The 26x36 Solution
Here's how the final test problem was constructed. No
complaints are allowed, for the POTM-Master is omnipotent.
1. I wanted a large number of squares ... a smaller problem
increases the likelihood of ties which I always
strive to avoid.
2. I wanted to be sure there was a solution that used most
of the squares for the same reasons.
In retrospect, It would have been hard to avoid having
a large solution - but perhaps that's the subject of
3. So. I constructed a quilt with the word POTM in it
(written in black on a white background in very
jagged lettering). It so happened that the quilt
was 36 squares wide by 26 tall for a total of 936.
4. Then I took the white edges of the 936 squares and
randomly assigned colors to them so that they
would continue to fit together.
5. Just for fun, I added 25 more squares - each of which
was a single color - thus arriving at a total of
961 squares to work with. (Coincidentally, 31*31
is 961, but I have no idea if such a quilt is
possible as I'm writing this on 11/24/98!)
6. Next step was to randomize the rotations. I decided to
leave them in the original order since it was a
pain in the neck to "shuffle" them. As a result,
the 36x26 solution I started with uses the squares
in order from 1 to 936.
Well ... that's about it ... I wondered if there's anything
better than the 36x26 solution buried in the squares!!!
SIZE AREA PERM AREA/PERM
31x31 961 124 7.75
32x30 960 124 7.7419
33x29 957 124 7.7177
34x28 952 124 7.6774
35x27 945 124 7.621
31x30 930 122 7.62
32x29 928 122 7.6066
36x26 936 124 7.5484 (as constructed)
30x30 900 120 7.50
29x30 870 118 7.37... (best found)