Two piles of matches, take from one

Question

Two piles hold 15 and 9 matches. Players alternate. On your turn you pick one pile and remove any number of matches you like from it (at least one). The player who takes the last match wins.

Should you move first or second, and what is your plan?

Accepted Answer

Move first, and your winning first move is to make the two piles equal: remove 6 matches from the pile of 15, leaving 9 and 9. Why equal piles are deadly for the opponent. Once the piles are equal, adopt a pure mirroring strategy: whatever your opponent removes from one pile, you remove the same number from the other pile. This restores equality after every one of your moves. Because the piles are always equal right after you move, they are never both empty on the opponent's turn while nonempty for you, you can always copy their move. The total number of matches strictly decreases, so eventua…

Two piles of matches, take from one

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