## Tuesday, October 28, 2008

### Within The Curve

Here is a game for any number of players.

Each player has m (n-by-n) grids, where m is the number of players. (So there are m^2 grids all together.)
I suggest an n of at least 12.

On one of their grids each player secretly draws a closed non-self-intersecting curve. (The curve is bounded within the n-by-n grid.) Each player's curve does not go through any intersections of the grid-lines.

Next, on one of each of the other player's blank n-by-n grids each player copies his/her curve over.
The copies of each curve must go through the same respective squares of each grid as the original curve did.
So, there are m copies each of m curves, each player in possession of one copy of each curve.

Next, secretly and simultaneously, each player fills in the squares each curve goes through on any particular grid with 1,2,3,...., the integers placed in order and next to each other along the curve. The numbers can start anywhere along a curve, and can go either clockwise or counterclockwise.

Next, each player secretly fills in the squares within each curve's interior with 1,2,3,..., the numbers placed in order, each number placed in any empty interior square such that all other numbers (including possibly numbers along the curve) above, right of, left of, or below the number are coprime to that number.

(Any pair of adjacent numbers that are both in squares a curve passes through don't have to be coprime. Only interior numbers have to be coprime to adjacent numbers along the curve, or to adjacent numbers that are also on the curve's interior.)

Players continue to fill the interior of each curve with numbers until the players can't fill in any more numbers under the rules.

When each player has filled in each curve as far as they can, the score for each player is the sum of the top numbers in the interior squares of each of the m curves the player filled (partially) in.

Highest score wins.

Players may check their opponents' grids after the game is over to make sure that all applicable pairs of adjacent numbers are actually coprime. If a player made a mistake, that player automatically loses the game.

Thanks,
Leroy Quet