Pile Partition

Alice and Bob are playing a game. At the start of the game, there are $N$ piles of stones, the $i$-th of which has $A_i$ stones. Alice and Bob alternate moves. On their move, a player does the following: - Choose one pile of stones, and discard the remaining $N-1$ piles. - Partition the stones in the chosen pile into **exactly** $N$ new piles of stones, each containing at least $1$ stone. Note th