## Sunday, September 21, 2008

### Row/Column Solitaire Game

Start by drawing an r-by-r grid on paper.
(I suggest an r of 5 or so for beginners.)
Put a "1" in each square of the left-most column
and in each square of the top-most row of the grid.
On each move the player chooses any one of the grid's
empty squares. This square is (m,n), which is the
square in the mth column from the left side of the
grid and in the nth row from the top.
The player then sums up all the integers already
written in the mth column and nth row.
So, if the grid looks like this, where the * is
the square the player has chosen to fill in next
with an integer,
1 1 1 1
1 3
1 * 6
1 6
then the sum of the column 2's terms and row 3's
terms is 1+3+6 +1+6 = 17. Let this sum be s.
Next the player counts the number of integers in the
squares to the left of (m,n) and above (m,n)
(ie, the squares with coordinates (j,k), 1 <=j <=m,
1 <=k <=n) which are coprime to s.
If the count is c integers in the given rectangle
which are coprime to s, then the player writes c
in square (m,n).
Play continues until there is an integer in every
square of the grid.
The player's score is the number in the last square
he/she fills in.