Dual Nim

Nim is a well-known combinatorial game, based on removing stones from piles. In this problem, we'll deal with a similar game, which we'll call Dual Nim. The rules of this game are as follows: - Initially, there are N piles of stones, numbered 1 through N. The i-th pile contains ai stones. - The players take alternate turns. If the bitwise XOR of all piles equals 0 before a player's turn, then th