# Kayles

Kayles was introduced by Dudeney and Loyd. Two players take turns knocking down skittles - pins here in the US. Usually skittles are arranged in a row. In the applet below they make a circular pattern. With one ball a player may knock either 1 (a direct hit) or 2 adjacent skittles (hitting just in-between the two.) The players are so good at playing the game they can knock down the desired skittles at will. The last player to move wins.

(The number of skittles is modified by clicking a little left or right off its center line. You can force the computer to make first move by clicking the *You start* button.)

What if applet does not run? |

The Kayles is by far more difficult than, say, Plainim. One variant is quite trivial, though. As described by M. Gardner (*Entertaining Mathematical Puzzles*, Dover, 1961), if the skittles are distributed uniformly over a circle, the second player can always force a win. After the first player's first move, the second player removes one or two skittles so as to leave to separate groups of skittles of equal size. After that, the second player simply leaves paired groups of skittles, with the groups in a pair having the same number of skittles. This is always possible by mimicing the moves of the first player. When the first player performs a move on one group in a pair, the second player answers with a corresponding move on another group in the same pair.

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