A 57 Integers 11 ( 2011 ) Diophantine Equations of Matching Games

We solve a family of quadratic Diophantine equations associated with a simple kind of game. We show that the ternary case, in many ways, is the most interesting and the least arbitrary member of the family. 1. The Matching Games An (n, d)-matching game (n, d ≥ 2) is a game in which the player draws d balls from a bag of balls of n different colors. The player wins if and only if the balls drawn are all of the same color. A game is non-trivial if there are at least d balls in the bag. It is faithful if there are balls in each of the n colors. A game is fair if the player has an equal chance of winning or losing the game. In this article, we only study the (n, 2)-matching games or simply the n-color games, leaving the study of the higher d case to [10]. An n-tuple (a1, . . . , an), where ai is the number of balls in the bag having the ith color, represents an n-color game. For m ≤ n, an m-color game (a1, . . . , am) can be regarded as the n-color game (a1, . . . , am, 0, . . . , 0). The only trivial n-color fair games, are the zero game (0, . . . , 0) and, up to permutation, the game (0, 0, . . . , 1). By considering the number of ways for the player to win the game, one sees that the n-color fair games are exactly the non-negative integral solutions of (∑n i=1 xi 2 ) = 2 ( n ∑