the player-selected dice container. 
         [0010]    Because casinos are extremely concerned about cheating, a dice game designed for play on a Blackjack-style table offers unique challenges for the casino. Most likely, casinos will require the dice to be “rolled” or shaken in either totally closed containers (e.g. Chuck-A-Luck cages) or in partially enclosed containers (e.g. dice cups). Dice cups that allow the dice to roll out on the table surface are not seen as a preferred method of rolling because of the security compromises such player access to the dice present. And while enclosed dice shakers or rolling devices already exist they tend to be quite expensive. Because the new non-transitive game requires three dice shakers, and since minimizing game cost to offering casinos is of great priority, one embodiment of the present invention uses proprietary, sealed, transparent low-cost dice shakers. Not only is the new dice shaker lower in cost than existing devices, it also “rolls” the dice in a more random manner. 
         [0011]    Optionally, a non-transparent sleeve or cover, placed over the shaker during the shaking process, provides additional concealment of the dice “rolls” or “outcomes” within the shakers both during and after the shaking process. The purpose of the sleeve is to prevent last moment “adjustments” to the dice outcome (should the player observe a low numerical outcome) and thus should minimize disputes with the casino personnel. Nobody can possibly know the outcome of the “roll” until the sleeve is completely removed from the shaker to reveal the dice outcome. The sleeve is removed only after the dice and the dice shaker container are completely at rest upon the table surface The dice shakers prevent players and the banker/dealer from directly handling the dice. This method of “rolling” the dice also virtually eliminates any physical contamination of the dice with drinks, cigarette ashes, nicks from jewelry, or any other foreign object interaction. Furthermore, the risk or appearance of cheating is virtually eliminated. This method of rolling dice also increases the duty cycle of the dice, thereby reducing the cost to the casino for dice replacement as well as reducing the time casino personnel are required to spend to perform periodic dice inspections. 
         [0012]    Some embodiments of the invention are directed to a dice shaker for randomly changing the positional state of dice, comprising: a sealable cylindrical body member constructed of a rigid and transparent material and defining an inner chamber for securely storing the dice therein; a sleeve defining a tube dimensioned for receiving the cylindrical body member therein and constructed of an non-transparent material; and a padding adhered to at least a portion of the inner surface of the cylindrical body member, wherein the pad is constructed of a resilient material having sound-reducing qualities, and wherein the combined weight of the dice shaker and sleeve ranges from 10 to 14 ounces and the length of the cylindrical body member ranges from about 4 inches to about 6 inches. 
         [0013]    The dice shaker body may be comprised of a clear transparent cylinder with two end caps. In some embodiments, both end caps are permanently attached to the cylindrical body member, and one or both may be removable. In other embodiments the end caps are permanently attached to the cylindrical body member. In some embodiments, the inside surface of one or both end caps is substantially covered with the padding. 
         [0014]    Other features, embodiments and variations will become evident from the following detailed description, drawings and claims. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0015]      FIG. 1  shows a gaming table layout of a live embodiment of the present invention; 
           [0016]      FIG. 2  shows a chart detailing indicia of each of three pair of non-transitive dice; 
           [0017]      FIG. 2   a  shows the single-roll probabilities associated with the dice numbering scheme detailed in  FIG. 2 ; 
           [0018]      FIG. 2   b  shows a chart detailing the associated game probabilities for a three-roll dice game embodiment using the three dice pair of  FIG. 2 ; 
           [0019]      FIG. 2   c  shows a chart detailing a three non-transitive card deck embodiment and the associated probabilities for a three-deal card game; 
           [0020]      FIG. 3  shows a dice shaker for facilitating one or more of the embodiments of the present invention; 
           [0021]      FIG. 4  shows a flow chart detailing one method of play of a live embodiment of the present invention; 
           [0022]      FIG. 5  shows a gaming device of the type which may facilitate an electronic embodiment of the present invention; 
           [0023]      FIG. 6  shows six faces with pips thereon, splayed out, of each die of a first or red pair of dice constructed in accordance with one embodiment of the invention; 
           [0024]      FIG. 7  shows six faces with pips thereon, splayed out, of each die of a second or blue pair of dice constructed in accordance with one embodiment of the invention; 
           [0025]      FIG. 8  shows six faces with pips thereon, splayed out, of each die of a third or amber pair of dice constructed in accordance with one embodiment of the invention; 
           [0026]      FIG. 9  shows the detailed calculation of all of the probabilities for the red pair vs. the blue pair combination of non-transitive dice of  FIGS. 6-8 ; 
           [0027]      FIG. 10  shows the detailed calculation of all of the probabilities for the amber pair vs. the red pair combination of non-transitive dice of  FIGS. 6-8 ; 
           [0028]      FIG. 11  shows the detailed calculation of all of the probabilities for the blue pair vs. the amber pair combination of non-transitive dice of  FIGS. 6-8 ; 
           [0029]      FIG. 12  shows six faces with pips thereon, splayed out, of each die of a first or red pair of dice constructed in accordance with another embodiment of the invention; 
           [0030]      FIG. 13  shows six faces with pips thereon, splayed out, of each die of a second or blue pair of dice constructed in accordance with another embodiment of the invention; 
           [0031]      FIG. 14  shows six faces with pips thereon, splayed out, of each die of a third or amber pair of dice constructed in accordance with another embodiment of the invention; 
           [0032]      FIG. 15  shows the detailed calculation of all of the probabilities for the blue pair vs. the amber pair combination of non-transitive dice of  FIGS. 12-14 ; 
           [0033]      FIG. 16  shows the detailed calculation of all of the probabilities for the amber pair vs. the red pair combination of non-transitive dice of  FIGS. 12-14 ; 
           [0034]      FIG. 17  shows the detailed calculation of all of the probabilities for the red pair vs. the blue pair combination of non-transitive dice of  FIGS. 12-14 ; 
           [0035]      FIG. 18  shows six faces with pips thereon, splayed out, of each die of a first or red pair of dice constructed in accordance with yet another embodiment of the invention; 
           [0036]      FIG. 19  shows six faces with pips thereon, splayed out, of each die of a second or blue pair of dice constructed in accordance with yet another embodiment of the invention; 
           [0037]      FIG. 20  shows six faces with pips thereon, splayed out, of each die of a third or amber pair of dice constructed in accordance with yet another embodiment of the invention; 
           [0038]      FIG. 21  shows the detailed calculation of all of the probabilities for the red pair vs. the blue pair combination of non-transitive dice of  FIGS. 18-20 ; 
           [0039]      FIG. 22  shows the detailed calculation of all of the probabilities for the amber pair vs. the red pair combination of non-transitive dice of  FIGS. 18-20 ; 
           [0040]      FIG. 23  shows the detailed calculation of all of the probabilities for the blue pair vs. the amber pair combination of non-transitive dice of  FIGS. 18-20 ; 
           [0041]      FIG. 24  shows the detailed calculation of all of the probabilities for the red card deck vs. the blue card deck combination of non-transitive gaming elements; 
           [0042]      FIG. 25  shows the detailed calculation of all of the probabilities for the blue card deck vs. the amber card deck combination of non-transitive gaming elements; 
           [0043]      FIG. 26  shows the detailed calculation of all of the probabilities for the amber card deck vs. the red card deck combination of non-transitive gaming elements; 
           [0044]      FIG. 27  shows another embodiment of a dice shaker for facilitating one or more of the embodiments of the present invention; 
           [0045]      FIG. 28  provides tables and information regarding the randomness measures for a dice shaker constructed in accordance with the invention versus dice thrown in the game of Craps; 
           [0046]      FIG. 29  provides tables and information regarding a weight analysis of a dice shaker and cover sleeve therefor constructed in accordance with the invention; 
           [0047]      FIGS. 30-32  provides various views of a particular embodiment of a dice shaker constructed in accordance with the invention; and 
           [0048]      FIGS. 33-34  provide various view including a cross-sectional view of a cover sleeve constructed in accordance with the invention and in particular for use with the dice shaker shown in  FIGS. 30-32 . 
       
    
    
     DETAILED DESCRIPTION 
       [0049]    Reference is now made to the figures wherein like parts are referred to by like numerals throughout.  FIG. 1  shows a gaming table layout generally referred to by reference numeral  100 . The layout  100  accommodates six player positions  110 - 1  through  110 - 6  and a dealer position  115 . The layout  100  depicts six player (P) and banker/dealer (B) wager areas  120 - 1  through  120 - 6 , a push wager area  130 , a triple-player-win wager area  140 , a double tie wager area  150  and a triple tie wager area  160 . The wager areas may also include associated payouts. To track play of the game, a series of result icons  170 , namely a player (P) win  180 , banker (B) win  190  or tie (T)  200 , are depicted near a center of the layout  100 . As described below, the result icons  170  permit the banker/dealer to temporarily record the results of three successive rolls of the game dice. It is conceivable that the results may be tracked using other means including an electronic display device similar to those used with conventional Roulette and Baccarat games. 
         [0050]    A first embodiment of the present invention is facilitated by a group of six-sided non-transitive dice. In a first embodiment, the group comprises three pairs of non-transitive dice. The group comprises three uniquely colored pairs of dice (e.g., red, blue and amber). One example of the non-transitive numbering of the dice is illustrated in chart  300  of  FIG. 2 . Specifically, the dice numbering is such that, on average, the score outcomes of the red pair are beaten by the score outcomes of the blue pair, the score outcomes of the blue pair are beaten by the score outcomes of the amber pair and the score outcomes of the amber pair are beaten by the score outcomes of the red pair. Specific probabilities related to the non-transitive numbering of  FIG. 2  are shown in the chart  310  of  FIG. 2   a.  Those skilled in the art will understand that many other non-transitive numbering schemes are possible without departing from the spirit and scope of the present invention. For a three-roll game, the house advantages for all of the game wagers are shown chart  312  in  FIG. 2   b.    
         [0051]    A second embodiment of the new non-transitive casino game employs three decks of specially constructed card decks. Exactly as in the three-dice-pair embodiment, there are three non-transitive objects, in this case, card decks: A, B and C. Arbitrarily, each card deck is constructed out of 75 cards but the three decks each have very different card compositions. The three card decks are constructed to have the non-transitive property so that, on average, a single card dealt from Deck B will beat a single card dealt from Deck A, and a single card dealt from Deck C will, on average, beat a single card dealt from Deck B. Similarly, on average, a single card dealt from Deck A will beat a single card dealt from Deck C. The exact deck compositions and the single card probabilities for but one example are shown in a top portion  315  of  FIG. 2   c.  Like the dice embodiment, there are only eleven possible integer results that can appear as an outcome. With the dice, the lowest number on each cube is a 1 so in the pair total, the lowest number to occur is a 2. And the largest numerical outcome that a dice pair can total (using standard dice pips) is 12. So, there are only eleven distinct integer outcomes that can occur. 
         [0052]    For the non-transitive three-card-deck embodiment, integers one through eleven inclusive, are used. One way to implement or distinguish the ones and elevens is to simply assign all red aces the value one and all black aces the value eleven. In this way, all three of the non-transitive three-deck embodiment can be constructed out of multiple, standard single card decks. Note, however, that Deck A consumes 5 standard card decks because it requires 19 sixes. 
         [0053]    The lower portion  320  of  FIG. 2   c  shows the house advantage for all of the wagers associated with the three-card-deck and three-card-deal embodiment of the new non-transitive casino game. The game and method of play is exactly the same game (whoever wins 2 out of 3 outcomes wins) as in the three-dice-pair embodiment. Accordingly, the dealer deals the player a single card from the player-selected deck and the dealer deals the house a single card from the dealer-selected deck. Thus, three successive card match-ups between the player and the dealer comprise a single game. 
         [0054]    When comparing the three non-transitive card deck house advantage results with the three non-transitive dice pair results of  FIG. 2   b,  it is apparent that there is considerably less fluctuation in the house advantages among the three non-transitive card combinations (B, A), (C, B) and (A, C). Of course, this should not be surprising since there are 75 independent and distinct elements (cards) to “play with” in constructing each of the three card decks. In the three-dice-pair embodiment there are only 6 elements, one for each face on each cube, which gives rise to 36 combinations, which are not independent. 
         [0055]      FIG. 3  shows a dice shaker  350  for facilitating the live table game embodiment of the present invention. The shaker  350  comprises a transparent cylindrical housing  355  having an open end. An end cap  360  acts to seal the dice  365  within the housing  355 . Although the end cap  360  may be joined to the housing  355  in any number of ways, as shown in  FIG. 3 , the end cap  360  includes a threaded lip  370  for receipt by a threaded upper portion  375  of the housing  355 . One key feature of the new dice shaker  350  is the rubber-like inserts used to cover both the fixed bottom  380  of the housing  355  and an underside  385  of the end cap  360  in which the rubber-like disk is recessed into the cap  360 . The material used in the present embodiment of the shaker is EVA (ethylene vinyl acetate), a copolymer member of the polyolefin family derived from random copolymerization of vinyl acetate and ethylene resulting in a resin with similar properties to that of polyethylene but with greater flexibility and resistance to impact and elongation. Because of EVA&#39;s resiliency, the dice achieve very high velocities during the shaking process which virtually guarantees randomness in the outcomes when the dice come to rest on the shaker bottom. This soft material also prevents damage to the dice  365  and reduces the noise level associated with operation of the shaker  350 . 
         [0056]      FIG. 4  shows a flow chart  400  detailing a first method of play of a live embodiment of the present invention. At step  410 , a player is designated as a player dice roller. The designated player acts as the proxy roller for all player bettors at the table. The casino offering the game will determine the number of games which any one designated player may roll. The casino may alternatively require that the dealer roll for the designated player, allowing the designated player to only select the dice that will be used for the player. If the casino permits the player to actually roll the dice, the player may be permitted to roll for only one game or a series of games. At step  420 , players place wagers on either the player or the banker/dealer. Optionally, players may also place proposition wagers on the occurrence of a push, double tie, triple tie and/or three consecutive player wins. An allowable range of wager amounts is established by individual casinos. Steps  410  and  420  may be reversed without impacting the game. However, by designating the player before wagers are placed, the other players at the table are able to use past results of the designated player roller to determine whether to wager on the player or the banker/dealer. While the past results have no scientific relevance to the upcoming rolls, players tend to be superstitious and look for reasons to justify their wager. The various wagers have corresponding payouts as follows: TABLE-US-00001 Player or Banker/Dealer 1 to 1 Push 5 to 1 Triple-player-win 10 to 1 Double Tie 30 to 1 Triple Tie 750 to 1. 
         [0057]    Next, at step  430 , the designated player selects one pair of non-transitive dice from the three available pair of dice. It is noted that the player is able to select new dice after each game. In fact, depending on the casino offering the game, the player may be able to change dice during a game. In any event, ideally, the dice are contained in a transparent dice shaker as shown in  FIG. 3 . Then, at step  440 , the banker/dealer specifically selects the one pair of remaining dice which, on average, has the advantage over the player-selected dice. That is, since the banker/dealer knows which pair of remaining dice has the advantage over the player-selected pair of dice, he or she is able to select the same. For example, assuming the dice hierarchy noted above, if the player selects the blue pair of dice, the banker/dealer must select the amber pair of dice. Once the dice pairs are selected, at step  450 , the player and the banker/dealer utilize their respective shakers  350  to roll the contained dice  365 . Ideally, the player and the banker/dealer roll the dice simultaneously. However, there is no reason that the banker/dealer cannot roll before the player or vice versa. Subsequent to the first roll, at step  460 , the banker/dealer records the outcome of the first roll by placing a marker on one of the result icons  170 . Consequently, the banker/dealer must place the marker on the player (P) win icon  180 , banker (B) win icon  190  or tie (T) icon  200 . Once three rolls have been completed and recorded, the banker/dealer resolves the player wagers at step  470 . 
         [0058]    The player or banker/dealer wagers are based on the scoring outcomes of the three rolls. More particularly, a player wager wins if the player outscores the dealer on at least two of the three rolls and the banker/dealer wager wins if the dealer outscores the player on at least two of three rolls. Winning player or banker/dealer wagers pay even money (i.e., 1 to 1). A push wager wins when neither the player nor the banker/dealer outscores the other on two of the three rolls. Specifically, a push occurs when the player wins one roll, the banker/dealer wins one roll and the other roll is a tie or when the player and banker/dealer tie on two or three rolls. A winning push wager pays 5 to 1. Since the non-transitive dice provide the house with the edge, there must be a mechanism for ensuring the player-placed banker/dealer wager favors the house. Thus, in every case except three, a push results in no action (i.e., the player retains his or her original wager) for the player and banker/dealer wagers. To create the house edge on the banker/dealer wager, any push outcome consisting of one of the following three roll sequences: PBT, PTB, and TPB, results in the banker/dealer bettor losing one half of their bet on the banker/dealer wager. Those skilled in the art will recognize that another sequence (e.g., BPT, BTP, and TBP) can be substituted for the above three banker/wager sequences. 
         [0059]    Other proposition wagers include wagers on the player outscoring the banker/dealer each of the three consecutive rolls, two ties occurring during the three rolls and three ties occurring during the three rolls. The aforementioned wagers pay 10 to 1, 30 to 1 and 750 to 1, respectively. It is unusual to find a 750 to 1 payout on a live table game. Moreover, considering the number of games which can be played over the course of one day, the three ties outcome should occur about once per eight hour shift. Clearly, the wagers and corresponding payouts may be manipulated to the satisfaction of the casinos offering the game. 
         [0060]      FIG. 5  illustrates an electronic gaming device, generally designated as reference numeral  600 , of the type that may be used to implement embodiments of the present invention, such as all of the embodiments discussed herein, in an electronic format. The external features of the gaming device  600  include a display  610 , wager selection buttons  620 , dice selection buttons  625 , card reader  630 , credit display  640 , bill reader  650  and coin input  660 . However, the display  610  may also comprise touch screen technology to facilitate simple player interaction. Device switches and similar physical components may also act as player interfaces. 
         [0061]    The operation of the gaming device  600  is controlled by a microprocessor that communicates with an internal memory device and the external features of the device  600 . The microprocessor also incorporates, or communicates with, a random number generator which ensures the randomness of the rolled dice or specially configured cards during the play of the game. Since the technology for operating and controlling gaming devices is well known to those skilled in the art, the subtle details are not described herein. 
         [0062]    Accordingly, in an electronic embodiment of the present invention, a player places or inputs his or her wagers and selects his or her pair of dice. Although this embodiment is discussed with reference to pairs of dice, it should be readily apparent that other elements may be employed such as a die or card decks instead of dice in accordance with this invention. The device processor then selects, according to the non-transitive hierarchy, the proper pair of dice from the two remaining pair of dice and simulates the three dice rolls for both the player and the device. The processor records the results of each roll and resolves the player wagers. Two inherent benefits of the electronic embodiment over a live game are the speed at which the game can be played and the elimination of cheating associated with physical dice. 
         [0063]    Other embodiments of the game are clearly possible. For example, three differently-colored electronic modules or “pucks” each having an embedded random number generator and a series of light-emitting diodes (LEDs) or digital displays can replace the three pairs of dice or three decks of cards. In the same manner as the dice or cards, the random number generators are programmed in a non-transitive manner. The player selects his or her electronic unit, followed by the banker/dealer selecting his or her unit. The electronic pucks, or units, are then activated and display their non-transitive outcomes. The outcomes may be akin to dice outcomes such that the display shows conventional dice pips. Alternatively, the electronic units may allow non-integer outcomes (e.g. 4.5) to be displayed. The use of non-integer outcomes allows for very precise manipulation of the probabilities and corresponding payouts. 
         [0064]    Also, three differently colored decks of non-transitive cards can be constructed to replace the three dice pairs. Just as the electronic puck embodiment allows more fine-tuning of the non-transitive probabilities, so does this embodiment of the game but to a somewhat lesser extent since the cards must still have integer values. While this more precise “fine tuning” is advantageous, there are some disadvantages with the card decks embodiment. One is that the three decks would have to be composed carefully each shift and checked routinely to verify that no modifications in composition have occurred. (That is, that no cheating has taken place.) Another is that the decks of cards would have to be shuffled after every game. This latter requirement would probably necessitate the use of two automatic shuffling machines so that the game is not slowed down significantly. 
         [0065]    While the description above focuses on three rolls per game, the number of rolls may be more or less. Also, the numbers on the non-transitive dice may be modified along with the disclosed payouts. 
         [0066]      FIGS. 6-23  provides three embodiments of gaming elements constructed in accordance with the invention in the form of three sets of three pairs of dice, each set of which exhibits a non-transitive relationship among the three pairs. In these three embodiments, each die is substantially cube-shaped having six sides defining six faces with pips or other marks disposed thereon. The faces of each die are shown schematically laid open in the figures primarily for convenience sake to provide a view of all of the faces of each die in one drawing. The dice are referred to by color herein but it should be readily apparent that the dice may be the other colors or the same color, or may include features or other symbols which render the dice as being distinguishable from each other, or be indistinguishable from each other aside from the pips. 
         [0067]      FIGS. 6 ,  7  and  8  illustrate an embodiment of non-transitive gaming elements  710  which includes a first or red pair of dice,  712   a  and  712   b,  a second or blue pair,  714   a  and  714   b  and a third or amber pair,  716   a  and  716   b,  respectively. Each die includes six faces  718  with one or pips disposes thereon. As shown in  FIGS. 6-8 , the configuration of pips on faces  718  of each die of this embodiment are as follows: 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 
               
               
                   
               
               
                 Pips on Gaming Elements 710 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Red Die 712a 
                 2 
                 2 
                 2 
                 2 
                 5 
                 6 
               
               
                   
                 Red Die 712b 
                 2 
                 2 
                 3 
                 4 
                 5 
                 6 
               
               
                   
                 Blue Die 714a 
                 1 
                 1 
                 3 
                 3 
                 4 
                 4 
               
               
                   
                 Blue Die 714b 
                 1 
                 2 
                 5 
                 5 
                 5 
                 6 
               
               
                   
                 Amber Die 716a 
                 2 
                 2 
                 3 
                 4 
                 5 
                 6 
               
               
                   
                 Amber Die 716b 
                 1 
                 1 
                 1 
                 4 
                 6 
                 6 
               
               
                   
                   
               
             
          
         
       
     
         [0068]      FIGS. 9-11  provide tables  720 ,  722  and  724 , respectively, which illustrate the relative probabilities of gaming elements  710 . Table  720  illustrates all possible roll outcomes for red pair  712  compared with all possible roll outcomes for blue pair  714  to determine the edge or advantage blue pair  714  has over red pair  712 . The advantage blue pair  714  has over red pair is the total amount of red wins subtracted from the total amount of blue wins divided by the total of all outcomes, which is “(586−572)/1296” and about 1.0802%. The probability of a tie is the total of red and blue tie outcomes divided by the total of all outcomes, which is “138/1296” and about 10.6481%. Table  722  illustrates all possible roll outcomes for amber pair  716  compared with all possible roll outcomes for red pair  712  to determine the edge red pair  712  has over amber pair  716 , in the manner as described above, to be about 1.1574% and probability of a tie to be about 10.2623%. Table  724  illustrates all possible roll outcomes for amber pair  716  compared with all possible roll outcomes for blue pair  714  to determine the edge amber pair  716  has over blue pair  712  to be about 0.9259% and probability of a tie to be about 10.6481%. It is noted that the total amount of pips on each of the three pairs are not equal, in that red pair  712  has 41 pips, blue pair  714  has 40 pips and amber pair  716  has 41 pips. The mean advantage between all gaming elements  710  is therefore 1.055%, and the deviation from the mean of any pair  712 ,  714  or  716 , is well less than 0.5% in this embodiment. The mean of the probabilities of any two elements resulting in a tie score is 10.520%. The deviation from that mean for any pair is less than 1% in this embodiment. 
         [0069]      FIGS. 12 ,  13  and  14  illustrate another embodiment of non-transitive gaming elements  810  which include a first or red pair of dice,  812   a  and  812   b,  a second or blue pair,  814   a  and  814   b  and a third or amber pair,  816   a  and  816   b,  respectively. Each die includes six faces  818  with one or pips disposes thereon. As shown in  FIGS. 12-14 , the configuration of pips on faces  818  of each die of this embodiment are as follows: 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 
               
               
                   
               
               
                 Pips on Gaming Elements 810 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Red Die 812a 
                 2 
                 3 
                 3 
                 3 
                 3 
                 6 
               
               
                   
                 Red Die 812b 
                 1 
                 1 
                 3 
                 4 
                 6 
                 6 
               
               
                   
                 Blue Die 814a 
                 2 
                 3 
                 4 
                 4 
                 5 
                 5 
               
               
                   
                 Blue Die 814b 
                 1 
                 1 
                 2 
                 3 
                 5 
                 6 
               
               
                   
                 Amber Die 816a 
                 1 
                 1 
                 5 
                 6 
                 6 
                 6 
               
               
                   
                 Amber Die 816b 
                 1 
                 1 
                 1 
                 3 
                 4 
                 5 
               
               
                   
                   
               
             
          
         
       
     
         [0070]      FIGS. 15-17  provide tables  820 ,  822  and  824 , respectively, which illustrate the relative probabilities of gaming elements  810 . Table  820  illustrates all possible roll outcomes for red pair  812  compared with all possible roll outcomes for blue pair  814  to determine the edge or advantage blue pair  814  has over red pair  812  to be about 0.6173% and probability of a tie to be about 10.6481%. Table  822  illustrates all possible roll outcomes for amber pair  816  compared with all possible roll outcomes for red pair  812  to determine the edge red pair  812  has over amber pair  816  to be about 0.5401% and probability of a tie to be about 10.7253%. Table  824  illustrates all possible roll outcomes for amber pair  816  compared with all possible roll outcomes for blue pair  814  to determine the edge amber pair  816  has over blue pair  812  to be about 0.6173% and probability of a tie to be about 10.4938%. It is noted that the total amount of pips on each of the three pairs are not equal, in that red pair  812  has 41 pips, blue pair  814  has 41 pips and amber pair  816  has 40 pips. 
         [0071]    The mean advantage between all gaming elements  810  is therefore 0.5916%, and the deviation from the mean of any pair  812 ,  814  or  816 , is less than the prior embodiment. The mean of the probabilities of any two elements resulting in a tie score is 10.6224%. 
         [0072]      FIGS. 18 ,  19  and  20  illustrate yet another embodiment of non-transitive gaming elements  910  which include a first or red pair of dice,  912   a  and  912   b,  a second or blue pair,  914   a  and  914   b  and a third or amber pair,  916   a  and  916   b,  respectively. Each die includes six faces  918  with one or pips disposes thereon. As shown in  FIGS. 18-20 , the configuration of pips on faces  918  of each die of this embodiment are as follows: 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 
               
               
                   
               
               
                 Pips on Gaming Elements 910 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Red Die 912a 
                 1 
                 3 
                 3 
                 3 
                 5 
                 6 
               
               
                   
                 Red Die 912b 
                 2 
                 2 
                 3 
                 3 
                 5 
                 5 
               
               
                   
                 Blue Die 914a 
                 1 
                 1 
                 3 
                 3 
                 4 
                 4 
               
               
                   
                 Blue Die 914b 
                 1 
                 1 
                 5 
                 5 
                 6 
                 6 
               
               
                   
                 Amber Die 916a 
                 1 
                 1 
                 3 
                 3 
                 6 
                 6 
               
               
                   
                 Amber Die 916b 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
               
               
                   
                   
               
             
          
         
       
     
         [0073]      FIGS. 21-23  provide tables  920 ,  922  and  924 , respectively, which illustrate the relative probabilities of gaming elements  910 . Table  920  illustrates all possible roll outcomes for red pair  912  compared with all possible roll outcomes for blue pair  914  to determine the edge or advantage blue pair  914  has over red pair  912  to be “(584−576)/1296” which is about 0.6173% and probability of a tie to be “136/1296” which is about 10.4938%. Table  922  illustrates all possible roll outcomes for amber pair  916  compared with all possible roll outcomes for red pair  912  to determine the edge red pair  912  has over amber pair  916  to be about 0.6173% and probability of a tie to be about 10.4938%. Table  924  illustrates all possible roll outcomes for amber pair  916  compared with all possible roll outcomes for blue pair  914  to determine the edge amber pair  916  has over blue pair  912  to be about 0.6173% and probability of a tie to be about 10.4938%. It is noted that the total amount of pips on each of the three pairs are not equal, in that red pair  912  has 41 pips, blue pair  914  has 40 pips and amber pair  916  has 41 pips. It is further noted that in this embodiment, the advantages and tie probabilities among the three pairs of gaming elements in this embodiment are identical (that is, “(584−576)/1296” and “136/1296”). 
         [0074]      FIGS. 24-26  illustrate another embodiment of non-transitive gaming elements  1010  comprising three decks of non-transitive cards, a red deck  1012 , blue deck  1014  and amber deck  1016 . In this embodiment, each deck includes  36  cards and each of the cards is assigned numerical values of one to eleven. It should be understood that a special deck or a conventional playing cards may be used in which additional cards of certain values may be added and non-numbered cards may be reassigned new values, such as for example, a configuration in which Aces have a value of one and face cards (Jacks, Queens, or Kings) have a value of eleven, or red Aces have a value of one and black aces have a value of eleven. Other variations may be employed with this embodiment so long as the number of cards of each value in each deck correspond to the number of cards of each value in each deck as set forth in the below table. 
         [0000]    
       
         
               
             
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 
               
             
             
               
                   
               
               
                 Gaming Elements 1010 
               
             
          
           
               
                   
                 Card Value 
               
             
          
           
               
                   
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
               
             
          
           
               
                 Deck 
                 Number of cards in each deck having the above value: 
               
               
                   
               
             
          
           
               
                 Red 
                 None 
                 2 
                 2 
                 6 
                 8 
                 2 
                 10 
                 2 
                 2 
                 2 
                 None 
               
               
                 Deck 
               
               
                 1012 
               
               
                 Blue 
                 4 
                 None 
                 4 
                 4 
                 4 
                 4 
                 4 
                 8 
                 4 
                 None 
                 None 
               
               
                 Deck 
               
               
                 1014 
               
               
                 Amber 
                 2 
                 2 
                 4 
                 4 
                 4 
                 6 
                 4 
                 4 
                 2 
                 2 
                 2 
               
               
                 Deck 
               
               
                 1016 
               
               
                   
               
             
          
         
       
     
         [0075]      FIGS. 24-26  shows the complete probability analysis for all possible single-card deals for all three possible non-transitive deck pairings, namely, red deck  1012  vs. blue deck  1014 , blue deck  1014  vs. amber deck  1016 , and amber deck  1016  vs. red deck  1012 . All three possible deck pairings exhibit identical edges of 8/1296 or about 0.62%, presuming cards are selected at random or dealt from a randomly-ordered deck. Also, each of the three possible deck pairings has exactly the same tie probability, namely, 136/1296 or about 10.49%, again presuming that the cards are selected at random or dealt from a randomly-ordered deck. 
         [0076]      FIGS. 27-32  illustrate another embodiment of the invention a dice shaker  1350  for facilitating the live table game embodiment of the invention. The shaker  1350  comprises a transparent cylindrical housing  1355  having an open end  1361  configured to mate with end cap  1360  for sealing the dice  1365  within housing  1355 . Dice shaker  1350  further includes a sleeve  1363  for being seated thereon and capable of bidirectional linear motion over the outside of cylindrical housing  1355 . 
         [0077]    While dice shaker housing  1355  may be formed from a transparent material so that the dice therein are visible, sleeve  1363  may be constructed of an opaque material to prevent the dice from being viewed while sleeve  1363  is in a position to cover housing  1355 . Sleeve  1363  may therefore be used to hide the action of the dice inside housing  1355  during shaking. Sleeve  1363  may thereafter be moved to reveal the outcome of the dice roll within housing  1355 . 
         [0078]    Shaker  350  and  1350  may take into account various considerations, such as: the mass of shaker and cover; the total weight, that is, lightweight is more likely to prevent dealer fatigue during a shift that requires many hundreds of shakings per hour; the rattling noise the dice and shaker may produce; and its physical size, that is, to the extent possible the shaker should be small enough to provide for an easy grip by any person with average sized hands, yet sufficiently large enough to allow for the dice to move and rotate to deliver a random result. 
         [0079]    In determining the appropriate sizing to address the above considerations, among other things, it has been found that increasing the separation between the main bouncing surfaces of dice shaker  350  and  1350  produce more and more dice rotations and therefore random results, as the die cubes rotate more unpredictably in colliding with the ends and inside walls while traversing the gap. 
         [0080]    The dimension of the dice being shaken is a factor affecting random results. One can increase or decrease the randomness of the results simply by changing the size of the dice within a fixed shaker size. Dice measuring only 0.25 inches per side will produce more random results in the above example, although their spots would be much more difficult to see. This discussion suggests that a better measure of randomness is expressed by the number of “dice units” traveled. For example, if a die with a 0.5 inch edge dimension travels 2.0 inches, the distance traveled is 4.0 dice units. 
         [0081]    Another factor to consider in obtaining random results is the length of time devoted to one shake, or more precisely, how many collisions a die cube will experience during the period of time that it is being shaken. The longer the time spent shaking generally translates to the more collisions the dice have, and the more kinetic and rotational energy is imparted to the dice. Increasing the number of collisions should therefore produce more random results. 
         [0082]    Assuming that a normal person will shake a dice shaker as disclosed herein at a rate of 3.9 full cycles per second. A shake “cycle” is defined herein as a transition from the dice starting at rest on the bottom of the shaker, then the dice hitting the top, and the dice hitting the bottom again. This 3.9 cycles per second rate was measured using a video recording of a dice shaker such as those disclosed herein in action. It is noted that this number is in good agreement with the calculated 3.7 cycle/sec. for a person&#39;s natural clapping rate. 
         [0083]    The first part of  FIG. 28  shows the distance that dice travel for “shake times” between 1 and 6 seconds. So, this shake time defines the length of time of a “roll”. For example, we see that for a recommended 3-seconds roll, dice travel 11.8 feet inside the dice shaker. For ⅝″ dice cubes, in “dice units”, this equates to a total of 206 dice units traveled. Now consider the lower part of  FIG. 28  where a Craps throw randomness measure is calculated. For an average length table, a dice toss of 7 feet, using ¾ inch standard casino dice, the dice thrown will undergo only about 112 dice units of travel, assuming that they stop rolling after hitting the energy-absorbing bumper at the end of the craps table. 
         [0084]    Thus, based on the foregoing, in three seconds of shaking a dice shaker constructed as disclosed herein, and as exemplified by shakers  350  and  1350 , produces about twice the “randomness”, as measured by dice units traveled, of thrown craps dice on a craps table. But there are even more randomness advantages of the new dice shaker. An unexpected result is that it is actually impossible to perform a single shake of the shaker disclosed herein such that the enclosed dice fail to collide and bounce at least once off of the shaker cylinder&#39;s wall. This is likely because the hand/arm shaking motion is not done 100% up and down, but it is rather part of a circular arc, with the radius of the arc equal to the shaker&#39;s arm length. This means the distance that the dice traverse is always greater than the minimum vertical distance traversed. In  FIG. 28  the displayed row of “more reasonable distance that . . . ” is simply 10% more than the minimum distances. This adjustment is a very low estimate of the actual path distance. Thus, it should be readily apparent that the shaker calculations of  FIG. 28  are conservative numbers. 
         [0085]    Dice shaken inside a dice shaker such as shaker  350  and  1350  continuously receive additional bursts of both kinetic and rotational energy after each collision that they undergo with the top/bottom of the shaker. In contrast, once the craps dice leave the shooter&#39;s hand, all of the dice energy begins to immediately dissipate with each bounce down the craps table and no added energy is ever transferred to the dice. They just bounce, rotate, bounce more, and then ultimately come to rest. In dice shaker  350  and  1350 , dice not only travel long distances (in dice units) but also they spin with much more rotational energy which increases their random orientation even more. 
         [0086]    Dice shakers  350  and/or  1350  may include pads made of ethylene vinyl acetate (EVA) which are affixed to the inside of the top and bottom of the shaker ends which increases the kinetic and rotational energy transfer to the dice. EVA is an excellent choice because of its high resiliency and it also provides good sound absorption, but other materials, like rubber, are also good choices. 
         [0087]    In the preferred embodiment the shape of the dice shaker is cylindrical with a combined diameter (shaker and cover or sleeve) being no larger than about 3.3 inches. For example, if the cover is about ⅛″ thick, then the shaker diameter should be no larger than about 3.0 inches. In order to attain the maximum amount of randomness, the shaker diameter should be as large as possible. With regard to length, the larger the separation of the dice shaker&#39;s top and bottom, the more random the dice rolls will be. Thus, the dice shaker should be as elongated as possible, that is, until other features such as weight become a limiting factor. Focus group studies show that both players and dealers are ergonomically very comfortable using a combined shaker/cover that weighs about 13 ounces. 
         [0088]      FIG. 29  provides a detailed weight analysis for seven different candidate shaker/cover heights, using the weight densities of acrylic and leather and simple solid geometry formulas for cylinders. The weights of the top, which may be a plastic, such as Delrin®, and the acrylic bottom were both weighed from actual manufactured parts. From  FIG. 29  it may be concluded that an acrylic tube height of about 5.5 inches results in a very acceptable total weight of the shaker and leather sleeve of 13.1 ounces, and an embodiment thereof is shown in  FIGS. 30-32 , which also shows the depth of the threading on the inside of the acrylic tube and the inner and outer diameters of the acrylic tube, the details of the machining for the Delrin® screw-on top, which allows for the EVA disk that measures about 2.5 inches in diameter and about ¼ inch in thickness, and a depiction of the opaque acrylic bottom part which is adhered onto the acrylic tube. 
         [0089]      FIGS. 33 and 34  shows detailed dimensions for an embodiment of a cover or sleeve for the shaker of the invention, and in particularly the shaker  1350  as shown in  FIGS. 30-32 . An air gap of 0.11 inches provides for enough clearance to easily drop on and lift off the leather cover over the shaker. Also, the leather cover should be about 0.14 inches thick for best performance of being easy to clasp. This thickness, together with a leather end cap, also provides the cylindrical cover enough resilience to maintain its shape. 
         [0090]    It will be appreciated by those skilled in the art that while the disclosure has been described above in connection with particular embodiments and examples, the scope of the invention is not necessarily so limited, and that numerous other embodiments, examples, uses, modifications and departures from the embodiments, examples and uses described herein are intended to be encompassed by the invention, claims attached hereto and equivalents thereof.