NON-TRANSITIVE GAMING ELEMENTS AND GAMING METHODS
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.
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 INVENTIONThe invention generally relates to groups of gaming elements, such as pairs of dice, customized to exhibit a non-transitive relationship, and games therefor.
BACKGROUNDAs 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.
SUMMARYThe 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Other features, embodiments and variations will become evident from the following detailed description, drawings and claims.
Reference is now made to the figures wherein like parts are referred to by like numerals throughout.
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
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
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.
The lower portion 320 of
When comparing the three non-transitive card deck house advantage results with the three non-transitive dice pair results of
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
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.
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.
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.
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.
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.
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.
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.
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%.
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.
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.
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.
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.
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.
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's natural clapping rate.
The first part of
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'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's arm length. This means the distance that the dice traverse is always greater than the minimum vertical distance traversed. In
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'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.
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.
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'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.
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.
Claims
1. A dice shaker for randomly changing the positional state of dice, comprising:
- a) a sealable cylindrical body member constructed of a rigid and transparent material and defining an inner chamber for securely storing the dice therein;
- b) a sleeve defining a tube dimensioned for receiving the cylindrical body member therein and constructed of an non-transparent material; and
- c) 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.
2. The dice shaker of claim 1 wherein the cylindrical body member is comprised of a clear transparent tube with two end caps.
3. The dice shaker of claim 2 wherein the transparent tube is comprised of a plastic type material.
4. The dice shaker of claim 3 wherein both end caps are permanently attached to the cylindrical body member.
5. The dice shaker of claim 3 wherein both end caps are removable.
6. The dice shaker of claim 3 wherein one end cap is permanently attached to the tube.
7. The dice shaker of claim 6 wherein the non-permanently attached end cap is threaded to screw into a threaded receiving end of the tube.
8. The dice shaker of claim 7 wherein the inside surface of one or both end caps is substantially covered with the padding.
9. The dice shaker of claim 8 wherein the padding comprises ethylene vinyl acetate (EVA).
Type: Application
Filed: Dec 3, 2012
Publication Date: Apr 11, 2013
Inventor: Stanley Klein (Irvine, CA)
Application Number: 13/692,805
International Classification: A63F 9/04 (20060101);