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"Shut The Box" Analysis

Casino Style Shut the Box

(game design by Bill Butler)

This proposed design may not be used on any device without  the written consent of the author (Bill Butler).

Picture of a casino  With a few modifications, Shut the Box could be modified into a game that appears to be perfect for casino (gaming, gambling, wagering, betting)  operations. Features would include: simple rules, high jackpot potential, fast execution, and the (false) illusion that it is not difficult to win. Also, it would have an intragame option that would allow the player to either cash out at the current fair value (and forfeit the remainder of the game), or continue in hopes of winning a large jackpot (e.g. $1,000,000). This feature is not offered on any other current casino game.

Actual play would be similar to the traditional game except:
1) The numbers (boxes) 1 through 12 are used instead of 1 through 9.
2) There is no single die option. (The player always uses two dice.)
3) The dice are weighted so that 1's, 2's, and 3's are more frequent and 4's, 5's and 6's are less frequent.
For electronic versions, this weighting is done by the machine. (i.e. The game would be played on a slot machine type device.)

   Perfect games that flip all 12 numbers (sum of the flipped numbers = 78) would be extremely rare, but payouts would have a sliding scale for partial results similar to the payout tables for Keno.

   The casino could set the payout table and dice weighting for each slot machine. These settings control the size of player winnings, the frequency of player winnings, and the expected rate of return for the casino. Existing computer programs can instantly calculate the expected results of any combination of these variables, and the following example is typical:

Sample Game

   First, the dice are weighted so that the 4, 5, and 6 on each die occur 75 percent less often than normal and the 1, 2, and 3 are 75 percent more frequent than normal. This is equivalent to having a 24-sided die with seven 1's, seven 2's, seven 3's, one 4, one 5, and one 6. It is probably advantageous to the casino to display this dice weighting as it gives the illusion of making the game only somewhat more difficult while in reality it is now extremely difficult to win a large jackpot.

   Then we define a payout table for a sample $1.00 game. The player's end-of-game score (sum of the flipped digits) is used to index into the following payout table.

    Score          Payout
   0 to 29        Nothing
  30 to 34          $1.00
  35 to 39           2.00
  40 to 44          10.00
  45 to 49          20.00
  50 to 54          50.00
  55 to 59         100.00
  60 to 64       1,000.00
  65 to 69      10,000.00
  70 to 74     250,000.00
  75 to 78   1,000,000.00

   To start the game, the player rolls the dice and uses the sum to flip (or otherwise mark) any of the numbers 1 through 12 as used. (The sum of the flipped numbers will be equal to the dice total.) The player then repeatedly rolls the dice and flips additional numbers until there are no usable combinations for the current dice roll. The sum of the previously flipped numbers is then used to index into the above table to see what payout (if any) is paid to the player by the casino.

   The actuarial value of this particular $1.00 game is $0.961868 to the player and $0.038132 to the casino. The player will actually win something 13.22 percent of the time, but $1,000,000 payouts (via any of the four possible scores) will hit only once every 4,761,905 games (long term average). In most games, the player will run out of usable combinations in five or less dice rolls, and would then have to wager another dollar for the next game.

   An additional option exists if the player manages to roll several of the more difficult dice rolls at any point in the game. For example, assume a player had rolls of 8, 9, 11, and 12 (and flipped the corresponding numbers). The long term actuarial value of this position is $10,447.02 even though the sum of the numbers (so far) is only 40. The player would have the option of cashing in the game at this point and receive payment of $10,447.02, or he could continue the game. The actuarial long-term value to players and the casino is not changed by whatever this decision might be. It is interesting to note that no other casino game gives this option to a player.

   If any party is interested in manufacturing this casino style game (or would like more information with no obligation (financial, rights, or other)) , please contact Bill Butler at:

(Remove the "w"'s for a valid E-mail)

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