What are the Galvagni, Reid, and Plover Figures for small polyforms? What can be proved about such things?

Many people worked on these problems and sent solutions. The attributions below indicate who did the most work on any given problem, but it was rare for one person to do it all.

Mike Reid pointed out that there are shapes with infinitely many minimal solutions:

Joseph DeVincentis and Mike Reid proved that all numbers larger than 1 are the Galvagni number of some polyomino. Mike's construction on the left uses unions of 2x2 squares to make Galvagni number 5. Mike Reid used the same idea (with 3x3 squares) to generate a shape whose only minimal configuration is asymmetrical, on the right. Corey Plover searched polyominoes for a shape that had a large number of possible minimal solutions. His best yielded 23 solutions. Then Mike Reid found this polyomino with 27 different minimal configurations:

You can also see George Sicherman's pages of Galvagni figures for pentacubes, tetrominoids, and tetrarhons.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 7/20/09.