I saw this game in a dream.

I post it here only as a prototype for others to improve upon.

(Half the fun of playing this game is not playing it, but is coming up

with rules for it.)

Game is for 2 players and is played using an n-by-n grid drawn on

paper.

Players take turns writing, in order, 1 through n^2 into the grid's

empty squares,

one integer per square,until the grid is filled.

(So, player 1 writes in the odd integers, player 2 writes in the even

integers.)

Player 1 gets the sum of the greatest common divisors (GCDs) of the

n*(n-1)

_horizontally_ adjacent pairs of integers in the grid.

Player 2 gets the sum of the greatest common divisors (GCDs) of the

n*(n-1)

_vertically_ adjacent pairs of integers in the grid.

Example:

If we have the 3-by-3 grid

1 2 4

3 9 8

6 5 7

Player 1 gets

GCD(1,2)+GCD(2,4)+

GCD(3,9)+GCD(9,8)+

GCD(6,5)+GCD(5,7)

= 9 points.

Player 2 gets

GCD(1,3)+GCD(3,6)+

GCD(2,9)+GCD(9,5)+

GCD(4,8)+GCD(8,7)

= 11 points.

Now, this would be a good place to stop so as to have a simple game,

but my dream took the game further.

If an adjacent pair had a GCD of 1, a player, instead of getting one

point for this pair, would get the minimum element of the pair added to

the score.

And, in addition, player 1 would get 5 bonus points added for every

row-score which added up to a prime, player 2 would get 5 bonus points

for every column-score which added up to a prime.

So, back to the example:

1 2 4

3 9 8

6 5 7

Player 1 gets

1 + 2 + 5 (add 5 because 1+2 is prime)

+ 3 + 8 (8 is minimum of 8 and 9) + 5

+ 5 + 5

= 34 points.

Player 2 gets

1 + 3

+ 2 + 5 + 5

+ 4 + 7 + 5

= 32 points

And any who wish to play this game can add their own rules for other

values of GCDs.

For example, if the GCD of a pair is 2, you can add the maximum of the

pair to the score;

if GCD = 3, you can add the product of the pair;

or you can add the floor of the maximum of the pair divided by the

minimum of the pair;

or you can add the number of divisors of the sum of the terms of the

pair;

etc...

Have fun!

thanks,

Leroy Quet

## Saturday, September 20, 2008

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