The Coin on the Table

The first player always wins. Place the first coin at the exact centre, then mirror every opponent move through the centre point. Since every legal opponent move has a symmetric counterpart, you always have a valid response and your opponent runs out of moves first.

This is a strategy-stealing argument combined with a geometric mirror symmetry. The first player wins by establishing control of the centre and maintaining a symmetric response to every opponent move. Step 1: On your very first turn, place a coin precisely at the centre of the square table. Step 2: Now the table has rotational symmetry of order 2 around the centre coin. For every point on the table, there is a unique mirror point directly opposite through the centre. Step 3: Whenever your opponent places a coin at any position P, you respond by placing your coin at position P' — the point diametrically opposite P through the centre. Step 4: Why does this always work? If your opponent's position P was a legal placement (no overlap, on the table), then P' is also legal by symmetry — it mirrors a legal position, so it too cannot overlap any existing coin and is also on the table. Step 5: This means: whenever your opponent can make a move, you can always make a move immediately after. The opponent can never be in a position where they cannot play while you still can. Step 6: Therefore your opponent runs out of moves first. You place the last coin and win. Key Insight: The centre coin is the key. It breaks the symmetry in your favour from the start — there is only one centre, and you claimed it. After that, the mirror strategy guarantees you always have a response. The game is won on move 1; the rest is just execution.