## Friday, May 29, 2009

### Convoluted Coprimality Game

Here is a game for any plural number of players.

Start by drawing an n-by-n grid on paper, where I suggest that n is >= 12.

Players take turns, going in a predetermined order (such as clockwise by how the players are seated).
The game starts when Player 1 places a 1 in the upper-left square of the grid.
Player 1 then places a 2 either in the square immediately to the right of the 1 or in the square immediately below the 1.
Then it is player 2's turn.

A "turn" is made up of a series of "moves". Only one player makes his/her moves during a single turn.

On the kth "turn", a player (whose turn it is to move) writes the numbers > than the (k-1)th prime and <= the kth prime in empty squares as follows:

On the kth turn, the integer j starts as (the {k-1}th prime)+1. The player continues moving until j equals the kth prime.
On the jth "move", a player places the number j in an empty square either immediately above, right of, below, or left of the square with a (j-1) in it. (The {k-1}th prime would have been placed in a square by the previous player to move.)
A number must be placed in a square bordered (above, right of, left of, below) by 2 or more squares that have already been filled in with numbers previously (filled in with numbers by any player). (So, the empty square to have the number j placed in it must be next to the square with the number (j-1) in it, plus the number j must be next to ONE OTHER square, at least, with a number already in it.)

BUT, if the square being filled in is in a row or column that is on the border of the grid, then the number j need only be next to ONE square that is already filled in (which is the square with the (j-1)).

When a player is forced to -- or does so by accident -- place an integer j in a square that is immediately next to (in the direction of above, below, right of, or left of) a square with a number that is NOT coprime to j, then that player is eliminated from competition.

Play continues until there is one player left, who is the winner.

If, during play, there are no empty squares where numbers can be placed, then the remaining players start again on a new empty grid, and j = 1 and k = 1 again.

Thanks,
Leroy Quet