# Dawson's Kayles

Dawson's Kayles was invented in 1930s and is played with one or more heaps of items. The nature of the items is not important, but as in the game of Kayles they are referred to as Skittles. In the applet below, the skittles are positioned on a circle, with the number of positions shown in the upper right portion of the applet. This number may be specified by clicking a little off its vertical axis. (Clicks to the left of the axis decrease the number, to the right increase it.)

In Dawson's Kayles a player always removes exactly two adjacent skittles. Thus a lone skittle is a "dead weight" with the Grundy number of 0. The presence of a lone skittle, with no neighbors, does not affect the course of the game. To remove a pair of adjacent skittles just click somewhere between the two.

### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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Explanation

### Dawson's Kayles

The game ends when no skittles are left over or when the remaining one are all solitary without immediate neighbors. The Grundy number for an empty board is 0. For a single skittle it is also 0. It is 1 for two skittles. For other sizes, the Grundy numbers are found via the Mex rule. Here is a partial table:

0-9 0 0 1 1 2 0 3 1 1 0 3 3 2 2 4 0 5 2 2 3 3 0 1 1 3 0 2 1 1 0 4 5 2 7 4 0 1 1 2 0 3 1 1 0 3 3 2 2 4 4 5 5 2 3 3 0 1 1 3 0 2 1 1 0 4 5 3 7 4 8 1 1 2 0 3 1 1 0 3 3 2 2 4 4 5 5 9 3 3 0 1 1 3 0 2 1 1 0 4 5

Remarkably, it is exactly the values of Dawson's chess with an additional 0 inserted at the beginning.

As usual, the P-positions are those whose Grundy number is 0. You are bound to win the game if, on every move, you leave a P-Position.

### References

1. E. R. Berlekamp, J. H. Conway, R. K. Guy, Winning Ways for Your Mathematical Plays, Volume 1, A K Peters, 2001
2. R. Guy, fair game, Comap's Explorations in Mathematics, 1989

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