Sicherman dice

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Sicherman dice are the only pair of 6-sided dice bearing only positive integers other than a pair of normal dice which have the same probability distribution for the sum. The faces on the dice are numbered 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8.

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[edit] Mathematics

Crazy dice refers to a standard mathematical problem or puzzle in elementary combinatorics, involving a re-labeling the faces of a pair of six-sided dice to reproduce the same frequency of sums as the standard labeling.

It is a standard exercise in elementary combinatorics to calculate the number of ways of rolling any given value with 2 fair six-sided dice (by taking the sum of the two rolls). The below table enumerates the number of such ways of rolling a given value n:

n # of ways n # of ways
2 1 8 5
3 2 9 4
4 3 10 3
5 4 11 2
6 5 12 1
7 6

A question arises whether there are other ways of re-labeling the faces of the dice with positive integers that generate these sums with the same frequencies. The surprising answer to this question is that there does indeed exist such a way. These are the Sicherman dice.

[edit] History

These dice were discovered by Colonel 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).

[edit] Mathematical justification

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. The product of this polynomial with itself yields the generating function for the throws of a pair of dice: x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 + 5x8 + 4x9 + 3x10 + 2x11 + x12. From the theory of cyclotomic polynomials, we know that

x^n - 1 = \prod_{d\,\mid\,n}^n \Phi_d(x).\;

where d ranges over the divisors of n and \Phi_d(x)\, is the d-th cyclotomic polynomial. We note also that

\frac{x^n -1}{x-1} = \sum_{i=0}^{n-1} x^i = 1 + x + \dots + x^{n-1}.

We therefore derive for the generating function of a single n-sided canonical die

x + x^2 + \dots + x^n = \frac{x}{x-1} \prod_{d\,\mid\,n}^n \Phi_d(x)

\Phi_1(x) = x - 1\, and is canceled. Thus the factorisation of the generating function of a six-sided canonical die is

x\,\Phi_2(x)\,\Phi_3(x)\,\Phi_6(x) = x\;(x+1)\;(x^2 + x + 1)\;(x^2 - x +1)

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 spots 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. Only one such partition exists:

x\;(x + 1)\;(x^2 + x + 1) = x + 2x^2 + 2x^3 + x^4

and

x\;(x + 1)\;(x^2 + x + 1)\;(x^2 - x + 1)^2 = x + x^3 + x^4 + x^5 + x^6 + x^8

This gives us the distribution of spots on the faces of a pair of Sicherman dice as being {1,2,2,3,3,4} and {1,3,4,5,6,8}, as above.

This technique can be extended for dice with an arbitrary number of sides.

[edit] References

  • Brunson, B. W. & Swift, Randall J. (1997/8), “Equally likely sums”, Mathematical Spectrum 30 (2): 34–36 

[edit] External links

This article incorporates material from Crazy dice on PlanetMath, which is licensed under the GFDL.