Could you explain what your algorithm was?

I have one of the puzzles, number 4, I believe, and spent a large amount
of time trying to find a solution that was not trial and error. I could
not.

The algorithm that I used to have the computer solve it for me was to
fill the 5x5 in the following manner, recursively, returning when no
possible pieces fit.

 1  2  4  7 11
   /  /  /  /
  /  /  /  /
 3  5  8 12 16
   /  /  /  /
  /  /  /  /
 6  9 13 17 20
   /  /  /  /
  /  /  /  /
10 14 18 21 23
   /  /  /  /
  /  /  /  /
15 19 22 24 25

(wrapping at the edges to keep incrementing properly)

I did that because given any pieces diagonal from one another, there are
at most two pieces that can fill the gap (line up with both correctly).
(When the four colors are different, there are two tiles
When there is a single repeated color, there is one tile
When there are 2 pairs of colors there is no tile
And in all these cases, if the tile(s) was already used, or didn't
exist, that is the bottom of that branch of the search tree)

Is this better or worse than the algorithm you used?

Has anyone found a non-brute-force solution scheme?

-Dale