Grid Game: 1 to 1 to 1

This is a game for, preferably, 3 or 4 players.

Start with an m-by-m grid drawn on paper. I suggest that m be at least 3 times the number of players.

Each player take turns writing odd numbers in the squares, each number being small enough that other numbers may also be written in any square if necessary.


In each player's first move, he/she writes a 1 in any empty square of the grid.

Thereafter, a player places a 3 then a 5 then a 7, etc, each number in a square. The number (2k+1) must be in the square adjacent and horizontal, vertical, or diagonal to the square (but not in the same square) where the SAME player last wrote the number (2k-1). Each number must either be written in an empty square, or be written in a square such that the new number is coprime (relatively prime) to all numbers previously written in that square (by any player).

The first player whose path of numbers visits all of her/his opponents' 1's and then lastly returns to her/his own 1 is the winner.

If a player cannot move, then he/she is out of the game.

A player may win if all other players forfeit by not being able to move.

Thanks,
Leroy Quet

PS: I have changed the rules slightly to have all the numbers be odd. -- 8-20-09