## Saturday, September 20, 2008

### Enumeration

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
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.
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