## Tuesday, November 9, 2010

### Multiplications Within The Permutation

This is a game for any plural number of players. Let the number of players be m.

Start with an n-by-n grid drawn on paper, where n = k*m + 1, k is some integer >= 3.

In the first part of the game, players take turns placing x's in empty squares of the grid, one x per turn, such that no more than one x is in each row and in each column of the grid.

After exactly n x's are placed in the grid, the first part of the game is over.

The second part of the game starts with a 1 being placed in the leftmost square with an x in it. Players then take turns.
On the jth move (starting at move # 1) of the second part of the game, the moving player places a (j+1) in any square with an x and without a number already in it.
He/she gets added to her/his score:
|x(j)-x(j+1)| * |y(j)-y(j+1)|,
where x(j) is the number of squares from the bottom of the grid where the square with the j in it is located, and y(j) is the number of squares from the left side of the grid where the square with the j in it is located.

So, what we are adding to the moving player's score (the score of the player writing a j+1 in a square) is the product of {the change in horizontal distance between the squares with j and j+1 in them} and {the change of vertical distance between the squares with j and j+1 in them}.

When the nth x is numbered with a n= m*k+1, the game is over.

Largest score wins.

Here is an example:
n=7. m =2.
. 3 . . . . .
. . 6 . . . .
. . . . . . 5
. . . . 7 . .
1 . . . . . .
. . . 4 . . .
. . . . . 2 .

Squares 1 to 2: 5*2 = 10
Squares 2 to 3: 4*6 = 24
Squares 3 to 4: 2*5 = 10
Squares 4 to 5: 3*3 = 9
Squares 5 to 6: 4*1 = 4
Squares 6 to 7: 2*2 = 4

Player 1 gets: 10+10+4 = 24 points.
Player 2 gets: 24 + 9 + 4 = 37 points.
Player 2 wins.

Thanks,
Leroy Quet