## Monday, March 21, 2011

### Sum Kind Of Game

This is a game for two players.
Start with an n-by-n grid drawn on paper, where n is even.

Players take turns placing numbers into the empty grid squares, one number per move.
Player 1 places into the grid a 1 then a 2 then 3 then... then the value of n^2/2-1 then finally the value of n^2/2.
Player 2, on the other hand, goes the opposite way, placing into the grid the value of n^2/2 then the value of n^2/2-1 then... then 3 then a 2 and finally a 1.

Player 1 gets as a score the number of squares with integers S where each is the sum of the values in two different squares of the same ROW (same row as their sum S).

Player 2 gets as a score the number of squares with integers S where each is the sum of the values in two different squares of the same COLUMN (same column as their sum S).

(The addends can be the same value, but must be in different squares.)

It doesn't matter who wrote down each number as far as scoring is concerned.

Largest score wins.

Sample game: (n=6)

01 02 03 04 05 18
12 17 05 15 11 10
10 14 13 17 03 06
09 14 06 04 09 02
08 16 07 15 08 12
16 18 07 13 11 01

Player 1 gets for each row (top to bottom):
3, 2, 2, 1, 2, 1
11 points.

Player 2 gets for each column (left to right):
2, 2, 1, 1, 3, 2
11 points.

(I may have made a mistake or two figuring these out.)

A tie, if I didn't err.

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Is there a bias in favor of one player in the game because strategies are different due to the two orders the numbers are placed into the grid?

Thanks,
Leroy Quet