# Explosive Nim

Logic Level 5

In explosive Nim, the usual rules of Nim are followed. However, whenever a pile is exhausted, all other piles double in size. For example, from the position $$1,2,3$$, taking a stone from the 2-pile acts usually to $$1,1,3$$, but taking another stone from the 2-pile "explodes" the other piles, turning to $$2,0,6$$.

Alpha and Beta play an explosive Nim on the piles $$1, 2, 3, 4, \ldots, 2015$$. Alpha plays first. Will the game end, and who wins?

• A. The game will end in finitely many moves, and Alpha wins
• B. The game will end in finitely many moves, and Beta wins
• C. The game will end in finitely many moves, but it cannot be known who wins
• D. The game might not end, but Alpha wins in a finite number of moves
• E. The game might not end, but Beta wins in a finite number of moves
• F. The game might not end, and with perfect play the game will last forever

Clarification:

This is two questions in one.

The first question, "will the game end", means that "will the game end for any sequence of moves that Alpha and Beta play?" If there's any sequence of moves where Alpha and Beta can keep playing indefinitely, then the game might not end (the answer is one of D, E, F); if there's no such sequence, then the game will always end (the answer is one of A, B, C).

The second question, "who wins", means that "who wins if both players play perfectly?" If there's any move that is not perfect (the moving player was winning before the move, but after the move it's their opponent that is winning), the whole play is not counted.

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