Tuesday, December 2, 2008

Co-Compositeness (a game)

This is a game for any number of players. Start with an n-by-n grid, where n is larger if there are more players. (I suggest an n of at least 16 if there are 2 players.)

The first player to move places a 1 in any of the grid's squares.

Players take turns placing numbers in the grid's squares as follows:

*Each player places in a grid-square the next higher odd integer than the (odd) integer previously put in a square by the previous player to move. (So, let m be the number of total moves made by all the players so far; then the next player to move places a {2m+1} in the next square.)

*Players place the (odd) integer in a square that is immediately adjacent (in any of the 8 directions of: above, below, left of, right of, or diagonally) to the square the previous (odd) number was last put inside.

*Each integer is either placed in an empty square or in a square that already contains just one number that is NOT coprime to the integer the player is now placing in the square. There may be no more than 2 integers in any one square.

Every time a player places an integer in a square with an integer already in it, then that player gets a point. (Any pair of integers in the same square must be "co-composite", ie non-coprime.)

Players continue filling in the squares until a player cannot move anywhere. (If a player can move, then the player must move.) Then the game is over.

Highest score wins.

(Note: Part of the strategy of this game might be for a player to try to force an early ending to the game if that player has the highest score so far.)

Leroy Quet

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