A die (plural ``dice'') is a Solid with markings on each of its faces. The faces are usually all the same shape, making Platonic Solids and Archimedean Solid Duals the obvious choices. The die can be ``rolled'' by throwing it in the air and allowing it to come to rest on one of its faces. Dice are used in many games of chance as a way of picking Random Numbers on which to bet, and are used in board or role-playing games to determine the number of spaces to move, results of a conflict, etc. A Coin can be viewed as a degenerate 2-sided case of a die.

The most common type of die is a six-sided Cube with the numbers 1-6 placed on the faces. The value of the roll is indicated by the number of ``spots'' showing on the top. For the six-sided die, opposite faces are arranged to always sum to seven. This gives two possible Mirror Image arrangements in which the numbers 1, 2, and 3 may be arranged in a clockwise or counterclockwise order about a corner. Commercial dice may, in fact, have either orientation. The illustrations below show 6-sided dice with counterclockwise and clockwise arrangements, respectively.

The Cube has the nice property that there is an upward-pointing face opposite the bottom face from which the value of
the ``roll'' can easily be read. This would not be true, for instance, for a Tetrahedral die, which
would have to be picked up and turned over to reveal the number underneath (although it could be determined by noting which
number 1-4 was *not* visible on one of the upper three faces). The arrangement of spots
corresponding to a roll of 5 on a six-sided die is called the Quincunx. There are also special names for certain rolls
of two six-sided dice: two 1s are called Snake Eyes and two 6s are called Boxcars.

Shapes of dice other than the usual 6-sided Cube are commercially available from companies such as Dice & Games, Ltd.

Diaconis and Keller (1989) show that there exist ``fair'' dice other than the usual Platonic Solids and duals of the Archimedean Solids, where a fair die is one for which its symmetry group acts transitively on its faces. However, they did not explicitly provide any examples.

The probability of obtaining points (a roll of ) on -sided dice can be computed as follows. The number of
ways in which can be obtained is the Coefficient of in

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

The most common roll is therefore seen to be a 7, with probability , and the least common rolls are 2 and 12, both with probability 1/36.

For six-sided dice,

(12) |

(13) |

For three six-sided dice, the most common rolls are 10 and 11, both with probability 1/8; and the least common rolls are 3 and 18, both with probability 1/216.

For four six-sided dice, the most common roll is 14, with probability 73/648; and the least common rolls are 4 and 24, both with probability 1/1296.

In general, the likeliest roll for -sided dice is given by

(14) |

(15) |

(16) |

(17) |

The probabilities for obtaining a given total using 6-sided dice are shown above for , 2, 3, and 4 dice. They can be seen to approach a Gaussian Distribution as the number of dice is increased.

**References**

Diaconis, P. and Keller, J. B. ``Fair Dice.'' *Amer. Math. Monthly* **96**, 337-339, 1989.

Dice & Games, Ltd. ``Dice & Games Hobby Games Accessories.'' http://www.dice.co.uk/hob.htm.

Gardner, M. ``Dice.'' Ch. 18 in
*Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American.*
New York: Vintage, pp. 251-262, 1978.

Robertson, L. C.; Shortt, R. M.; Landry, S. G. ``Dice with Fair Sums.'' *Amer. Math. Monthly* **95**,
316-328, 1988.

Sloane, N. J. A. Sequence A030123 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

© 1996-9

1999-05-24