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2 dice with same sum as 2 ordinary ones (Chile NMO 1998 P7)

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November 24, 2021
algebracombinatorics

Problem Statement

When rolling two normal dice, the set of possible outcomes of the sum of the points is 2,3,3,4,4,4,...,11,11,122, 3, 3, 4,4, 4,..., 11, 11,12. Notice that this sequence can be obtained from the identity (x+x2+x3+x4+x5+x6)(x+x2+x3+x4+x5+x6)=x2+2x3+3x4+...+2x11+x12.(x + x^2 + x^3 + x^4 + x^5 + x^6) (x + x^2 + x^3 + x^4 + x^5 + x^6) = x^2 + 2x^3 + 3x^4 +... + 2x^{11} + x^{12}. Design a crazy pair of dice, that is, two other cubes, not necessarily the same, with a natural number indicated on each face, such that the set of possible results of the sum of its points is equal to of two normal dice.
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