Twenty coins in a circle

Question

Twenty coins lie in a circle on a table. Two players take turns. On a turn, a player removes either one coin, or two coins that are adjacent (next to each other with no gap between them). Coins do not move after others are removed, so gaps stay gaps. The player who removes the last coin wins. Which player wins with best play, and what is the strategy? Does the answer depend on the number of coins?

Accepted Answer

The second player wins, for any number of coins from 3 upwards, by a symmetry strategy. The first move breaks the circle. Whatever the first player removes (one coin or two adjacent ones), the remaining coins form a single arc, a row with a gap at one place. The row has either 19 or 18 coins. The second player splits the row in half. If the row has 19 coins (odd), the second player removes the middle coin, leaving two separate rows of 9 each. If it has 18 coins (even), the second player removes the middle two coins, leaving two rows of 8 each. Either way, the position is now two identical row…

Twenty coins in a circle

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