Urabeth,
I just finished this game a couple of days ago and this puzzle isn't really a chess game. I had trouble with it too because I couldn't figure out what I was supposed to do and kept clicking on the circles and squares and nothing happened.
If Mac doesn't still have a saved game, go here
http://www.uhs-hints.com/uhsweb/hints/ndrew4/1006.php Scroll down to "How to solve the checkerboard puzzle" and click on it. What I did was click on each "next hint" until I had them all on the page and then did a copy and paste into a Word document.
If you still have problems solving the puzzle, the last hint will take you to step-by-step directions for one solution.
Good luck.

Jema