1297: Return of the Nim

Sherlock and Watson are playing the following modified version of Nim game:

• There are n piles of stones denoted aspiles0,piles1,...,pilesn-1, and n is a prime number;
• Sherlock always plays first, and Watson and he move in alternating turns. During each turn, the current player must perform either of the following two kinds of moves:
1. Choose one pile and remove k(k >0) stones from it;
2. Remove k stones from all piles, where 1≤k≤the size of the smallest pile. This move becomes unavailable if any pile is empty.
• Each player moves optimally, meaning they will not make a move that causes them to lose if there are still any better or winning moves.

The first contains an integer, g, denoting the number of games. The 2×g subsequent lines describe each game over two lines:
1. The first line contains a prime integer, n, denoting the number of piles.
2. The second line contains n space-separated integers describing the respective values of piles₀,piles₁,...,piles_n-1.

• 1≤g≤15
• 2≤n≤30, where n is a prime.
• 1≤piles_i≤10⁵ where 0≤i≤n−1

2
3
2 3 2
2
2 1

Sherlock
Watson