## Saturday, September 20, 2008

### Dream Math Game

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.
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