# Problem of the Month (November 2004)

A couple months ago, when we studied polypolyforms, Jeremy Galvagni suggested a variant of the problem that I thought was worth being a problem of the month in its own right. Given a shape S, find the smallest shape that can be tiled by S in more than one way. We call these shapes Galvagni Figures. Mike Reid was interested in such tilings without holes. Thus we call such figures Reid Figures. Corey Plover was interested in such tilings that did not allow reflections of the tile. Thus we call such figures Plover Figures.

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:

POLYIAMONDS
SizeGalvagniReidPlover
1-5 Erich
Friedman
George
Sicherman
George
Sicherman
6 Erich
Friedman
George
Sicherman
George
Sicherman
7 Mike
Reid
George
Sicherman
George
Sicherman
8 Mike
Reid
Mike
Reid
George
Sicherman
9 George
Sicherman
George
Sicherman
George
Sicherman

The generalized 3-iamonds are here.

POLYABOLOES
SizeGalvagniReidPlover
1-3 Erich
Friedman
? George
Sicherman
4 Mike
Reid
George
Sicherman
George
Sicherman
5 George
Sicherman
George
Sicherman
George
Sicherman
6 George
Sicherman
George
Sicherman
George
Sicherman

POLYOMINOES
SizeGalvagniReidPlover
1-4 Erich
Friedman
same Corey
Plover
5 Erich
Friedman
George
Sicherman
Corey
Plover
6 George Sicherman
& Corey Plover
George
Sicherman
Corey
Plover
7 George
Sicherman
George
Sicherman
George
Sicherman
8 George
Sicherman
George
Sicherman
George
Sicherman
9 George
Sicherman
(3 sets)
George
Sicherman
George
Sicherman

POLYKINGS
SizeGalvagniReidPlover
1-4 George
Sicherman
? George
Sicherman
5 Corey
Plover
? George
Sicherman
6 George
Sicherman
? George
Sicherman
POLYPENTS
SizeGalvagniPlover
1-3 Erich
Friedman
same
4 George
Sicherman
George
Sicherman
5 Scott
Reynolds
George
Sicherman
6 George
Sicherman
(3 sets)
George
Sicherman
7 George
Sicherman
(4 sets
total)
George
Sicherman

POLYHEXES
SizeGalvagniReidPlover
1-3 Erich
Friedman
George
Sicherman
same
4 Erich
Friedman
George
Sicherman
George
Sicherman
5 Erich
Friedman
George
Sicherman
George
Sicherman
6 Mike Reid George
Sicherman
George
Sicherman
7 George
Sicherman
George
Sicherman
George
Sicherman
8 George
Sicherman
(there
are
seven
sets
total)
George
Sicherman
George
Sicherman
POLYHEPTS
SizeGalvagniPlover
1-3 George
Sicherman
same
4 George
Sicherman
George
Sicherman
5 George
Sicherman
George
Sicherman

POLYOCTS
SizeGalvagniPlover
1-3 George
Sicherman
same
4 George
Sicherman
George
Sicherman
5 George
Sicherman
George
Sicherman
POLYNONS
SizeGalvagniPlover
1-3 George
Sicherman
same
4 George
Sicherman
George
Sicherman
5 George
Sicherman
George
Sicherman
POLYDECS
SizeGalvagniPlover
1-3 George
Sicherman
same
4 George
Sicherman
George
Sicherman

George Sicherman has many pages devoted to this problem here.

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.