### Three Checker Game on a One Row Board

The setup for the 2-players game described below consists of three checkers placed on a K×1 board:

A move consists in picking one of the outside checkers and placing it anywhere between the other two. Here's a sequence of two successive moves.

As usual, players take turns. The one who can't make a move loses the game. Is there a right strategy? How should one play? Would you want to make the first move?

The game has two parameters: the number of free squares between the first and the second, and the second and the third checkers. Let's call them M and N. A position is completely described by the set

Two small configurations will give a clue for a possible strategy.

You would certainly like to be faced with the {1, N} configuration, whatever

You would certainly not like to be faced with the {2, 2} configuration because, the only configuration you may leave is

Placing a checker into an odd gap (unless it's of length 1) may create either an

The general theory of combinatorial games applies to the game at hand. From the foregoing discussion we can identify positions *P*-Positions - position that one would like to leave after one's move. Configurations *N*-Positions; these are the positions you would like to face.

From an *P*-position, every move leads to a *N*-position. Every *N*-position admits moves that leave *P*-position.

It follows that the first player who starts with a configuration

### References

- XXX M. V. Lomonosov Competition, 2007, MCCME.