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This application is related to the application Ser. No. 691,944, filed on Aug. 5, 1996, now U.S. Pat. No. 5,795,226, granted on Aug. 18, 1998. The inventor's name was misprinted as Chen Yi. A certificate of correction was issued on Nov. 24, 1998.
This invention relates to games of chance, more specifically, to methods of playing a betting game of chance determined by one or several rounds of random drawing.
About the time of U.S. Pat. No. 5,795,226 being granted, I started to ask myself, how I would like to market my patented games. For the automatic version, I set up a 3½ diskette entitled “NINE DICE”—which became U.S. registered trademark No. 2,322,258 on Feb. 22, 2000—. But it is only a PC game, not yet what I desire to make available at casinos simply because of using invisible computer random numbers. Nine Dice at casinos should be equipped with a visible mechanical random number generator. I have figured out a few such devices and keep on testing prototypes. The non-automatic version requires a 8′ by 8′ table, a rolling dice box, and so on, all made-to-order only. Its operation requires several workers. All this means high cost which will result in high house edge, something I hate. So, why not replace the big table by a monitor display? Why not let a keno bowl of balls to generate random numbers? Why not allow the players to determine the track length, and to start a race anytime? Why not drop the racing characteristic and bet on movers moving to certain sites? Besides, if a bet is to be determined by more than one round of random numbers and players are allowed to place make-up bets between two rounds, why not allow the hanging ticket holder to earn credit placing free make-up bets. Placing free bets reduces the payout of an original bet, but eliminates bettor's worry about bankroll and saves fund transaction handling. Answering these questions has resulted in filing this patent application and another one, both to be deposited with the U.S. Postal Service on the same day.
As far as playing surface is concerned, every game with moving pieces is prior art. As far as gambling nature is concerned, all games of chance such as keno, lottery, craps, roulette, bingo are prior art. As far as technology is concerned, any game requiring computer data processing is prior art. However, none of those regarding moving pieces or data processing deserves a description as prior art here. Probably, based on operation equipment, most people will consider keno or lottery as quite relevant prior art. But some might argue that craps is more relevant, because odds bet is make-up wagering, a characteristic of the game of this invention.
Many casino gambling books call keno a game to avoid, while live keno is no more available in every big Las Vegas casino. In the master chart of all casino bets in ‘MENSA Guide to Casino Gambling’ by Andrew Brisman, keno is listed last with 30% highest house edge. It's well known that California lotteries, from scratch-off to Super Lotto Plus, are mostly played by poor laborers, except when there is jackpot approaching 100 million. In the U.S. close to forty state governments run various lotteries to produce revenue. According to ‘The Lottery book’ by Don Catlin, their house edges are 45 to 55%. One may think that the low popularity and the high house edge interact as both cause and effect. Then, can lowering house edges attract more players? Probably not, because of three hardly changeable weak points:
1. Winning probabilities are too low. The highest winning probability in keno and California lotteries is 1in 4 while most casino games have higher probabilities. Looking into statistic on annual casino revenue and state lottery ticket sale, one sees that the common gambling intention is inclined by far to catching more probable wins rather than getting rich fast.
2. Too troublesome to place large amount of bets. Example one: David W. Cowles mentioned in ‘Complete Guide to winning keno’ that besides straight, combination, way, king, top/bottom, high-low tickets there are individual casino keno folders offering varieties to place bets. All of the above are usually pay-any-catch, but some of them can also be catch-all (one-pay). A $2 6-spot pay-any-catch ticket mostly pays $2 for catching 3/6, $8 for catching 4/6, $175 for catching 5/6, and $3,000 for catching 6/6. A $2 6 spot catch-all ticket, in one casino—no name given in the book—pays $11,000 for catching 6/6—house edge 29.0583%—. Now, you may find the $11,000 casino, and tell a keno writer that you want to purchase all possible $2 6-spot catch-all combination tickets with numbers 1, 2, 3 plus any three numbers from 4 to 80. This set of bets costs 73150*$2 with probability (20*19*18)/(80*79*78)=0.013875% to win 73150*$11,000. Your request will probably lead you to the keno manager, who will probably say that no such betting is available in their keno folders; so you must mark to purchase all tickets separately. One may call the betting Double-King for two groups of spots instead of one group in the usual King ticket, and write (3,3)×(77,3) together with one circle around 1,2,3, and another around 4 to 80. If the spots of a group are not adjacent to each other, or if there are more than two groups to circle, confusion will likely occur. Besides, any bet slip with complex circled groups of spots is difficult to apply computerized digital scanning. Example two: If you have $42 million at hand, it's not a bad idea to purchase all possible combination tickets, when the California Super Lotto Plus jackpot is over $100 million. But, every marked playslip results in 5 tickets only; and mechanically most lottery machines need a few seconds to handle a playslip. Applying ‘touch screen’ is hardly faster. Missing one purchase or a defective sorting can be disastrous. Well, maybe you can find a way to request a single ticket for all combinations.
3. Every bet is determined by drawing once. Those so-called multi-game, multi-play, multi-race are all one draw game except exacta keno. The rule of Exacta at Gold Coast, Las Vegas, is to mark the same number of spots, from one to ten, in two consecutive games, paying $1 per game plus $0.25 for exacta. The simplest and best payout Gold Coast Exacta is to mark one spot in each game which pay $3 for first game, $3 for second game and $4 for exacta. The house edges are 25%, 25% and 0% respectively. Unfortunately, due to 0% on 25 cents and 25% on $2, there is no way to take advantage of a hanging ticket by placing make-up bets after the first draw. Changing house edges and exacta amount can make keno exacta bets working similarly to some bets of the invention; but difficulties in marking (and scanning—if available—) keno tickets can hardly be overcome. Besides, the choices of keno bets remains by far very restricted in comparison to that of this invention, simply due to applying a single selection of 20 out of 80, rather than multiple selections of one out of n, where n<10.
Craps is a high operation cost game requiring boxman, stickman and two more dealers to serve a table of 10 to 12 players. Nevertheless, this fast action, low house edge game takes place in every Las Vegas casino. There are always people around fond of playing about 50 rounds in less than 2 hours, to place odds bets from $40 to $100, and to win or lose a few hundred bucks. But, at crap, both bet varieties and winning probabilities are quite limited. Besides, many people like crap, but never Actual play for the reason that they don't like others watching their gamble actions, winning or losing expressions.
From 1976 to present there are 611, 328, 129, 184, 293, 307 and 528 US patents granted related to Classes 463/16, 463/17, 463/18, 463/22, 273/243, 273/269 and 273/274 respectively. A few of them will be listed in the Information Disclosure Statement.
The principal object of the invention is to provide a low operation cost multi-draw game of chance, which can be carried out by easily made to order equipments or existing keno/lottery facilities with minor changes.
Here is a game without those keno/lottery weaknesses discussed above. Besides, although manipulation at keno/lottery is out of the question, occasionally, one may not win due to a numbered ball missing. According to ‘Complete Guide to Winning Keno’ by David Cowles, it occurred that a costumer remarked to a keno manager at a popular Las Vegas casino, “I've been playing keno for hours, and Number 29 never has come up . . . ”. . . Indeed, Number 29 was missing. Using a large amount of balls, missing may take a while to notice. Besides, even though all balls are whirling around, within a short period of selection time, some of them never have a chance to enter a suitable region to be pushed by hot air force into the selection tube. Thus, it's better if the game, as in this invention, requires less than 30 balls, and with several copies of the same kind, such as four copies each of seven different ones.
Here is a wagering game with any desired winning probabilities. It allows the operator to set a wide range of house edges so that players can either try to win big money with high negative expectation or enjoy gambling excitement with minimal negative expectation.
Assume a die will roll once a while regularly to let you bet on the outcomes on one or several rolls. The house edge is 1/60. Say you spend $1 to purchase a 2-Roll ticket on certain two numbers and expect payout $35.40. Say you do catch the first roll correctly. Would you have second thought to play safely? How about to purchase five $6 tickets on other numbers so that no matter what the second roll may be you earn $4.40? Since such a sure win does exist, why not allow you to win without that $30 and thus avoid fund transaction handling? The invention is to provide a game more interesting than rolling a die with the feature of credit for hanging ticket to place free make-up bets.
It is also an object to provide a game which may cause players to dream of being accompanied by loved ones.
The invention provides a game of chance with a playing surface. The playing surface is a map of sites on which there are circled numbers called movers. A computer monitor will be required to display the playing surface with movers.
The invention provides a plurality of ruled movements directing movers to move from one site to another. A random draw device functioning like the one used at keno will be required. While there a number is printed on each ball, here a symbol representing the ruled movement. There are equally many, four or five, copies for the same movement.
The game requires a player called operator to conduct. Other players are bettors. The operator executes random draw of ruled movements, one round after another, moving each mover accordingly once per round. Henceforth every round of drawing will be simply called a draw. The bettors can place bets during the break between two draws.
The invention provides a plurality of bet slips showing either the whole playing surface with all movers or a part of the playing surface with a single mover. As at a racetrack bettors mark paper bet slips to place bets. The bettor can select one or several movers together with one or several sites which the bettor expects to match the outcomes of forthcoming one or several draws.
Wagering machines connected to a computer with database wagering system software, similar to those used at a racetrack, will be required to examine marked bet slips, to store, process betting data and to print bet tickets showing all officially accepted bets. Bet slips can also be on ‘touch screen’.
A multi-draw ticket becomes or remains hanging if it contains a selection matching the last draw outcomes, and thus has a chance to be a winner later on. The invention provides the option that any hanging ticket holder earns credit to place free make-up bets, henceforth called credit bets.
The winning probability of every bet as well as how to calculate payouts and credits is included.
The invention provides a video/computer version of the game.
FIG. 1 is a flowchart illustrating the game process.
FIGS. 2 to 4 show similar playing surfaces.
FIG. 2A is a universal bet slip using playing surface as shown in FIG. 2.
FIGS. 3A, 4A are plain bet slips using playing surface as shown in FIGS. 3, 4 respectively.
FIGS. 3AA, 4AA are chain bet slips using playing surface as shown in FIGS. 3, 4 respectively.
FIGS. 2B, 3B and 4B are each a ‘simple’ bet ticket using playing surface as shown in FIGS. 2 to 4 respectively.
FIGS. 2C, 3C and 4C are each a ‘site’ bet ticket using playing surface as shown in FIGS. 2 to 4 respectively.
FIGS. 2D, 3D and 4D are each a ‘mixed’ bet ticket using playing surface as shown in FIGS. 2 to 4 respectively.
FIG. 2E is a 4-Draw chain bet ticket using playing surface as shown in FIG. 2.
FIGS. 3E and 4E are each a 3-Draw chain bet ticket using playing surface as shown in FIGS. 3 and 4 respectively.
FIGS. 2F, 3F and 4F are each a Draw 2 revised chain bet ticket using playing surface as shown in FIGS. 2 to 4 respectively.
FIGS. 2G, 3G and 4G are each a Draw 3 revised chain bet ticket using playing surface as shown in FIGS. 2 to 4 respectively.
FIG. 2H is a Draw 4 revised chain bet ticket using playing surface as shown in FIG. 2.
FIG. 5 shows a betting activity statement.
Playing surface 10 in FIG. 2 contains seven sites 11 on which there are six movers 12.
The ruled movements for this playing surface are denoted ‘A’, ‘B’, ‘C’, ‘D’, ‘E’, ‘F’, and ‘G’, and defined below:
For any mover in any location there are always the same number w of movements, namely w=7.
A ruled movement aiming at a specific location will be called jump; otherwise, non-jump. Thus, all movements on playing surface as shown in FIG. 2 are jump, while all those declared below for FIGS. 3 and 4 are non-jump.
Playing surface 10 in FIG. 3 contains ten sites 11 on which there are six movers 12. Similarly to most computer games, it is necessary to regard the top border line as identical to the bottom line. Thus, site ‘A’ lies one site downward to site ‘L’, two sites downward to site ‘K’, site ‘B’ lies two sites downward to site ‘L’, site ‘L’ lies one site upward to site ‘A’, two sites upward to site ‘B’ and three sites upward to site ‘C’, site ‘K’ lies two sites upward to site ‘A’ and three sites upward to site ‘B’, site ‘H’ lies three sites upward to site ‘A’. The modified playing surface 100 shown in FIG. 3AA visualizes this crossing border down/up situation.
The ruled movements for this playing surface are denoted ‘00’, ‘U1’, ‘U2’, ‘U3’, ‘D1’,and ‘D2’, and defined below:
For any mover in any location there are always the same number w of movements, namely w=6.
Playing surface 10 in FIG. 4 contains twenty sites 11 on which there are six movers 12. It is necessary to regard the top border line as identical to the bottom one, the left border line identical to the right one. Thus, site ‘AA’ lies surrounded by site ‘DA’ in the north, by site ‘DB’ in the northeast, by site ‘DE’ in the northwest, by site ‘AB’ in the east, by site ‘AE’ in the west, by site ‘BA’ in the south, by site ‘BB’ in the southeast, and by site ‘BE’ in the southwest; site ‘AB’ lies surrounded by site ‘DB’ in the north, by site ‘DC’ in the northeast, by site ‘DA’ in the northwest, by site ‘AC’ in the east, by site ‘AA’ in the west, by site ‘BB’ in the south, by site ‘BC’ in the southeast, and by site ‘BA’ in the southwest; and so on, as visualized in the modified playing surface 100 shown in FIG. 4AA.
The ruled movements for this playing surface are denoted ‘00’ ‘N’, ‘E’, ‘W’, ‘S’, ‘NE’, ‘NW’, ‘SE’, and ‘SW’, and defined below:
For any mover in any location there are always the same number w of movements, namely w=9.
There are Draw 1 to 4 plain bets, further classified as ‘simple’, ‘site’ or ‘mixed’. There are 2- to 4-Draw chain bets, further classified as ‘linked’ or ‘unlinked’. All bets made on one bet slip are of the same class/type.
It is intended solely to enhance fun unrelated to game rules that on the bet slips movers will be given names of popular persons, and sites that of popular places.
The bet slip as shown in FIG. 2A will be used for any bet concerning the playing surface as shown in FIG. 2. For playing surface as shown in FIG. 3, bet slip as shown in FIG. 3A will be used for plain bets while that of FIG. 3AA for chain ones. For playing surface as shown in FIG. 4, bet slips as shown in FIG. 4A will be used for plain bets while that of FIG. 4AA for chain ones. All of the above bets on a bet slip will be referred to as ‘regular’. Other bets, explained later on, are ‘credit’ bet. A mover once selected in a draw will be referred to as a bet-on mover of that draw. A site in which a bet-on mover is located will be referred to as selected site of that mover.
A common action on every bet slip is to mark a type, and either an ‘amount per bet’ or a ‘total bet amount’ except in the case of credit bet where ‘credit’ must be marked.
When movements are jump, all sites 11 in FIG. 2 are reachable by a single ruled movement and become sites 31, 41, 51 and 61 in FIG. 2A.
When movements are non-jump, referring to FIG. 3A or 4A, not every site 31 in Draw 1 is reachable by a single ruled movement. Thus, if a mover selected lies in an unreachable site, it will be simply cancelled by the computer, when the bet slip is submitted for approval. By two or more movements, all sites 41, 51 and 61 are always reachable from any start location. Referring now to FIGS. 3AA, 4AA, instead of a complete playing surface, we use partial ones, for each mover, with one site 31, 41, 51 or 61 in gray color to represent the location of the concerning mover before Draw 1, 2, 3 and 4 Draw respectively. All sites 11 in FIGS. 3 or 4 reachable by a single ruled movement become exactly those sites 31, 41, 51 or 61 in FIGS. 3AA or 4AA. The location of the gray site 31 in the complete playing surface is known at the time of betting. The locations of the gray sites 41, 51 and 61 in the complete playing surface will naturally be determined by Draws 1 to 3 respectively.
In the following, * is the multiplication operator, and ˆ the exponent operator. Π(f(M)) is multiplication of f(M) over all M to be specified, and Σ(f(M)) is summation of f(M) over all M to be specified. Mathematically in general, M is a variable of function f, where f remains to be defined whenever needed.
To place plain bets, the bettor marks to select one or several movers 12 located in sites 31, 41, 51 and/or 61. Every selected 12 becomes a bet-on mover. The bettor can play any one or more draws on the bet slip. All draws are independent. It is allowed, for example, to select some movers in sites 31, some in sites 51, but none in 41 or 61 for playing Draws 1 and 3.
Let #31(M), #41(M), #51(M) and #61(M) denote respectively the number of selected sites 31, 41, 51, and 61 of mover M.
In the ‘simple’ case, every selected site with a bet-on mover counts a bet. The same site will be counted as many times as the number of bet-on movers lying inside, draw by draw independently. In other words, the numbers of Draw 1, 2, 3, and 4 ‘simple’ bets are Σ(#31(M)), Σ(#41(M)), Σ(#51(M)), and Σ(#61(M)) respectively. The bettor wins, draw by draw independently, whenever there is a selected site 31 with one bet-on mover matching the outcomes of Draw 1, a selected site 41 with one bet-on mover matching the outcomes of Draw 2, a selected site 51 with one bet-on mover matching the outcomes of Draw 3, a selected site 61 with one bet-on mover matching the outcomes of Draw 4.
In the ‘site’ case, every selected site of all bet-on movers inside counts a bet. The bettor wins, draw by draw independently, whenever there is one selected site 31 with all bet-on movers inside matching the outcomes of Draw 1, one selected site 41 with all bet-on movers inside matching the outcomes of Draw 2, one selected site 51 with all bet-on movers inside matching the outcomes of Draw 3, one selected site 61 with all bet-on movers inside matching the outcomes of Draw 4.
In the ‘mixed’ case, every bet contains a complete set of all bet-on movers, each connected with one selected site. For example, in Draw 1, there are three bet-on movers, one with 4 selected sites, another 3 selected sites, and the third 5 selected sites. Here are Π(#31(M))=4*3*5=60 bets. In other words, the numbers of Draw 1, 2, 3, and 4 ‘mixed’ bets are Π(#31(M)), Π(#41(M)), Π(#51(M)), and Π(#61((M)) respectively. The bettor wins, draw by draw independently, if, each time, for each one of all bet-on movers, there is one selected site 31 matching the outcomes of Draw 1, one selected site 41 matching the outcomes of Draw 2, one selected site 51 matching the outcomes of Draw 3, or one selected site 61 matching the outcomes of Draw 4. The mixed bets on one bet slip can bring in for each draw one winner only.
To place 2-Draw bets, the bettor marks to select first one or several movers in site 31 for Draw 1, and then one or several movers in sites 41 for Draw 2. Every Draw 1 bet-on mover must be bet-on in Draw 2 and vice versa. As mentioned above, the location of the gray site 41 is yet unknown. So, for one and the same bet-on mover, any selected site 41 is valid for all selected sites 31, no matter what the outcomes of Draw 1 may be. If the bettor wants a certain selected site 41 just for a certain selected site 31, then it is necessary to use separate bet slips. —For example, using one bet slip you can bet a mover moves first either to east or west and then either to north or south. This is four bets on one slip. If you want to bet that the mover moves either first to east then to north or first to west then to south. This is two bets, and you need to place them separately using two bet slips.—In the ‘unlinked’ case, the number of bets is Σ(#31(M)*#41(M)). —For example, there are bet-on movers A, B and C with #31(A)=4, #31(B)=3, #31(C)=5, #41(A)=2, #41(B)=6, and #41(C)=1, then :(#31(M)*#41(M))=4*2+3*6+5*1. —The bettor wins if, with respect to one and the same bet-on mover, there is one selected site 31 matching the outcomes of Draw 1, and one selected site 41 matching the outcomes of Draw 2. The unlinked bets in one bet slip can bring in as many winners as the number of bet-on movers. In the ‘linked’ case, the number of bets is Π(#31(M)*#41(M)).—For example, there are bet-on movers A, B and C with #31(A)=4, #31(B)=3, #31(C)=5, #41(A)=2, #41(B)=6, and #41(C)=1, then Π(#31(M)*#41(M))=(4*2)*(3*6)*(5*1).—The bettor wins only if for each one of all bet-on movers, there are one selected site 31 matching the outcomes of Draw 1, and one selected site 41 matching the outcomes of Draw 2. The linked bets in one bet slip can bring in one winner only.
To place 3-Draw bets the bettor marks to select first just as explained in the 2-Draw case; then one or several movers in sites 51 for Draw 3. Every Draws 1 to 2 bet-on mover must be bet-on in Draw 3 and vice versa. As mentioned above, the location of the gray site 51 is yet unknown. So, for one and the same bet-on mover, any selected site 51 is valid for all selected sites 31 and 41, no matter what the outcomes of Draws 1 and 2 may be. If the bettor wants a certain selected site 51 just for a certain selected sites 31 and 41, then it is necessary to use separate bet slips. In the ‘unlinked’ case, the number of bets is Σ(#31(M)*#41(M)*#51(M)). The bettor wins if, with respect to one and the same bet-on mover, there is one selected site 31 matching the outcomes of Draw 1, one selected site 41 matching the outcomes of Draw 2, and one selected site 51 matching the outcomes of Draw 3. The unlinked bets in one bet slip can bring in as many winners as the number of bet-on movers. In the ‘linked’ case, the number of bets is Π(#31(M)*#41(M)*#51(M)). The bettor wins only if, for each one of all bet-on movers, there are one selected site 31 matching the outcomes of Draw 1, one selected site 41 matching the outcomes of Draw 2, and one selected site 51 matching the outcomes of Draw 3. The linked bets in one bet slip can bring in one winner only.
To place 4-Draw bets the bettor marks to select first just as explained in the 3-Draw case; then one or several movers in sites 61 for Draw 4. Every Draws 1 to 3 bet-on mover must be bet-on in Draw 4 and vice versa. As mentioned above the location of the gray site 61 is yet unknown. So, for one and the same bet-on mover, any selected site 61 is valid for all selected sites 31, 41, and 51, no matter what the outcomes of Draws 1 to 3 draws may be. If the bettor wants a certain selected site 61 just for a certain selected sites 31, 41, and 51, then it is necessary to use separate bet slips. In the ‘unlinked’ case, the number of bets is Π(#31(M)*#41(M)*#51(M)*#61(M)). The bettor wins if, with respect to one and the same bet-on mover, there is one selected site 31 matching the outcomes of Draw 1, one selected site 41 matching the outcomes of Draw 2, one selected site 51 matching the outcomes of Draw 3, and one selected site 61 matching the outcomes of Draw 4. The unlinked bets in one bet slip can bring in as many winners as the number of bet-on movers. In the ‘linked’ case, the number of bets is Π(#31(M)*#41(M)*#51(M)*#61(M)). The bettor wins only if, for each one of all bet-on movers, there are one selected site 31 matching the outcomes of Draw 1, one selected site 41 matching the outcomes of Draw 2, one selected site 51 matching the outcomes of Draw 3, and one selected site 61 matching the outcomes of Draw 4. The linked bets in one bet slip can bring in one winner only.
Every marked bet slip will be checked and approved by the computer in order to issue one or several bet tickets as shown in FIGS. 2B, 3B and 4B for simple bets and FIGS. 2C, 3C and 4C for site bets and FIGS. 2D, 3D and 4D for mixed bets and FIGS. 2E, 3E and 4E for chain bets, The ticket shows type of bets, per bet amount (if marked by the bettor), total number of bets, total bet amounts, the Draw # of Draw 1. A playing surface with start locations of all movers before Draw 1 will be printed unless the ruled movements are jumps. In the case of one draw bets, only bet-on movers in selected sites will show up on the bet ticket. In the case of chain bets, bet-on movers in selected sites will be marked with “X”. It is for the sake of convenience to allow unlinked bets on one slip. In order to avoid confusion and to make credit bets simple, there will be no unlinked chain bet ticket. Any unlinked bet slip of n bet-on movers will result in issuing n chain bet tickets, one for each bet-on mover. Thus, every chain bet ticket with more than one bet-on mover is always ‘linked’ without stating so.
After Draw 1, a hanging chain bet ticket can be used as bet slip to place credit bets as follows: The player marks to select ‘credit percentage’ and either ‘new slip’ or not. —The calculation of credit amount will be explained later on. What a bettor has to do is to select a certain percentage.—In the case of using a new bet slip, the bettor places bets as explained before except that instead of ‘Amount per bet’ or ‘Total bet amount’ now ‘credit’ must be marked. The credit modified by selected percentage on the hanging bet ticket will be evenly applied to all bets on the new slip. The bettor will receive a regular bet ticket for the new slip and a revised bet ticket just as the original one except that there is a “X” on the selected ‘credit percentage’. The revised ticket remains hanging if one selection matches the outcomes of Draw 2. In the case of no new bet slip, the player marks to select for each bet-on mover one or several sites 41. The credit modified by selected percentage will be evenly applied to be all credit bets. Let #41′(M) denote the number of both originally and newly selected sites 41 for mover M. The number of credit bets will be #41(cr)=Π(#41′(M))−Π(#41(M)). This ticket as bet slip will be checked by the computer so that a revised ticket as shown in FIG. 2F, 3F or 4F can be issued. The revised ticket shows all original data. It also shows selected credit percentage, all new selections marked with “=”. Besides, each mover in the Draw I drawn site will be printed in gray. All credit bets marked on a hanging ticket are Draw 2 ‘mixed’, which naturally become ‘simple’ if there is only one bet-on mover.
After Draw 2 a hanging chain ticket—whether revised or not—can be used as bet slip to place credit bets as follows: The player marks to select ‘credit percentage’ and either ‘new slip’ or not. In the case of using a new bet slip, the bettor places bets as explained before except that instead of ‘Amount per bet’ or ‘Total bet amount’ now ‘Credit’ must be marked. The credit modified by selected percentage on the hanging bet ticket will be evenly applied to all bets on the new slip. The bettor will receive a regular bet ticket for the new slip and a revised bet ticket just as the original one except that there is a “X” on the selected ‘credit percentage’. The revised ticket remains hanging if one selection matches the outcomes of Draw 3. In the case of no new bet slip, the player marks to select for each bet-on mover one or several sites 51. The credit modified by selected percentage will be evenly applied to be all credit bets. Let #51′(M) denote the number of both originally and newly selected sites 51 for mover M. The number of credit bets will be #51(cr)=Π(#51′(M))−Π(#51(M)). This ticket used as bet slip will be checked by the computer so that a revised ticket as shown in FIG. 2G, 3G or 4G can be issued. The revised ticket shows all original data. It also shows selected credit percentage, all new selections marked with “=”. Besides, each mover in the Draw 2 drawn site will be printed in gray. All credit bets marked on a hanging ticket are Draw 3 ‘mixed’, which naturally become ‘simple’ if there is only one bet-on mover.
After Draw 3 a hanging chain ticket—whether revised or not—can be used as bet slip to place credit bets as follows: The player marks to select ‘credit percentage’ and either ‘new slip’ or not. In the case of using a new bet slip, the bettor places bets as explained before except that instead of ‘Amount per bet’ or ‘Total bet amount’ now ‘Credit’ must be marked. The credit modified by selected percentage on the hanging bet ticket will be evenly applied to all bets on the new slip. The bettor will receive a regular bet ticket for the new slip and a revised bet ticket just as the original one except that there is a “X” on the selected ‘credit percentage’. In the case of no new bet slip, the player marks to select for each bet-on mover one or several sites 61. The credit modified by selected percentage will be evenly applied to be all credit bets. Let #61′(M) denote the number of both originally and newly selected sites 61 for mover M. The number of credit bets will be #61(cr)=Π(#61′(M))−Π(#61(M)). This ticket as bet slip will be checked by the computer so that a revised ticket as shown in FIG. 2H can be issued. The revised ticket shows all original data. It also shows selected credit percentage, all new selections marked with “=”. Besides, each mover in the Draw 3 drawn site will be printed in gray. All credit bets marked on a hanging ticket are Draw 4 ‘mixed’, which naturally become ‘simple’ if there is only one bet-on mover.
In the case of plain bet using bet slip as shown in FIG. 2A, any ‘simple’ bet has winning probability p=1/w. Any ‘site’ bet has winning probability p=1/wˆm where m is the number of bet-on movers in that site of the concerning draw. Any ‘mixed’ bet has winning probability p=1/wˆm where m is the number of bet-on movers of the concerning draw.
In the case of plain bet using bet slip as shown in FIG. 3A or 4A we need to specify a bet by the relative start-to-end locations explained below.
Referring to FIG. 3A, the identification of top with bottom border lines allows us to assign any one of the ten sites with 1-dimensional coordinates x and others with coordinates x+i, where every calculation involving i or x is modulo 10 arithmetic. Now we can replace A, B, etc by x, x+1, etc. respectively.—For example, A:0, B:9, C:8, D:7, E:6, F:5, G:4, H:3, K:2, L:1.—And we can also say that x+i lies i sites away from x. A movement from 0 to i is equivalent to a movement from x to x+i. Thus,
There is no other 1-movement path d1(i).
Let #d1(i) denote the number of all d1(i) for i. Obviously we have d1(0)=#d1(1)#d1(2)=#d1(3)=#d1(8)=#d1(9)=1 and #d1(4)=#d1(5)=#d1(6)=#d1(7)=0; in total 6.
p1M=#d1(i(M))/w is the probability of mover M from its start location to get on a d1(i(M)) path to reach the site 31 lying i(M) sites away. Here we use i(M) to specify i for the concerning M.
Let d2(i) be any d1(x) followed by any d1(i−x), defining a 2-movement path from any site to a site lying i sites away.
Let #d2(i) denote the number of all d2(i) for i. #d2(i) is the sum of #d1(x)*#d1(i−x) over all x; explicitly, we have #d2(0)=5, #d2(1)=6, #d2(2)=5, #d2(3)=4, #d2(4)=3, #d2(5)=2, #d2(6)=2, #d2(7)=2, #d2(8)=3, #d2(9)=4; in total 36, that is 6ˆ2.
p2M=#d2(i(M))/wˆ2 is the probability of mover M from its start location to get on a d2(i(M)) path to reach the site 41 lying i(M) sites away. Here we use i(M) to specify i for the concerning M.
Let d3(i) be any d1(x) followed by any d2(i−x), defining a 3-movement path from any site to a site lying i sites away.
Let #d3(i) denote the number of all d3(i) for i. #d3(i) is the sum of #d1(x)*#d2(i−x) over all x; explicitly, we have #d3(0)=25, #d3(1)=27, #d3(2)=27, #d3(3)=25, #d3(4)=22, #d3(5)=18, #d3(6)=16, #d3(7)=16, #d3(8)=18, #d2(9)=22; in total 216, that is 6ˆ3.
p3M=#d3(i(M))/wˆ3 is the probability of mover M from its start location to get on a d3(i(M)) path to reach the site 51 lying i(M) sites away. Here we use i(M) to specify i for the concerning M.
Let d4(i) be any d1(x) followed by any d3(i−x), defining a 4-movement path from any site to a site lying i sites away.
Let #d4(i) denote the number of all d4(i) for i. #d4(i) is the sum of #d1(x)*#d3(i−x) over all x; explicitly, we have #d4(0)=135, #d4(1)=144, #d4(2)=148, #d4(3)=144, #d4(4)=135, #d4(5)=124, #d4(6)=115, #d4(7)=112, #d4(8)=115, #d4(9)=124; in total 1296, that is 6ˆ4.
p4M=#d4(i(M))/wˆ4 is the probability of mover M from its start location to get on a d4(i(M)) path to reach the site 61 lying i(M) sites away. Here we use i(M) to specify i for the concerning M.
A Draw r simple bet, where r=1 to 4, on mover M lying i(M) sites away will be denoted by dr(i(M)). It has winning probability p=prM=#dr(i(M))/wˆr.
A Draw r site bet, where r=1 to 4, on movers M lying each i(M) sites away from site S will be denoted by drS( . . . ,i(?), . . . ), where ? goes from mover #1 to #6, and i(?) is i(M) if ? is a bet-on mover, otherwise ‘-’ (a dash). For example, d3B(-,2,1,-,-,8) is a Draw 3 site bet on site B with bet-on movers #2, #3 and #6, lying respectively 2, 1 and 8 sites away from site B. Or, d4E(3,2,1,-,7,-) is a Draw 4 site bet on site E with bet-on movers #1, #2, #3 and #5, lying respectively 3, 2, 1 and 7 sites away from site E. The drS( . . . ,i(?), . . . ) bet has the winning probability of p=Π(prM) where multiplication is over bet-on movers in site S of Draw r.
A Draw r mixed bet, where r=1 to 4, on movers M lying each i(M) sites away will be denoted by drX( . . . ,i(?), . . . ), where ? goes from mover #1 to #6, and i(?) is i(M) if ? is a bet-on mover, otherwise ‘-’ (a dash). For example, d2X(2,5,-,-,-,6) is a Draw 2 mixed bet with bet-on movers #1, #2 and #6, lying respectively 2, 5 and 6 sites away. Or, d3X(-,-,3,2,3,-) is a Draw 3 mixed bet with bet-on movers #3, #4 and #5, lying respectively 3, 2 and 3 sites away. The drS( . . . ,i(?), . . . ) bet has the winning probability of p=Π(prM) where multiplication is over all bet-on movers of Draw r.
Referring now to FIG. 4A, the identification of top with bottom border lines and left with right border lines allows us to assign any one of the twenty sites with 2-dimensional coordinates (x,y) and others with coordinates (x+i,y+j), where every calculation involving i or x is modulo 4 arithmetic, involving j or y is modulo 5 arithmetic. Now we can replace AA, AB, etc. by (x,y), (x,y+1) etc. respectively. —For example, AA:(0,0), AB:(0,1), AC:(0,2), AD:(0,3), AE:(0,4), BA:(1,0), BB:(1,1), BC:(1,2), BD:(1,3), BE:(1,4), CA:(2,0), CB:(2,1), CC:(2,2), CD:(2,3), CE:(2,4), DA:(3,0), DB:(3,1), DC:(0,2), DD:(0,3), DE:(0,4).—And we can also say that (x+i,y+j) lies (i,j) sites away from (x,y). A movement from (0,0) to (i,j) is equivalent to a movement from (x,y) to (x+i,y+j). Thus,
There is no other 1-movement path d1(i,j).
Let #d1(i,j) denote the number of all d1(i,j) paths from (0,0) to (i,j). Obviously, we have #d1(0,0)=#d1(0,1)=#d1(0,4)=#d1(1,0)=#d1(1,1)=#d1(1,4)=#d1(3,0)=#d1(3,1)=#d1(3.4)=1 and #d1(i,j)=0 for all other (i,j); in total 9.
p1M=#d1(i(M),j(M))/w is the probability of mover M from its start location to get on a d1(i(M),j(M)) path to reach the site 31 lying (i(M),j(M)) sites away. Here we use (i(M),j(M)) to specify (i,j) for the concerning M.
Let #d2(i,j) denote the number of all d2(i,j) paths from (0,0) to (i,j). #d2(i,j) is the sum of #d1(p,q)*#d1(i−p,j−q) over all p and p; explicitly, we have #d2(0,0)=9, #d2(0,1)=6, #d2(0,2)=3, #d2(0,3)=3, #d2(0,4)=6, #d2(1,0)=6, #d2(1,1)=4, #d2(1,2)=2, #d2(1,3)=2, #d2(1,4)=4, #d2(2,0)=6, #d2(2,1)=4, #d2(2,2)=2, #d2(2,3)=2, #d2(2,4)=4, #d2(3,0)=6, #d2(3,1)=4, #d2(3,2)=2, #d2(3.3)=2, #d2(3,4)=4; in total 81, that is 9ˆ2.
p2M#d2(i(M),j(M))/wˆ2 is the probability of mover M from its start location to get on a d2(i(M),j(M)) path to reach the site 41 lying (i(M),j(M)) sites away. Here we use (i(M),j(M)) to specify (i,j) for the concerning M.
Let d3(i,j) be any d1(x,y) followed by any d2(i−x,j−y), defining a 3-movement path from any site to a site lying (i,j) sites away.
Let #d3(i,j) denote the number of all d3(i,j) paths from (0,0) to (i,j). #d3(i,j) is the sum of #d1(p,q)*#d2(i−p,j−q) over all p and p; explicitly, we have #d3(0,0)=49, #d3(0,1)=42, #d3(0,2)=28, #d3(0,3)=28, #d3(0,4)=42, #d3(1,0)=49, #d3(1,1)=42, #d3(1,2)=28, #d3(1,3)=28, #d3(1,4)=42, #d3(2,0)=42, #d3(2,1)=36, #d3(2,2)=24, #d3(2,3)=24, #d3(2,4)=36, #d3(3,0)=49, #d3(3,1)=42, #d3(3,2)=28, #d3(3,3)=28, #d3(3,4)=42, in total 729, that is 9ˆ3.
p3M=d3(i(M),j(M))/wˆ3 is the probability of mover M from its start location to get on a d3(i(M),j(M)) path to reach the site 51 lying (i(M),j(M)) sites away. Here we use (i(M),j(M)) to specify (i,j) for the concerning M.
Let d4(i,j) be any d1(x,y) followed by any d3(i−x,j−y), defining a 4-movement path from any site to a site lying (i,j) sites away.
Let #d4(i,j) denote the number of all d4(i,j) paths from (0,0) to (i,j). #d4(i,j) is the sum of #d1(p,q)*#d3(i−p,j−q) over all p and q; explicitly, we have #d4(0,0)=399, #d4(0,1)=357, #d4(0,2)=294, #d4(0,3)=294, #d4(0,4)=357, #d4(1,0)=380, #d4(1,1)=340, #d4(1,2)=280, #d4(1,3)=280, #d4(1,4)=340, #d4(2,0)=380, #d4(2,1)=340, #d4(2,2)=280, #d4(2,3)=280, #d4(2,4)=340, #d4(3,0)=380, #d4(3,1)=340, #d4(3,2)=280, #d4(3,3)=280, #d4(3,4)=340; in total 6561, that is 9ˆ4.
p4M=d4(i(M),j(M))/wˆ4 is the probability of mover M from its start location to get on a d4(i(M),j(M)) path to reach the site 61 lying (i(M),j(M)) sites away. Here we use (i(M),j(M)) to specify (i,j) for the concerning M.
A Draw r simple bet, where r=1 to 4, on mover M lying each (i(M),j(M)) sites away will be denoted by dr((i(M),j(M)). It has winning probability p=prM.
A Draw r site bet, where r=1 to 4, on movers M lying each (i(M),j(M)) sites away from site S will be denoted by drS( . . . ,(i(?),j(?)), . . . ), where ? goes for each (i,j) from mover #1 to #6, and (i(?),j(?)) is (i(M),j(M)) if ? is a bet-on mover, otherwise ‘-’ (a dash). For example, d3BB(-,(2,3), (3,1),-,-,(0,4)) is a Draw 3 site bet on site BB with bet-on movers #2, #3 and #6, lying respectively (2,3),(3,1) and (0,4) sites away from site BB. Or, d4DA((3,3),(0,2),(1,1),-,(0,0),-) is a Draw 4 site bet on site DA with bet-on movers #1, #2, #3 and #5, lying respectively (3,3),(0,2),(1,1) and (0,0) sites away from site DA. The drS( . . . ,(i(?),j(?)), . . . ) bet has winning probability p=Π(prM) where multiplication is over all bet-on movers in site S of Draw r.
A Draw r mixed bet, where r=1 to 4, on movers M lying each (i(M),j(M)) sites away will be denoted by drX( . . . ,(i(?),j(?)), . . . ), where ? goes for each (i,j) from mover #1 to #6, and (i(?),j(?)) is (i(M),j(M)) if ? is a bet-on mover, otherwise ‘-’ (a dash). For example, d2X((3,2),(2,3),-,-,-, (1,1)) is a Draw 2 mixed bet with bet-on movers #1, #2 and #6, lying respectively (3,2),(2,3) and (1,1) sites away. Or, d3X(-,-,(0,3),(1,2),(3,4),-) is a Draw 3 mixed bet with bet-on movers #3, #4 and #5, lying respectively (0,3),(1,2) and (3,4) sites away. The drX( . . . ,(i(?),j(?)), . . . ) bet has winning probability p=Π(prM) where multiplication is over all bet-on movers of Draw r.
For all playing surfaces 10:
A 2-Draw chain bet with m bet-on movers has probability 1/wˆm to become hanging, and probability 1/wˆ2 m to win.
A 3-Draw chain bet with m bet-on movers has probability 1/wˆm to become hanging, and probability 1/wˆ2 m to remain hanging, and probability 1/wˆ3 m to win.
A 4-Draw chain bet with m bet-on movers has probability 1/wˆm to become hanging, and probability 1/wˆ2 m to remain hanging, and probability 1/wˆ3 m to remain hanging again, and probability 1/wˆ4 m to win.
The basic rule is that every $a bet with winning probability p pays $a/p. Every $a bet with probability p to become or remain hanging earns credit $a/p.
Due to unequal probabilities, it's laborious and unnecessary to find out all possible payouts of a plain bet ticket containing numerous bets. Anyone, including the computer, needs only to calculate the payout of a bet which eventually becomes winner.
Due to equal probabilities and single winner, it's convenient and practical to consider at once the payout and credit of all bets in a chain bet ticket.
In the following, let $b denote the total bet amount in a chain bet ticket. Let p1=Π(#31(M)/w), p2=Π(#41(M)/w), p3=Π(#61(M)/w), p4=Π(#61(M)/w), q2=Π(#41′(M)/w)−Π(#41(M)/w), q3=Π(#51′(M)/w)−Π(#51(M)/w), and q4=Π(#61′(M)/w)−Π(#61(M)/w).
Let r2, r3 and r4 denote respectively the selected Draw 2, 3 and 4 credit percentage.
A 2-Draw ticket has probability p1*p2 to win payout $b/(p1*p2).
A 3-Draw ticket has probability p1*p2*p3 to win payout $b/(p1*p2*p3).
A 4-Draw ticket has probability p1*p2*p3*p4 to win payout $b/(p1*p2*p3*p4).
Any chain ticket has probability p1 to earn Draw 2 credit $b/p1.
A 3- or 4-Draw ticket has probability p1*p2 to earn Draw 3 credit $b/(p1p2).
A 4-Draw ticket has probability p1*p2*p3 to earn Draw 4 credit $b/(p1p2*p3).
A revised 2-Draw ticket with #41(cr) r2-bets has probability q2 to win payout $b*0.01*r2/(p1*q2) and probability p2 to win payout $b*0.01*(100-r2)/(p1*p2).
A revised 3- or 4-Draw ticket with #41(cr)) r2-bets has probability q2 to win payout $b*0.01*r2/(p1*q2) and probability p2 to Draw 3 credit $b*0.01*(100-r2)/(p1*p2).
A revised 3- or 4-Draw ticket with #51(cr)) r3-bets has probability q3 to win payout $b0.01*r3/(p1*p2*q3) and probability p3 to win payout $b0.01*(100-r3)/(p1*p2*p3) or to earn that amount Draw 4 credit.
A revised 3- or 4-Draw ticket with #41(cr) r2-bets and #51(cr) r3-bets has probability q3 to win payout $b*0.01*(100-r2)*0.01*r3/(p1*p2*q3) and probability p3 to win payout $b0.01*(100-2)*0.01*(100-r3)/(p1*p2*p3) or to earn that amount Draw 4 credit.
A revised 4-Draw ticket with #41(cr) r2-bets and #61(cr) r4-bets has probability q4 to win payout $b*0.01*(100-r2)*0.01*r4/(p1*p2*p3*q4) and probability p4 to win payout $b0.01*(100-r2)*0.01*(100-r4)/(p1*p2*p3*p4).
A revised 4-Draw ticket with #51(cr)) r3-bets and #61(cr) r4-bets has probability q4 to win payout $b*0.01*(100-r3)*0.01*r4/(p1*p2*p3*q4) and probability p4 to win payout $b0.01*(100-r3)*0.01*(100-r4)/(p1*p2*p3*p4).
A revised 4-Draw ticket with #41(cr)r2-bets, #51(cr))r3-bets and #61(cr)r4-bets has probability q4 to win payout $b*0.01*(100-r2)*0.01*(100-r3)*0.01*r4/(p1*p2*p3*q4) and probability p4 to win payout $b0.01*(100-r2)*0.01*(100-r3)*0.01*(100-r4)/(p1*p2*p3*p4).
The ‘simple’ bet ticket as shown in FIG. 2B has in Draw 1 the probability of 2/7, 3/7, 2/7, 1/7, 5/7, 3/7 respectively to win $7 each on movers #1 to #6; in Draw 2 the probability of 5/7, 2/7, 2/7, 2/7, 3/7, 1/7 respectively to win $7 each on movers #1 to #6; in Draw 3 the probability of 2/7, 2/7, 3/7, 3/7, 3/7, 2/7 respectively to win $7 each on movers #1 to #6; and in Draw 4 the probability of 4/7, 1/7, 2/7, 2/7, 3/7, 2/7 respectively to win $7 each on movers #1 to #6.
The ‘simple’ bet ticket as shown in FIG. 3B has in Draw 1 the probability of 4/6, 3/6, 2/6, 5/6, 2/6, 3/6 respectively to win $12 each on movers #1 to #6. In Draw r, where r=2 to 4, every dr(i(M)) bet has probability p=#dr(i(M))/wˆr to win payout $2/p. For example, d2(8(3)) bet has p=3/6ˆ2; d3(4(4)) bet has p=22/6ˆ3; d4(2(2)) bet has p=148/6ˆ4.
The ‘simple’ bet ticket as shown in FIG. 4B has in Draw 1 the probability of 7/9, 0/9, 8/9, 6/9, 2/9, 4/9 respectively to win $9 each on movers #1 to #6. In Draw r, where r=2 to 4, every dr(i(M),j(M)) bet has probability p=#dr(i(M),j(M))/wˆr to win payout $1/p. For example, d2(0(2),1(2)) bet has p=6/9ˆ2; d3(2(5),1(5)) bet has p=36/9ˆ3; d4(3(6),3(6)) bet has p=280/9ˆ4.
Every ‘site’ bet in the ticket as shown in FIG. 2C has winning probability p=1/7ˆm, where m is the number of bet-on movers in the side, to win payout $2/p.
In the ‘site’ bet ticket as shown in FIG. 3C, every drS( . . . ,i(M), . . . ) bet, where r=1 to 4, has probability p=Π(prM), where multiplication is over all bet-on movers of Draw r, to win payout $2/p. For example, d1C(1,-,-,-,0,-) bet has p=#d1(1)*#d1(0)/6ˆ2=1/6ˆ2; d2H(6,0,8,3,0,0) bet has p=#d2(6)*#d2(8)*#d2(3)/6ˆ6=2*3*4/6ˆ6; d3F(-,-,-,-,7,0) bet has p=#d3(7)*#d3(0)/6ˆ6=16*25/6ˆ6; d4D(-,4,2,7,9,2) bet has p=#d4(4)*#d4(2)*#d4(7)*#d4(9)*#d4(2)/6ˆ20=135*148*112*124*148/6ˆ20.
In the ‘site’ bet ticket as shown in FIG. 4C, every drS( . . . ,(i(M),j(M)), . . . ) bet, where r=1 to 4, has probability p=Π(prM), where multiplication is over all bet-on movers in site S, to win payout $2/p. For example, d1 BB(-,-,-,(3,1),(1,0),-) bet has p=#d1(3,1)*#d1(1,0)/9ˆ2=1/9ˆ2; d2CB(-,-,(1,2),-, (2,0),-) bet has p=#d2(1,2)*#d2(2,0)/9ˆ4=2*6/9ˆ4; d3BA((1,2),-,-,-,-,(2,2)) bet has p=#d3(1,2)*#d3(2,2)/9 ˆ6=28*24/9ˆ6; d4DE((3,1),-,(2,0),-,-,(0,1)) bet has p=#d4(3,1)*#d4(2,0)*#d4(0,1)/9ˆ12=340*380*357/9ˆ12.
Every ‘mixed’ bet in the ticket as shown in FIG. 2D has probability p=1/7ˆm, where m is the number of bet-on movers of the concerning draw, to win payout $100/(622*p).
In the ‘mixed’ ticket as shown in FIG. 3D every drX( . . . ,i(M), . . . ) bet, where r=1 to 4, has probability p=Π(prM), where multiplication is over all bet-on movers of the concerning draw, to win payout $0,10/p. For example, d1X(0,3,1,3,8,2) bet has p=d1(0)*#d1(3)*#d1(1)*#d1(3)*#d1(8)*#d1(2)/6ˆ6=1/6ˆ6; d2X(1,4,8,2,7,2) bet has p=#d2(1)*#d2(4)*#d2(8)*#d2(2)*#d2(7)*#d2(2)/6ˆ12=6*3*5*2*5/6ˆ12; d3X(1,3,8,4,9,9) bet has p=#d3(1)*#d3(3)*#d3(8)*#d3(4)*#d3(9)*#d3(9)/6ˆ18=27*25*18*22*22*22/6ˆ18; d4X(8,2,9,3,0,2) bet has p=#d4(8)*#d4(2)*#d4(9)*#d4(3)*#d4(0)*#d4(2)/6ˆ24=115*148*124*144*135*148/6ˆ24.
In the ‘mixed’ ticket as shown in FIG. 4D every drX( . . . ,(i(M),j(M)), . . . ) bet, where r=1 to 4, has probability p=Π(prM), where multiplication is over all bet-on movers of that draw, to win payout $0,10/p. For example, d1X((0,4),-,(3,4),(3,0),(1,0),(3,0)) bet has p=#d1(0,4)*#d1(3,4)*#d1(3,0)*#d1(1,0)*#d1(3,0)/9ˆ5=1/9ˆ5. d2X(-,(3,4),(1,3),(0,0),(0,1),-) bet has p=#d2(3,4)*#d2(1,3)*#d2(0,0)*#d2(0,1)/9ˆ8=4*2*9*6/9ˆ8; d3X((0,2),(3,4),-,-,(1,0),(0,0)) bet has p=#d3(0,2)*#d3(3,4)*#d3(1,0)*#d3(0,0)/9ˆ12=28*42*49*49/9ˆ12; d4X((1,2),(2,4),(1,4),(3,2),-, (3,0)) bet has p=#d4(1,2)*#d4(2,4)*#d4(1,4)*#d4(3,2)*#d4(3,0)/9ˆ20=280*340*340*280*380/9ˆ20.
The 4-Draw ‘chain’ bet ticket as shown in FIG. 2E has probability p1=(4*4*5*5/7ˆ4)=400/2401 to become hanging and earn Draw 2 credit $200*p1=$1,200.50, and probabilities p2=3*2*4*4/7ˆ4=96/2401, p3=3*4*3*3/7ˆ4=108/2401, and p4=2*2*2*2/7ˆ4=16/2401 to remain hanging repeatedly, and to win payout $200/(p1*p2*p3*p4)=$100,166,700.86.
The revised 4-Draw ‘chain’ ticket as shown in FIG. 2F with #41(cr)=1008-96 credit 70% bets has probability q2=#41(cr)/7ˆ4=912/2401 to win payout $200*0.01*70/(p1*q2)=$2212.37, and probability p2=96/2401 to earn Draw 3 credit $200*0.01*(100-70)/(p1*p2)=$9,007.50.
The revised 4-Draw ‘chain’ bet ticket as shown in FIG. 2G with #51(cr)=1512-108 60% credit 70% bets has probability q3=#51(cr)/7ˆ4=1404/2401 to win payout $200*0.01*(100-70)*0.01*60/(p1*p2*q3)=$9,242.31 and probability p3 to earn Draw 4 credit $200*0.01*(100-70)*0.01*(100-60)/(p1*p2*p3)=$80,100.04.
The revised 4-Draw ‘chain’ bet ticket as shown in FIG. 2H with #61(cr)=2401-16 credit 80% bets has probability q4=#61(cr)/7ˆ4=2385/2401 to win payout $200*0.01*(100-70)*0.01*(100-60)*0.01*80/(p1*p2*p3*q4)=$64,509.92 and probability p4 to win payout $200*0.01*(100-70)*0.01*(100-60)*0.01*(100-80)/(p1*p2*p3*p4)=$10,020.01.
The 3-Draw ‘chain’ bet ticket as shown in FIG. 3E has probability p1=5*5*5*5*5/6ˆ5=3125/7776 to become hanging and earn Draw 2 credit $200*p1=$497.66, and then probabilities p2=4*4*4*4*4/6ˆ5=1024/7776, and p3=3*2*3*2*3/6ˆ5=108/7776 to remain hanging repeatedly, and to win payout $200/(p1*p2*p3)=$272,097.79.
The revised 3-Draw ‘chain’ bet ticket as shown in FIG. 3F with #41(cr)=7776-1024 credit 70% bets has probability q2=#41(cr)/6ˆ5=6752/7776 to win payout $200*0.01*70/(p1*q2)=$401.20, and probability p2 to earn Draw 3 credit $200*0.01*(100-70)/(p1*p2)=$113.37.
The revised 3-Draw ‘chain’ bet ticket as shown in FIG. 3G with #51(cr)=7776-108 credit 80% bets has probability q3#51(cr)/6ˆ5=7668/7776 to win payout $200*0.01*(100-70)*0.01*80/(p1*p2*q3)=$919.77 and probability p3 to win payout $200*0.01*(100-70)*0.01*(100-80)/(p1*p2*p3)=$16,325.87.
The 3-Draw ‘chain’ bet ticket as shown in FIG. 4E has probability p1=8*8*8*7/9ˆ4=4704/6561 to become hanging and earn Draw 2 credit $200/p1=$278.95, and probabilities p2=7*7*6*6/9ˆ4=1764/6561, and p3=6*5*5*5/9ˆ4=750/6561 to remain hanging repeatedly, and to win payout $200/(p1*p2*p3)=$9,076.39
The revised 3-Draw ‘chain’ bet ticket as shown in FIG. 4F with #41(cr)=5832-1764 credit 60% bets has probability q2=#41(cr)/9ˆ4=4068/6561 to win payout $200*0.01*60/(p1*q2)=$269.94, and probability p2 to earn Draw 3 credit $200*0.01*(100-60)/(p1*p2)=$415.02.
The revised 3-Draw ‘chain’ bet ticket as shown in FIG. 4G with #51(cr)=5184-750 credit 70% bets has probability q3=#51(cr)/9ˆ4=4434/6561 to win payout $200*0.01*(100-60)*0.01*70/(p1*p2*q3)=$429.87 and probability p3 to win payout $200*0.01*(100-60)*0.01*(100-70)/(p1*p2*p3)=$1,089.17.
To start the very first game, the operator displays a playing surface 10 with movers 12 in arbitrarily chosen sites 11 as shown in FIG. 2 to 4. The bettors use bet slips as shown in FIGS. 2A, 3A, 3AA, 4A and 4AA to place bets until the operator signals to execute the first round of random drawing of ruled movements, called Draw #1, one movement draw for each mover. The outcomes will be displayed on the monitor, and input into the computer to determine if any bet ticket contains selections matching the outcomes so that its holder can obtain payout or credit. Next, anyone, whether having placed bets before Draw #1 or not, can place bets just like before Draw #1. Besides, it is an option that the hanging ticket holder can place credit bets. The operator executes the next round of drawing, called Draw #2, for all movers. The outcomes will be displayed and data processed just like after Draw #1. As the flowchart in FIG. 1 shows, the above steps repeat twice so that any bettor starting with a 4-Draw bet at the very beginning can complete the game. Unless pause or stop has been regulated ahead, the operator will let it go on indefinitely, while any player may start or stop betting anytime. The Draw # will grow accordingly. But players don't need to pay attention to it. For the sake of reference, ‘Draw 1 is Draw # so and so’ will be printed on every bet ticket. A regulated stop must allow every existing bet ticket to reach final results.
To play the automatic game, one needs either a video game machine or a personal computer equipped with made-to-order software inclusive random number generator to take care of drawing ruled movements. The computer is connected to a pointing device or touch screen monitor so that the action ‘select’ below can be executed by means of the pointing device or finger touching. Selecting any item on the display screen will either highlight it or result in a new display. Selecting a highlighted item is to cancel that selection. In the non-automatic game, all figures printed on paper are supposed to be black, white and gray. Now, on monitor they can be quite colorful.
The game starts with the display of a playing surface as shown in FIGS. 2, 3 or 4 with additional icons/items named “Another playing surface”, “Bet slip” and “Account”.
Selecting “Another playing surface” will result in the display of another one. All playing surfaces as shown in FIGS. 2 to 4, or maybe some one not given here, will be displayed cyclically one after another if the selection of “Another playing surface” continues.
Selecting “Bet slip” will display a bet slip as shown in FIGS. 2A, 3A, 3AA, 4A or 4AA with additional icons “Ticket” and “Account”; and “Alternative slip” if playing surface as shown in FIGS. 3 or 4 is used.
Selecting “Alternative slip” will switch to a chain bet slip if the displayed one is for plain bet, or conversely.
The player places bets on screen just as on paper in the non-automatic game; then selects “ticket” to submit. If the submitted slip is incomplete or contains error, there will be a message like ‘Incomplete! Please select per bet amount.’, requiring the player to make amendment. If the submission is approved, a bet ticket without a ticket number as shown in FIGS. 2B to 2D, 3B to 3D or 4B to 4D with additional icons “Go back”, “Ticket #”, “Cancel”, “Bet slip”, “Draw”, “Account” shows up.
Facing a bet ticket the player must select “Ticket #”, “Go back” or “Cancel”. Otherwise, there will be a message to remind the player to do so. Selecting “Ticket #” finalizes the bet so that a certain ticket number will be issued and shown on the ticket. Selecting “Go back” allows the player make changes on the submitted bet slip. Selecting “Cancel” is to abandon the submission.
After ‘Ticket # so and so’ or ‘Cancelled’ being displayed, the player can select “Bet slip”, “Draw”, or “Account”.
Selecting “Bet slip” will display a blank one to take bet.
Selecting “Draw” will cause one round of drawing followed by an outcome display as shown in FIGS. 2 to 4 with an additional icon “Account”. At the same time, the computer will update and process all bet ticket data.
Selecting “Account” will result in a display as shown in FIG. 5. It shows the available balance, and all activities since the start of the game or the opening of that account. There is one account for each playing surface.
Here the player can select “Ticket # so and so” to view that ticket as well as to use it for placing credit bets just as in the non-automatic game.
“Playing surface”, “Bet slip” or “Draw” allows the player to continue in whichever way preferred, while “Exit” to end the game.
The invention described above provides an extremely low operation cost game to be easily run by an existing or future keno/lottery kind of operator. The automatic version can be easily integrated into some existing video game machines.
Although the derivation of some probabilities has been performed by means of modular arithmetic, neither the operator nor any player needs to understand it. Since all #d1(i) to #d4(i) and #d1(i,j) to #d4(i,j) are explicitly provided, no one including the computer will be required to do any modular arithmetic. Besides, I also give some numerical examples to help everyone get acquaintance with practical calculations.
This is a betting game of manifold probability. Thus, besides the winning probability of every single bet, I have also shown the calculation of payout of a whole bet ticket. The operator can easily modify the payout by house edge, which should be on a whole ticket instead of each single bet, and based on the ratio of payout to total bet amount so that lower ratio tickets enjoy lower house edges.
To make the bettor without regret, every hanging bet earns credit equal to the payout of a bet which is up to that point equivalent to the hanging one, and just ends there. The operator can make house edge effective simply on the final payout only. Charging house edge only on final payout makes purchasing a multi draw ticket more incentive than purchasing tickets draw by draw.
There are occasions that skillful players would like to place make-up bets not using credit. Due to the existence of house edge, the operator always welcomes any additional bets.
Here is a game operative under one management with both house bankroll and pari-mutuel jackpot. The operator may rule that a bet of certain huge payout with extremely tiny winning probability is a jackpot bet, and that the bettor can choose whether to be a jackpot participant or not. For all non-jackpot wagering, printed house edge formulas must be available all the times.
Naturally, setting house edges is not inventor's business, but I would like to give a sample set for reference:
Let x be the ratio of payout to total bet amount, and e % be house edge.
e=x for 0<x<=5
e=4.5+(x−5)/10 for 5<x<=10,
e=4+n(n+1)/2+(n+1)*(x−10ˆn)/9 for 10ˆn<x<=10ˆ(n+1) where n is a non-zero natural number.
Every ticket of total bet amount $b>$10 gets an f % discount on e, where f=4*log(b−10), and the logarithm base is 10. Thus, the actual payout is $b*(100-e*(100-4*log(b−10))%)% unless there is a minimal service charge.
Besides, the operator can always by the way run contest as follows: Anyone paying an entry fee gets a non-cashable voucher for say $1M to play. The player must make a number of certain kinds of bets, including some credit ones. Every payout will be added to the voucher. Reaching a certain winning results will grant the player a prize which may include some percentage of the voucher. The computer handles contestants just like regular bettors.
Before a concerning draw takes place, it doesn't matter when a selection is made or changed. Thus, it can be an option that the bettor is allowed to change selections any time before the draw, or that the bettor may purchase a ticket stating the number of certain kinds of bets without detailed selections, and submit the details anytime ahead of the concerning draw. Computer random detailed selections may also be made available as an option.
There are only three similar playing surfaces with ruled movements given here. Obviously, the method can be applied to many other similar playing surfaces with other similar ruled movements. The number of sites and movers can easily be made different from those given above. Other types of betting can be added into the game. They can also be modified by some racing characteristics such as mover A reaching site B ahead of mover C reaching site D. Chain bets can be more than four draws.
Thus, the scope of the invention should be determined by the appended claims and their legal equivalents, rather than by examples given. and thus has a chance to be a winner at the end. The game provides the option that the holder of a hanging ticket receives credit to place free make-up bets, which will be henceforth referred to as credit bets.
Probability formulae as well as how to calculate payouts and credits will be provided.
The invention also includes an automatic video/computer version of the game.
FIG. 1 is a flowchart illustrating the game process.
FIG. 2 is a Win/Place/Show bet slip.
FIG. 2A is a 3-Race Win/Place/Show bet ticket.
FIG. 2B is a Race 2 revised 3-Race Win/Place/Show bet ticket.
FIG. 2C is a Race 3 revised 3-Race Win/Place/Show bet ticket.
FIG. 3 is a Win/Exacta/Tricta bet slip.
FIG. 3A is a 3-Race Win/Exacta/Tricta bet ticket
FIG. 3B is a Race 2 revised 3-Race Win/Exacta/Tricta bet ticket.
FIG. 3C is a Race 3 revised 3-Race Win/Exacta/Tricta bet ticket.
FIG. 4 shows a display of random numbers.
FIG. 5 shows a betting activity statement.
FIG. 6 shows a Win/Place/Show probability table.
FIG. 7 is a Win-Win probability table.
FIG. 8 is an Exacta probability table.
FIGS. 9A and 9B are two parts of a Tricta probability table.
As shown in FIGS. 2 and 3, the invention provides a race course 10 as playing surface, on which there are nine racers 12, all shown on every bet slip. The race course contains seventeen numbered strips 11 called lines. Line 0 is where all racers are located. Each line except lines 0 contains one spot F lying straight in front of every racer. All and only those F lying straight ahead of a racer form the racetrack for that racer to advance. The player marks to select one F for each racer as finish spot/line. After a round of nine random numbers being generated, racers will advance one after another, staring from racer 1, as many spots as random numbers indicating. The race ends when three racers have reached selected finish lines. Since a racer's advancement is equivalent to the retreat of its finish line, to record a race, it's practical, to keep all racers in line 0, and reset their finish lines based on advancements. case of Exacta bets, every selected racer i in spot 13 and every selected racer j in spot 14 with i≠j will be combined to form a 13(i)14(j) bet, which wins if racer i finishes first and racer j finishes second. If the bettor wants only specific combinations of selected spots 13 with 14 instead of all possible, then it is necessary to use separate slips. —For example, using one bet slip, you can bet racers 1 or 2 finishing first and racers 3 or 4 finishing second, This is four bets. If you want to bet EITHER racer 1 finishing first and racer 3 finishing second OR racer 3 finishing first and racer 4 finishing second. This is two bets; and you need to use two bet slips to place them separately.—In the case of Tricta bets, every selected racer i in spot 13 and every selected racer j in spot 14 and every selected racer k in spot 15 with i≠j≠k≠i will be combined to form a 13(i)14(j)15(k) bet. All bets on a slip can bring in one winner only.
To place 2-race Win/Exacta/Tricta bets, the bettor does first just as placing 1-race bets; then marks to select spots 23, 24, and/or 25. If one spot 25 is selected, then at least one spot 24 must be selected. If one spot 24 is selected, then at least one spot 23 must be selected. Every selection extends all bets placed in Race 1. If the bettor wants only specific Race 1 bets to be extended in specific ways instead of all possible, then it is necessary to use separate slips. In the case of only spots 23 being selected, every selected i′ in spot 23 will form Win-Win 13(i)23(i′) bets, Exacta-Win 13(i)14(j)23(i′) bets, and/or Tricta-Win 13(i)14(j)15(k)23(i′) bets. In the case of only spots 23 and 24 being selected, every selected racer i′ in spot 23 and selected racer j′ in spot 24 where i≠j′ will form Win-Exacta 13(i)23(i′)24(j′) bets, Exacta-Exacta 13(i)14(j)23(i′)24(j′) bets, and/or Tricta-Exacta 13(i)14(j)15(k)23(i′)24(j′) bets. If the bettor wants only specific combinations of selected racer i′ with racer j′ instead of all possible, then it is necessary to use separate slips. In the case of spot 25 being selected, every selected racer i′ in spot 23 and selected racer j′ in spot 24 and selected racer k′ in spot 25 where i′≠j′≠k′≠i′ will form Win-Tricta 13(i)23(i′)24(j′)25(k′) bets, Exacta-Tricta 13(i)14(j)23(i′)24(j′)25(k′) bets, and/or Tricta-Tricta 13(i)14(j)15(k)23(i′)24(j′)25(k′) bets. If the bettor wants only specific combinations of selected racer i′ with racer j′ and racer k′ instead of all possible, then it is necessary to use separate slips. Regardless of bet type, let #Race1 and #Race2 denote the number of bets in Race 1 and Race 2 respectively. The total number of 2-Race bets is Race1*#Race2. All bets on a slip can bring in one winner only.
To place 3-race Win/Exacta/Tricta bets, the bettor does first just as placing 2-race bets; then marks to select spots 33, 34, and/or 35. If one spot 35 is selected, then at least one spot 34 must be selected. If one spot 34 is selected, then at least one spot 33 must be selected. Every selection extends all bets placed in Races 1 and 2. If the bettor wants only specific Races 1 and 2 bets to be extended in specific ways instead of all possible, then it is necessary to use separate slips. In the
Here is a game operative under one management with both house bankroll and pari-mutuel jackpot. The operator may rule that a bet of certain huge payout with tiny winning probability is a jackpot bet, or that the bettor can choose whether to be a jackpot participant or not. For all non-jackpot wagering, printed precise house edge formulas must be available all the times.
Naturally, setting house edges is not inventor's business, but I would like to give a sample set for reference:
Let x be the ratio of payout to total bet amount, and e % be house edge.
e=2+x/2 for 0<x<=5,
e=4.5+(x−5)/10 for 5<x<=10,
e=4+n(n+1)/2+(n+1)*(x−10ˆn)/9 for 10ˆn<x<=10ˆ(n+1) where n is a non-zero natural number.
Every ticket of total bet amount $b>$10 gets an f % discount on e, where f=4*log(b−10), and the logarithm base is 10. Thus, the actual payout is $b*x*(100-e*(100-4*log(b−10))%)%/o unless there is a minimal service charge.
Following the step by step derivation of probability formulae one can easily set up games similar to the one precisely described above. First, rolled numbers are not necessarily 1 to 6, it can be 0 to 5 or any other positive or negative integers where negative ones mean backward motion. Second, a roll is not necessarily to generate six numbers, it can be more or less. One can similarly define (n,s)-sequences, R(n,s), stand-by and winning sequences, and P(n,s) etc. Third, since Tr(i) is a variable, the maxim track length is not necessarily equal to 16. Fourth, the number of racers is not necessarily 9. If it is q instead of 9, then in A(n,i)=P(n,Tr(i))*ch(n,i) we let i goes from 1 to q instead of 1 to 9; —everywhere modulo 9 becomes modulo q—. One can similarly form Bm(n,i,j) and Cm(n,i,j,k) to calculate all kinds of probabilities for racers i, j, k finishing first, second and third. Besides, in the same art of forming Cm(n,i,j,k), one can form Dm(n,i,j,k,l), Em(n,i,j,k,l,h) etc. to calculate the probability of a racer finishing 4th, 5th, etc. so that betting can be more exotic than Tricta. And, multi races can be 4-race, 5-race, etc.
The rule that only regular bets earn credit can be changed so that, for example, the bettor may choose whether credit bets also earn credit instead of payout.
Thus, the scope of the invention should be determined by the appended clams and their legal equivalents, rather than by examples given.
e=2+x/2 for 0<x<=5
e=4.5+(x−5)/10 for 5<x=10,
e=4+n(n+1)/2+(n+1)*(x−10ˆn)/9 for 10ˆn<x<=10ˆ(n+1) where n is a non-zero natural number.
Every ticket of total bet amount $b >$10 gets an f % discount on e, where f=4*log(b−10), and the logarithm base is 10. Thus, the actual payout is $b*x*(100-e*(100-4*log(b−10))%)% unless there is a minimal service charge.
Besides, the operator can always by the way run contest as follows: Anyone paying an entry fee gets a non-cashable voucher for say $1M to play. The player must make a number of certain kinds of bets, including some credit ones. Every payout will be added to the voucher. Reaching a certain winning results will grant the player a prize which may include some percentage of the voucher. The computer handles contestants just like regular bettors.
Before a concerning draw takes place, it doesn't matter when a selection is made or changed. Thus, it can be an option that the bettor is allowed to change selections any time before the draw, or that the bettor may purchase a ticket stating the number of certain kinds of bets without detailed selections, and submit the details anytime ahead of the concerning draw, Computer random detailed selections may also be made available as an option.
There are only three similar playing surfaces with ruled movements given here. Obviously, the method can be applied to many other similar playing surfaces with other similar ruled movements. The number of sites and movers can easily be made different from those given above. Other types of betting can be added into the game. They can also be modified by some racing characteristics such as mover A reaching site B ahead of mover C reaching site Do Chain bets can be more than four draws.
Thus, the scope of the invention should be determined by the appended claims and their legal equivalents, rather than by examples given.