This proposed design may not be used on any device without
the written consent of the author (Bill Butler).
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
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
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
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
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.
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
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: firstname.lastname@example.org
(Remove the "w"'s for a valid E-mail)
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