This is a game for 2 players. It is played on an n-by-n grid, where n is at least 8 or higher, I suggest.
Player 1 starts the game by placing a 1 in any square of the grid.
The game consists of "moves" alternately taken by each player. Each move is made up of a series of "motions", where a single player makes all of the motions in any particular move.
A player on move n (where player 1 placing the 1 any square is move #1 and is motion #1) makes motions p(n-1) through p(n)-1 (for moves n>=2), where p(n) is the nth prime.
Player 1 makes the odd numbered moves, while player 2 makes the even numbered moves.
On MOTION m, a player places the number m in any EMPTY square that is adjacent to the square with a (m-1) in it (which was placed in the (m-1) square by either player), such that:
*If m is an even composite, the player places m immediately either left of or right of the square with an (m-1) in it.
*If m is an odd composite, the player places m immediately either above or below the square with an (m-1) in it.
* If m is a prime (ie. If this is the first motion of a player's move), then the player can place m in the square that is immediately either above, below, right of, left of, or diagonal to the square with the (m-1) in it.
The last player that can make a motion loses.
Variation: The first player that cannot make a motion loses.
(The difference between the original version and the variation is simply that in the original version, if a player places a p-1, where p is a prime, but the other player can't place a p, then the player who placed the p-1 loses. In the variation, the player who cannot place a p loses.)
I leave it up to players to decide amongst themselves which version they prefer.