## Monday, September 22, 2008

### Turn It Around -- Multiply/Add Grid

Here is another game played on an n-by-n grid drawn on paper,
where n is even.
(Actually, it would be MUCH easier to play this game on a computer.)
2 players.
Player 1 starts the game by placing a 1 in any square of the grid.
The players take turns placing numbers in the empty squares of the
grid, one number in one square per move.
Integers can only be placed in any square that is adjacent (in the
direction of up, right, down, left) to a square that is already filled
in with a number. The integer placed in a square must be exactly 1
more than the number in any filled-in square that is adjacent to the
square being filled in on the move.
Play continues until all squares are filled in.
Scoring:
For player 1, imagine another grid filled in the same way as the game
grid. Rotate the the second grid 180 degrees and place on top of the
first grid. Now multiply each number in the second grid by the number
immediately below it, getting n^2 total products. Player 1 gets the
sum of these products as the score. (So, the score is relatively
large, as far as my games go.)
For player 2, do the same thing, but rotate the top grid by only 90
degrees. (It doesn't matter if you rotate clockwise or
counterclockwise.)
Here is a small example:
n = 3.
Finished grid looks like this:
1 2 3
4 3 6
5 4 5
Rotated 180 degrees, multiply and add:
Player 1's score =
1*5 + 2*4 + 3*5 +
4*6 + 3*3 + 6*4 +
5*3 + 4*2 + 5*1 =
113.
Rotated 90 degrees, multiply and add:
Player 2's score =
1*5 + 2*4 + 3*1 +
4*4 + 3*3 + 6*2 +
5*5 + 4*6 + 5*3 =
73.
Player 1 wins.

Thanks,
Leroy Quet