Number Superposition Game

Here is a simple game for any plural number of players.

This game is played on an m-by-m grid drawn on paper.


The first player to move places a 1 in any square of the grid.

Players thereafter take turns, each player placing the integer k in a square adjacent to (and in the direction of above, below, right of, or left of -- with restrictions {see below}) the square where the previous player to move wrote the integer (k-1). (k increases: 1,2,3,4,5,...)

A player may put the integer in an empty square OR may write the integer in a square that already has a single number in it -- but may not write a number in a square that already has two numbers written in it.

A player may also not write a number in the square previously written in by the player who moved before the last player to move.
(In other words: No U-turns. Any integer k cannot be placed in the same square as the integer (k-2).)

A player who writes the second number in a square gets {the absolute difference between the two numbers in the square} added to his/her score.

The game continues until no more moves can be made.

Highest score wins, of course.

Thanks,
Leroy Quet