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

Rectangles Of Distinction

This is a game for any plural number of players.

First, draw a (2n)-by-(2n) array of dots (where the dots correspond to the vertices of a grid of (2n-1)-by-(2n-1) squares), where 2n is at least 6, I suggest.

Players take turns drawing a rectangle each move, each rectangle using 4 of the dots as corners. Each rectangle must have a unique non-zero area and have unique corners (see below).

The sides of the rectangles may overlap those of previously drawn rectangles, but no corner should be the corner of a previously drawn rectangle (drawn by any player).
After drawing a rectangle, mark the 4 dots which are its corners with x's so that it is known which dots have been used already.

Also, after drawing a rectangle, write down in a (growing) list the area of this triangle (the area in terms of the "squares" of the original array of dots).
No rectangle may have the same area as any previously drawn rectangle (drawn by any player).

The last player able to successfully draw a rectangle using 4 previously-unused corners and having a unique area is the winner.
In other words, the player wins who moved just before the first player who THINKS he or she cannot move.

Thanks,
Leroy Quet