On Monday 30 July 2007, Bill Gosper wrote:
A die having five sides, 4,0,0,0,0, is paired with a seven sided die, 3,2,2,1,1,0,0. Construct a different pair of dice with the same statistics. --rwg
[answer and comments follow after spoiler space] ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... x^4+4 = (x^2-2x+2)(x^2+2x+2) [easy to find by considering (x^2+2)^2] x^3+2x^2+2x+2 is irreducible [by Eisenstein or, more elementarily, by seeing that there'd have to be a linear factor, etc.] so the only thing to do (other than boring things; see below) is to calculate (x^2-2x+2)(x^3+2x^2+2x+2), which turns out to be x^5+2x^2+4, all coeffs non-negative, yay. Hence {5,2,2,0,0,0,0} x {2,1,1,0,0} will do and nothing else will. If this was really a contest problem and was really stated as given, then it's dicey indeed, because an easier solution to the problem as stated is (e.g.) to replace the first die with one whose sides are 4,4,0,0,0,0,0,0,0,0. Or, almost as easy, to replace them with one all of whose sides are 0 and one with the obvious 35 sides. Given honest contestants, the question could have been rewritten as "Have you come across generating functions?" without much loss of discriminating power. -- g