Two players alternately place identical circular coins on a circular table. Coins may not overlap or hang over the edge, and a player who cannot place a coin loses. Who wins with correct play and what is the strategy?

Two players alternately place identical circular coins on a circular table. Coins may not overlap or hang over the edge, and a player who cannot place a coin loses. Who wins with correct play and what is the strategy?

Approach: Look for a move that makes the position invariant under a rotation, so that every reply the opponent has can be answered by the image of his move.

The first player wins by placing his first coin at the centre and then mirroring every opponent move through the centre. Once the centre is taken the position is invariant under a 180 degree rotation about it. If the opponent can place a coin at some spot, the diametrically opposite spot is free, because the image of every placed coin is itself placed and the centre coin maps to itself. So the first player always has a reply and never runs out first. The table holds finitely many coins, so the game ends, and it ends with the opponent stuck. The argument needs a centre of symmetry: on a half disc table it fails and the first player has no such guarantee.

Follow-up: Two players alternately place non-overlapping dominoes on an 8 by 8 board, one horizontally and one vertically. Who wins?

Key concepts: symmetry strategy, mirroring, finite game.