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Ed Pegg Jr

Sicherman Dice

July 13, 2010 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project

Is it possible to have a pair of nonstandard dice with the same odds as regular dice?

Sure. You just need to know how to calculate the odds, and how to determine what different numbers could be on the faces to give the same odds. Let’s start with some tables.

The addition table is one of the first tables learned in school. Here is
one way to present an addition table in Mathematica.

Creating an addition table in Mathematica

The addition table can be used to examine the values obtained by rolling two standard dice. For example, rolling a 7 occurs six times in the table. A 12 occurs once. The number of times each value occurs is known as the distribution.

Addition table for {1, 2, 3, 4, 5, 6} and {1, 2, 3, 4, 5, 6}
Addition table for {1, 2, 3, 4, 5, 6} and {1, 2, 3, 4, 5, 6}

If the two dice have different values on their faces, an addition table can be used to see the new distribution. Note, though, that the distribution of values in this new set and the two standard dice are identical—the numbers are the same, they’ve just been moved around.

Addition table for {1, 2, 2, 3, 3, 4} and {1, 3, 4, 5, 6, 8}
Addition table for {1, 2, 2, 3, 3, 4} and {1, 3, 4, 5, 6, 8}

That’s kind of neat, maybe. But how was it found? For that, a multiplication table will be useful.

Creating a multiplication table in Mathematica

Instead of integers, we’re using powers of x to multiply together. As might be expected, when two powers of an identical variable are multiplied, the powers add. Thus, a multiplication table can look something like an addition table.

Multiplication table for powers of x
Multiplication table for powers of x

Another way to find the distribution is to transform each set of values into a polynomial, and then to multiply the polynomials together. Both sets of dice above give the same polynomial.

Expand [(x^1 + x^2 + x^3 + x^4 + x^5 +x^6) (x^1 + x^2 + x^3 + x^4 + x^5 +x^6)]
x^2 + 2x^3 + 3x^4 + 4x^5 + 5x^6 + 6x^7 + 5x^8 + 4x^9 + 3x^10 + 2x^11 +x^12
Expand [(x^1 + 2x^2 + 2x^3 + x^4) (x^1 + x^3 + x^4 + x^5 + x^6 +x^8)]
x^2 + 2x^3 + 3x^4 + 4x^5 + 5x^6 + 6x^7 + 5x^8 + 4x^9 + 3x^10 + 2x^11 +x^12

One method for finding an alternative set of dice for a given distribution is to factor it.

poly = x^2 +2x^3 + 3x^4 + 4x^5 + 5x^6 + 6x^7 + 5x^8 + 4x^9 + 3x^10 + 2x^11 + x^12; Factor[poly]
x^2 (1 + x)^2 (1 -x + x^2)^2 (1 + x +x^2)^2

Ideally, code is needed that gives all the polynomial divisors with a coefficient sum of 6, as each of the dice has 6 faces. The following Mathematica code accomplishes that.

Mathematica code to find polynomial divisors with a coefficient sum of 6
Mathematica code to find polynomial divisors with a coefficient sum of 6

These alternative distributions were brought to the attention of Martin Gardner by George Sicherman in 1978. More explorations of dice distributions can be found at the Sicherman Dice Demonstration.

Sicherman Dice Demonstration

Download this notebook

Posted in: Mathematics
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One Comment


gion

Dear Prof.
I come from China, my major is theory physics, so the Mathematica is very useful.
Recently, i meet a small question. All the captions (including the axis labels) for 3D figs in Mathematica are horizontal. How to change captions to be parallel to the axis?

Posted by gion    July 19, 2010 at 3:51 am


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