# 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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Polyform Curiosities

Col. George Sicherman
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