Sicherman dice are a pair of 6-sided dice with non-standard numbers—one with the sides 1, 2, 2, 3, 3, 4 and the other with the sides 1, 3, 4, 5, 6, 8. They are notable as the only pair of 6-sided dice that are not normal dice, bear only positive integers, and have the same probability distribution for the sum as normal dice. They were invented in 1978 by George Sicherman of Buffalo, New York.
thumb|Comparison of sum tables of and dice. If zero is allowed, normal dice have one variant and Sicherman dice have two Each table has
A standard exercise in elementary combinatorics is to calculate the number of ways of rolling any given value with a pair of fair six-sided dice (by taking the sum of the two rolls). The table shows the number of such ways of rolling a given value
n
| n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Number of ways | 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |
Crazy dice is a mathematical exercise in elementary combinatorics, involving a re-labeling of the faces of a pair of six-sided dice to reproduce the same frequency of sums as the standard labeling. The Sicherman dice are crazy dice that are re-labeled with only positive integers. (If the integers need not be positive, to get the same probability distribution, the number on each face of one die can be decreased by k and that of the other die increased by k, for any natural number k, giving infinitely many solutions.)
The table below lists all possible totals of dice rolls with standard dice and Sicherman dice. One Sicherman die is colored for clarity:
1–2–2–3–3–4, and the other is all black, 1–3–4–5–6–8.| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Standard dice | 1+1 | 6+6 | ||||||||||
| Sicherman dice | 1+1 | 4+8 |
The Sicherman dice were discovered by George Sicherman of Buffalo, New York and were originally reported by Martin Gardner in a 1978 article in Scientific American.
The numbers can be arranged so that all pairs of numbers on opposing sides sum to equal numbers, 5 for the first and 9 for the second.
Later, in a letter to Sicherman, Gardner mentioned that a magician he knew had anticipated Sicherman's discovery. For generalizations of the Sicherman dice to more than two dice and noncubical dice, see Broline (1979), Gallian and Rusin (1979), Brunson and Swift (1997/1998), and Fowler and Swift (1999).
Let a canonical n-sided die be an n-hedron whose faces are marked with the integers [1,n] such that the probability of throwing each number is 1/n. Consider the canonical cubical (six-sided) die. The generating function for the throws of such a die is
x+x2+x3+x4+x5+x6
x2+2x3+3x4+4x5+5x6+6x7+5x8+4x9+3x10+2x11+x12
xn-1=\prodd\midn\Phid(x).
\Phid(x)
| xn-1 | |
| x-1 |
=
| n-1 | |
| \sum | |
| i=0 |
xi=1+x+ … +xn-1
x+x2+ … +xn=
| x | |
| x-1 |
\prodd\midn\Phid(x)
\Phi1(x)=x-1
x\Phi2(x)\Phi3(x)\Phi6(x)=x (x+1) (x2+x+1) (x2-x+1)
x (x+1) (x2+x+1)=x+2x2+2x3+x4
x (x+1) (x2+x+1) (x2-x+1)2=x+x3+x4+x5+x6+x8
This technique can be extended for dice with an arbitrary number of sides.