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 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.
In 1787, Mozart wrote the measures and instructions for a musical composition dice game. The idea is to cut and paste pre-written measures of music together to create a Minuet (Chuang).
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 above show 6-sided dice with counterclockwise and clockwise arrangements, respectively, when viewed from along the three-fold rotation axis towards the center of the die.
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 five 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 (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. 23-24).
Consider now 
.
For 
six-sided 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 
six-sided dice,
 |
(17)
|
and
 |  |  |
(18)
|
 |  |  |
(19)
|
 |  |  |
(20)
|
 |  |  |
(21)
|
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
 |
(22)
|
which can be written explicitly as
![p_L(n,s)={1/2n(s+1) for n even; 1/2[n(s+1)-1] for n odd, s even; 1/2n(s+1) for n odd, s odd.](/images/equations/Dice/NumberedEquation12.svg) |
(23)
|
For 6-sided 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 closed-form expression in terms of 
and 
.
However, the probabilities of obtaining the likeliest roll totals can be found explicitly
for a particular 
.
For 
6-sided 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 
6-sided 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.
