## Wednesday, September 8, 2010

### Plusses And Minuses Grid Game

This game is for two players.

Start with an n-by-n grid drawn on paper. I suggest that n be an odd integer >= 9.

The first part of the game doesn't involve the grid. In this part of the game, the players take turns each contributing to their own "prediction list" of +'s and -'a. On each move, a player appends to the end of their list either a + or a -. Each player is aware of the other player's list as it is being made. After both players' lists are n-1 symbols long, this part of the game is over. (See below for the significance of the lists.)

I suggest at this point that a dot be put at the lower left corner of the grid so as to keep this corner straight from the others.

In the next part of the game, the players take turns, each move filling in an empty square of the grid. The filled in square must not be in the same row or column as any other filled in square.

(I suggest that the player who moved first in the first part of the game moves second in the second part of the game. Just to be fair.)

After n squares total have been filled in, this part of the game is over.
(There will be exactly one filled in square in each row and in each column.)

Now, we determine the scores.
Take the grid, with the dot in the lower left corner.
Going from the left to the right, the filled in squares represent a permutation P = (p(1),p(2),...p(n)) of (1,2,3,...,n). Player 1 forms a "truth list" of +'s and -'s, where the kth symbol is the sign of p(k+1)-p(k).

Going from bottom to the top, the filled in squares represent a permutation Q = (q(1),q(2),...q(n)) of (1,2,3,...,n), where Q is the inverse permutation of P. Player 2 forms a truth list of +'s and -'s, where the kth symbol is the sign of q(k+1)-q(k).

(The lower-left square represents 1 both in respect to permutation P and permutation Q. And the upper right square represents n for both permutations.)

Write each players truth list below their prediction list so that respective signs are lined up. Each player gets a point for each corresponding pair of signs that match.

Largest score wins.

Here is a sample game:

n = 8.

Grid:
. * . . . . . .
. . . . . . . *
. . . . . . * .
. . . . . * . .
. . * . . . . .
. . . . * . . .
. . . * . . . .
* . . . . . . .

Player 1's prediction list:
+ - + - + - +
Player 1's truth list:
+ - - + + + +
The first pair, second pair, 5th pair, and 7th pair match. So, player 1 gets 4 points.

Player 2's prediction list:
+ + - + + - +
Player 2's truth list:
+ + - + + + -
The first pair, second pair, 3rd pair, 4th pair, and 5th pair match. So, player 2 gets 5 points.

Player 2 wins.

Thanks,
Leroy Quet