Sicherman dice
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.
Mathematics
[file:Sicherman dice table.svg|thumb|Comparison of summation|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. The table shows the number of such ways of rolling a given value :
| 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.
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 |
Properties other than sum need not mimic regular dice; for example, the probability of rolling doubles is 1/6 with regular dice, but 1/9 with Sicherman dice.
History
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, Gallian and Rusin, Brunson and Swift, and Fowler and Swift.
Mathematical justification
Let a canonical ''n-sided die be an n''-hedron whose faces are marked with the integers such that the probability of throwing each number is 1/n. Consider the canonical cubical die. The generating function for the throws of such a die is. The product of this polynomial with itself yields the generating function for the throws of a pair of dice:.We can analyze this polynomial using either cyclotomic polynomials, or elementary factoring.
Option 1: cyclotomic polynomials:
We know that :
where d ranges over the divisors of n and is the d-th cyclotomic polynomial, and
We therefore derive the generating function of a single n-sided canonical die as being
and is canceled. Thus the factorization of the generating function of a six-sided canonical die is
Option 2: Elementary factoring:
Thus,
The generating function for the throws of two dice is the product of two copies of each of these factors:. How can we partition them to form two legal dice whose pips are not arranged traditionally? Here legal means that the coefficients are non-negative and sum to six, so that each die has six sides and every face has at least one spot. That is, the generating function of each die must be a polynomial with all positive exponents and no constant term, and with positive coefficients.
Plugging in in the factors gives:,, and. To make both products of factors equal to 6, each factor must be paired with. The remaining pair of terms must either be separated, or be combined, representing Sicherman dice:
and
This gives us the distribution of pips on the faces of a pair of Sicherman dice as being and, as above.
This technique can be extended for dice with an arbitrary number of sides.