Holeless Plover Figures for Polyominoes

A Galvagni Figure is a minimal figure that can be tiled with a given polyform shape in two or more ways. A Plover Figure is a Galvagni Figure in which the tiles are not merely congruent but identical; that is, they are not distinct mirror images. For tiles with reflective symmetry, a Plover Figure is the same as a Galvagni Figure.

Corey Plover has found Plover Figures (and most of the Galvagni Figures) for polyominoes up to order 6. You may see the tetrominoes here, the pentominoes here, and the hexominoes here at Erich Friedman's Math Magic. Here are Plover figures without holes for pentominoes, hexominoes, and heptominoes. The gray polyominoes have not been solved. The black ones are known to have no Plover figures with or without holes.

Pentominoes

The following picture shows holeless Plover Figures for pentominoes without mirror symmetry. If you find smaller solutions, please let me know! The solutions for the P, F, and Y pentominoes are known to be minimal with or without holes.

Hexominoes

The following picture shows holeless Plover Figures for hexominoes without mirror symmetry. If you find smaller solutions or solve an unsolved case, please let me know! Of the gray hexominoes, only the second is known to have a Plover figure with holes.

Heptominoes

The following picture shows holeless Plover Figures for heptominoes without mirror symmetry. If you find smaller solutions or solve an unsolved case, please let me know!

Last revised 2011-01-23.


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Col. George Sicherman [ HOME | MAIL ]