Question

I decided to do an image to explain this better, I just want to check my thinking is ok, and that I can reduce total permutations by 75%:

alt text http://www.freeimagehosting.net/uploads/45e5c6b05e.gif

Answer 1

you are reducing the number of permutations, but not by 75%, since all possible positions of the small square fill up a 6×6 square, and your “quarter” fills up a 4×4 square.

Since there are “overlaps” to your quarters, you are actually adding a little permutations.
Since your quarter is 4×4, you have 4 squares overlapping in the middle column, and another four in your middle row.

Still, this is less than actually computing for each small square.

also, you could further increase performance with 2 squares by doing this:

let’s say you have 2 squares, 1 & 2. if your square is:

11110000

11110000

00000000

02000000

this will be equivalent to:

00001111

00001111

00000000

00000020

and

00000020

00000000

00001111

00001111

so, you could loop through all permutations of 1 in the first quarter of the grid, against all permutations of 2 in the FIRST HALF (left) of the grid. do this for quarters 1 and 2 (where quarter 1 is upper-left, and quarter 2 is upper right).