The game of coins on a round table

Idea. A symmetry strategy: after the first move, always answer by the symmetric move.

Strategy. Let $O$ be the centre of the table and $r$ the radius of a coin. The first player puts his first coin with its centre at $O$. Afterwards, each time the opponent places a coin with centre $P$, the first player places a coin with centre $P'$, the reflection of $P$ in $O$.

Invariant. Just before each move of the second player, the position is symmetric with respect to $O$.

The reply is legal. Let the second player place a coin at $P$ (legal, so $P\ne O$).
- *On the table:* the table is symmetric about $O$, so the coin at $P'$ lies completely on the table.
- *No overlap with the coin at $P$:* two coins overlap iff their centres are at distance less than $2r$. Here $PP'=2\,OP\ge2r$ because the move at $P$ was legal and $O$ is occupied.
- *No overlap with the other coins:* by symmetry, if the coin at $P'$ overlapped some coin $C$, then the coin at $P$ would overlap the reflection of $C$: impossible.

Conclusion. The first player always has a legal reply. Since the table has finite area, the game ends after finitely many moves, and the player who cannot move is the second one.

Lesson. Symmetry strategy: if the board has a central symmetry, occupy the centre and mirror every move of the opponent.