Here is yet another game of mine played using an n-by-n

grid drawn on paper and involving coprimality.

(I suggest that n be at least 6, maybe much higher.)

In the top row of the grid and in the grid's left-most

column write the integers 1,2,3,... through n, one integer

per grid-square. (1 is in the upper-left corner square.)

Now during play, players (2 in number) take turns writing

integers into the n^2 squares of the grid.

Only one integer is in each square, and a player can only

put an integer in an empty square which is both

immediately below and immediately right of squares which

already have integers in them.

The number that a player writes in an empty square

(square {j,k}) is either:

The number of integers

above and in the same COLUMN (j) as the square being

filled in

that are coprime to the number of the ROW (k) of the

square being filled in;

or

The number of integers

left of and in the same ROW (k) as the square being

filled in

that are coprime to the number of the COLUMN (j) of

the square being filled in.

(The original integers written in the top row and

left-most column indicate the j and k coordinates of

the columns and rows.)

(And the GCD(m,0) will be considered to be m,

for our purposes, so no number but 1 is coprime to

0.)

I have made up two variations of how to score in the game.

(I invite readers of this post to reply with their

own rules for scoring, if they desire.)

Variation 1:

The goal is for one player to get the highest value

in the lower right square (which is the last square

filled in) they can, while the other player tries

to minimize this value.

After playing one round, the players switch order

of play and switch who wants a high score in the

lower right square and who wants the low score.

(The players play the same sized grids for each round.)

The winner is the player who gets the highest score

for her/his round.

Also a solitaire version can be played where a player

simply tries to get the highest score possible.

Variation 2:

Player 1 gets a point for every odd integer in the

grid at game's end. Player 2 gets a point for every

even integer in the grid at game's end.

With the solitaire version of variation 2, a player

simply tries to maximize the number of odd (or even)

integers in the grid at game's end.

Sample partial game:

1 2 3 4 5 6

2 0 1 1 3 2

3 1 1 2 1 *

4 1 3 2

5 3

6

The next player to place in integer in a square,

if he/she will put the integer at position *,

can write a 1 (because 3 {row number} is coprime to

the 2 above the *, but not to 6)

or can write a 3 (because 6 {column number} is coprime

to the three 1's to the left of *, but not to the 2 or 3).

Thanks,

Leroy Quet

## Saturday, September 20, 2008

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