Coins on a Table

You and a friend want to settle who is the smarter of you two with a game. The game is played on a circular table, with you and your friend taking turns to put identically-sized coins on the table. The coins must be laid flat on the table without extending beyond the table or overlapping another coin. The first person who cannot put down a coin loses. Your friend lets you decide if you’d like to go first or second.

Should you go first or second? And how do you ensure that you can win?

Hint

If you choose to go first, there’s a strategy for you to guarantee a win.

Hint

The table doesn’t have to be a circle for this strategy to work. In fact, it can be an oval, a square, a rectangle, or even a hexagon. However, this strategy would fail on a triangular table.

Solution

You should go first. On your first move, place a coin in the middle of the table. After your friend places a coin, place your coin on the opposite side.

Your strategy involves keeping the table symmetric by a 180° rotation. This means that wherever your friend places a coin, there’ll be an identical spot on the opposite side where you can place a coin too. You can always guarantee a space for your coin until your friend runs out of spots to place his.

However, if a coin was placed over the midpoint of the table but offset to the side, then you can no longer keep the table symmetric, and the strategy breaks down. This is why you have to go first to place a coin in the exact middle of the table.

This strategy works on any table with a shape that is symmetric by a 180° rotation. This is why it works on shapes like ovals, squares, rectangles, hexagons, and so on, but not shapes like triangles and pentagons.

You can face off with this strategy as the second player. You play as the blue player against the red player.

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