Placing coins on a round table

Question

Two players alternate placing identical circular coins on a round table. Each coin must lie flat, fully on the table, and must not overlap any coin already placed. The player who cannot make a legal move loses (the table is too full).

With perfect play, does the first or the second player win? Give the winning strategy.

Accepted Answer

The first player wins, using a symmetry (mirroring) strategy. Move 1. The first player places a coin exactly at the center of the table. The center is the unique point invariant under 180° rotation about the table's center. Every move after. Whatever the second player does, placing a coin at some location, the first player responds by placing a coin at the diametrically opposite point (the 180° rotation of the opponent's coin about the center). Why the mirror move is always legal. The table is symmetric under 180° rotation about its center. If the second player's coin fits (on the table, not …

Placing coins on a round table

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