Dice (singular "die") have been used in gameplaying (whether for money or not) for millennia. They typically take the form of a polyhedron bearing on each face a different symbol, usually a number.

The purpose of dice is to generate random values in order to make the outcome of a game unpredictable to some degree. Many games that use dice do give players meaningful strategic choices, thus rewarding experience and shrewdness.

By far the most common shape for a die is a cube, with a different number from one to six on each face. Most commonly, each of the twelve edges is rounded as approximately one-fourth of a cylinder, while each of the eight corners is rounded as approximately one-eighth of a sphere.

The purpose of this report is to propose two sets of three dice each, in the traditional cubic shape, bearing the numbers one through nine. Figure one shows the pip arrangements for their faces.

figure one
			traditional for dice
			introduced for this proposal

For comparison, figure two contains five example tiles from an ordinary 55-piece set of double-nine dominoes. It reveals the basis for the 7-, 8-, and 9-pip patterns of figure one.

figure two

Table one shows, row by row, the 8 combinations of 6 different integers in the range 1 to 9 that sum to 30. Each combination is the 6 numbers that would appear on a die. For reference, shown also for each combination are the three numbers omitted, which must total 15. For example, die A has numbers 1, 2, 5, 6, 7, and 9 on its faces; hence omitted are 3, 4, and 8.

Three dice have a total of 18 faces. The goal of this report is to find sets of dice such that each number from 1 to 9 appears exactly twice in each set, but never twice on one die. As it turns out, there are only two such sets: the set containing dice "A", "B", and "C"; and the set containing "D", "E", and "F". Dice "G" and "H" will not work. (It is much easier to investigate the possibilities by studying the omitted numbers rather than the numbers in use.)

There are 30 ways to distribute 6 different symbols among the faces of a cube, if the orientation of a symbol within its face is not considered. There is a long-standing tradition in dice that opposite pairs of faces should all sum to the same number. For instance, on a standard cubic die, the opposite faces are 1-6, 2-5, and 3-4; each totaling 7. That property cannot be achieved with the dice proposed here, but it can be approximated. Table two shows the recommended opposite-face pairs, each of whose sums will be 9, 10, or 11:

Each of sets ABC and DEF can reasonably be considered as balanced; combining then into a 6-die set will be balanced as well. Not apparent to the present author is any elegant way to work dice "G" and "H" into the system.

When the three dice of a set are rolled, the sum of the values on their upward faces can take many values. Tables three and four show the distribution of values for the sets ABC and DEF respectively. Both distributions are symmetrical and bimodal.

Table five shows that when all six are rolled together, the distribution is unimodal and symmetrical.