"Shut The Box" Analysis
Mathematical Analysis and Optimal Strategy for Several
Variations of the Dice Game "Shut The Box"
Game Rules:
The game consists of a pair of dice and markers (boxes) for the digits
1 to 9. A player rolls the dice and uses the total in any way that he
sees fit to "shut" or otherwise mark one or more of the digits (boxes)
as used. For example, if the dice total is 9, any of the following
combinations may be used: 9, 1&8, 2&7, 3&6, 4&5,
1&2&6, 1&3&5, 2&3&4.
Then the dice are rolled again and the new total is used to mark additional digits as used. However, each digit may be used only once with the result the choices become increasingly constricted as the game progresses. If a player has remaining digits totaling 6 or less (sum of the flipped digits >= 39), then only one die is used. The game ends when the current dice (die) total has no remaining combinations that can be used. A perfect game occurs if all nine digits (sum = 45) are flipped.
Scoring: Scoring for the game consists of adding together all the flipped (used) digits. If all nine digits are used, then their sum is 45, which represents the best possible score.
Casino style payouts are used with a default payout of 100 awarded for a perfect score of 45. All other scores are awarded zero. This payout schedule optimizes the game for an "all or nothing win", and the calculated "Expected Value" for any mid-game position becomes the probability (in percent) that the game will be won given the current status of the flipped digits. Alternately, smaller payouts could be awarded for "close but not quite". (Changing the payout table also changes the optimal strategy).
Game Variations: In addition to the standard game described above, there are several variations of "Shut the Box". In one variation, the player always uses two dice. Another variation uses one die if the sum of the flipped digits is >= 39, but the number of digits that can be flipped on any dice roll is limited to two. Thus, if the player rolls a "9" the combinations 1&2&6, 1&3&5, 2&3&4 can not be used.
Finally, it is possible to design a high-jackpot casino style game (use real money).
Statistics Data and Graph: The Statistical Data link displays a table showing the probability of any given final (end of game) score assuming the player is using optimal strategy to maximize the winning probability. The above graph summarizes the probabilities for all three different game variations.
If you are using the “use 1 die at 39 or higher game option”, the red line shows the probabilities for ending at any given score assuming that you are using optimal strategy to play for a win. Using optimal strategy, there is a 9.7614% chance that you will win. If you always use two dice (blue line), there is only a 7.1432% chance that you will win. Note that without the “1 die” option, there are higher probabilities of getting stuck just short of winning. Finally, the green line shows the probilities if you can never flip more than two boxes at a time, but are allowed to shift to 1 die if your score is 39 or higher. Here, there is a 7.4194% chance of winning.
Optimal Strategies: The optimal strategy pages for the Standard Game, Always use two dice, and Max 2 flips variations show the best flip strategy given any interim status of flipped digits and the current dice roll. The "Expected Value" for each position shows the probability of winning (expressed as a percent) given the current combination of flipped digits. (Allow a minute to download the table.)
Casino Style "Shut the Box": This link describes a possible variation of the game that could be played using real money in a high-jackpot casino game (or possibly a web based equivalent). (Any manufacturer or web developer interested?)
Another variation of Shut the Box uses 12 boxes instead of 9. There are no standard rules for the 12 box version, but the analysis given here assumes that you always use 2 dice. As shown below, it is much more difficult to shut all 12 boxes. The probability of winning is only 0.3622 percent. (One chance in 276.) A possible variation of the game might instead use 3 dice until a score of 50 is reached, then use 2 dice up through a score of 71, and then use 1 die if your score is 72 or higher.
The graph above shows the probability of various outcomes for the 12-box game. (Assumes that you are using optimal strategy to maximize your chance of winning the game.) The dashed line shows the probability that your game will end by the time you get to the indicated score. For example, there is about an 83.355 percent chance that your game will end on or before the total of your flipped boxes reaches the spike at 55.
The solid line shows the probability that your game will end at a specific score. (The score is the sum of the flipped boxes.) For example, if you are using optimal strategy to try to win the game, there is a 3.83 percent probability that your game will end when the sum of your flipped boxes is exactly 45.
The spikes in the probabilities for scores of 36, 45, 55, and 66 are of interest. First of all, the spikes are real and not computational or plotting errors. The theoretical probabilities (plotted above) were tested by running tens of millions of random games using different random dice sequences. The random tests produced similar spikes. (At 10,000,000 games the standard statistical test of +/- 2 standard deviations has a magnitude of about 0.01 on the left graph axis.) The probability spikes also show up at the same locations when the dice are weighted and the “Payoff Table” is set for the previously mentioned “Casino Style Shut the Box” game.
We note that 36 is the sum of the digits from 1 through 8, 45 is sum of the digits from 1 through 9, etc. If you toss a pair of dice, two out of three times you will get a 5, 6, 7, 8, or 9. It is easy to use these results to flip boxes with mid-sized or low numbers. It is significantly more difficult to flip higher numbered boxes. Thus the game tends to end after you have flipped all the low to mid sized numbers, and never have a chance to flip the higher numbers.
There is still another variation of Shut the Box that uses 10 boxes. As with the 12-box version, there are no standard rules for the 10 box version. Again, the analysis given here assumes that you always use 2 dice. It is more difficult to win the 10 box game than the 9-box game, but it is no where near as difficult as the 12 box game. If you use optimal strategy, the probability of winning is 4.0361 percent. (About one chance in 25.)
The graph above shows the probability of various outcomes for the 10-box game. (Assumes that you are using optimal strategy to maximize your chance of winning the game.) The dashed line shows the probability that your game will end by the time you get to the indicated score. For example, there is about a 74.625 percent chance that your game will end on or before the total of your flipped boxes reaches 45.
The solid line shows the probability that your game will end at a specific score. (The score is the sum of the flipped boxes.) For example, if you are using optimal strategy to try to win the game, there is a 4.665 percent probability that your game will end when the sum of your flipped boxes is exactly 45.
As we observed in the 12-box game, spikes in the probable results show up at 36 and 45. The spike at 36 indicates that you may have flipped all the boxes from 1 to 8, but you probably were not able to flip boxes 9 or 10. The spike at 45 is similar, but it is the sum of flipping boxes 1 to 9.
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Then the dice are rolled again and the new total is used to mark additional digits as used. However, each digit may be used only once with the result the choices become increasingly constricted as the game progresses. If a player has remaining digits totaling 6 or less (sum of the flipped digits >= 39), then only one die is used. The game ends when the current dice (die) total has no remaining combinations that can be used. A perfect game occurs if all nine digits (sum = 45) are flipped.
Scoring: Scoring for the game consists of adding together all the flipped (used) digits. If all nine digits are used, then their sum is 45, which represents the best possible score.
Casino style payouts are used with a default payout of 100 awarded for a perfect score of 45. All other scores are awarded zero. This payout schedule optimizes the game for an "all or nothing win", and the calculated "Expected Value" for any mid-game position becomes the probability (in percent) that the game will be won given the current status of the flipped digits. Alternately, smaller payouts could be awarded for "close but not quite". (Changing the payout table also changes the optimal strategy).
Game Variations: In addition to the standard game described above, there are several variations of "Shut the Box". In one variation, the player always uses two dice. Another variation uses one die if the sum of the flipped digits is >= 39, but the number of digits that can be flipped on any dice roll is limited to two. Thus, if the player rolls a "9" the combinations 1&2&6, 1&3&5, 2&3&4 can not be used.
Finally, it is possible to design a high-jackpot casino style game (use real money).
Statistics Data and Graph: The Statistical Data link displays a table showing the probability of any given final (end of game) score assuming the player is using optimal strategy to maximize the winning probability. The above graph summarizes the probabilities for all three different game variations.
If you are using the “use 1 die at 39 or higher game option”, the red line shows the probabilities for ending at any given score assuming that you are using optimal strategy to play for a win. Using optimal strategy, there is a 9.7614% chance that you will win. If you always use two dice (blue line), there is only a 7.1432% chance that you will win. Note that without the “1 die” option, there are higher probabilities of getting stuck just short of winning. Finally, the green line shows the probilities if you can never flip more than two boxes at a time, but are allowed to shift to 1 die if your score is 39 or higher. Here, there is a 7.4194% chance of winning.
Optimal Strategies: The optimal strategy pages for the Standard Game, Always use two dice, and Max 2 flips variations show the best flip strategy given any interim status of flipped digits and the current dice roll. The "Expected Value" for each position shows the probability of winning (expressed as a percent) given the current combination of flipped digits. (Allow a minute to download the table.)
Casino Style "Shut the Box": This link describes a possible variation of the game that could be played using real money in a high-jackpot casino game (or possibly a web based equivalent). (Any manufacturer or web developer interested?)
12-Box Shut the Box
Another variation of Shut the Box uses 12 boxes instead of 9. There are no standard rules for the 12 box version, but the analysis given here assumes that you always use 2 dice. As shown below, it is much more difficult to shut all 12 boxes. The probability of winning is only 0.3622 percent. (One chance in 276.) A possible variation of the game might instead use 3 dice until a score of 50 is reached, then use 2 dice up through a score of 71, and then use 1 die if your score is 72 or higher.
The graph above shows the probability of various outcomes for the 12-box game. (Assumes that you are using optimal strategy to maximize your chance of winning the game.) The dashed line shows the probability that your game will end by the time you get to the indicated score. For example, there is about an 83.355 percent chance that your game will end on or before the total of your flipped boxes reaches the spike at 55.
The solid line shows the probability that your game will end at a specific score. (The score is the sum of the flipped boxes.) For example, if you are using optimal strategy to try to win the game, there is a 3.83 percent probability that your game will end when the sum of your flipped boxes is exactly 45.
The spikes in the probabilities for scores of 36, 45, 55, and 66 are of interest. First of all, the spikes are real and not computational or plotting errors. The theoretical probabilities (plotted above) were tested by running tens of millions of random games using different random dice sequences. The random tests produced similar spikes. (At 10,000,000 games the standard statistical test of +/- 2 standard deviations has a magnitude of about 0.01 on the left graph axis.) The probability spikes also show up at the same locations when the dice are weighted and the “Payoff Table” is set for the previously mentioned “Casino Style Shut the Box” game.
We note that 36 is the sum of the digits from 1 through 8, 45 is sum of the digits from 1 through 9, etc. If you toss a pair of dice, two out of three times you will get a 5, 6, 7, 8, or 9. It is easy to use these results to flip boxes with mid-sized or low numbers. It is significantly more difficult to flip higher numbered boxes. Thus the game tends to end after you have flipped all the low to mid sized numbers, and never have a chance to flip the higher numbers.
10-Box Shut the Box
There is still another variation of Shut the Box that uses 10 boxes. As with the 12-box version, there are no standard rules for the 10 box version. Again, the analysis given here assumes that you always use 2 dice. It is more difficult to win the 10 box game than the 9-box game, but it is no where near as difficult as the 12 box game. If you use optimal strategy, the probability of winning is 4.0361 percent. (About one chance in 25.)
The graph above shows the probability of various outcomes for the 10-box game. (Assumes that you are using optimal strategy to maximize your chance of winning the game.) The dashed line shows the probability that your game will end by the time you get to the indicated score. For example, there is about a 74.625 percent chance that your game will end on or before the total of your flipped boxes reaches 45.
The solid line shows the probability that your game will end at a specific score. (The score is the sum of the flipped boxes.) For example, if you are using optimal strategy to try to win the game, there is a 4.665 percent probability that your game will end when the sum of your flipped boxes is exactly 45.
As we observed in the 12-box game, spikes in the probable results show up at 36 and 45. The spike at 36 indicates that you may have flipped all the boxes from 1 to 8, but you probably were not able to flip boxes 9 or 10. The spike at 45 is similar, but it is the sum of flipping boxes 1 to 9.
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