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 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 roleplaying games to determine the number of spaces to move, results of a conflict, etc. A coin can be viewed as a degenerate 2sided case of a die.
In 1787, Mozart wrote the measures and instructions for a musical composition dice game. The idea is to cut and paste prewritten measures of music together to create a Minuet (Chuang).
The most common type of die is a sixsided cube with the numbers 16 placed on the faces. The value of the roll is indicated by the number of "spots" showing on the top. For the sixsided 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 above show 6sided dice with counterclockwise and clockwise arrangements, respectively, when viewed from along the threefold rotation axis towards the center of the die.
The cube has the nice property that there is an upwardpointing 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 14 was not visible on one of the upper three faces). The arrangement of five spots corresponding to a roll of 5 on a sixsided die is called the quincunx. There are also special names for certain rolls of two sixsided dice: two 1s are called snake eyes and two 6s are called Boxcars.
Shapes of dice other than the usual 6sided 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 (i.e., isohedra). There are 30 isohedra.
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)

since each possible arrangement contributes one term. can be written as a multinomial series
(2)
 
(3)

so the desired number is the coefficient of in
(4)

Expanding,
(5)

so in order to get the coefficient of , include all terms with
(6)

is therefore
(7)

But only when , so the other terms do not contribute. Furthermore,
(8)

so
(9)

where is the floor function, and
(10)

(Uspensky 1937, pp. 2324).
Consider now . For sixsided dice,
(11)

and
(12)
 
(13)
 
(14)
 
(15)
 
(16)

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 sixsided dice,
(17)

and
(18)
 
(19)
 
(20)
 
(21)

For three sixsided 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 sixsided 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
(22)

which can be written explicitly as
(23)

For 6sided dice, the likeliest rolls are given by
(24)

or 7, 10, 14, 17, 21, 24, 28, 31, 35, ... for , 3, ... (OEIS A030123) dice. The probabilities corresponding to the most likely rolls can be computed by plugging into the general formula together with
(25)

Unfortunately, does not have a simple closedform expression in terms of and . However, the probabilities of obtaining the likeliest roll totals can be found explicitly for a particular . For 6sided dice, the probabilities are 1/6, 1/8, 73/648, 65/648, 361/3888, 24017/279936, 7553/93312, ... for , 3, ....
The probabilities for obtaining a given total using 6sided dice are shown above for , 2, 3, and 4 dice. They can be seen to approach a normal distribution as the number of dice is increased.