# Problem of the Month(January 2008)

The problem of tiling equal polyominoes in squares has been well-studied. This month we investigate tiling equal polyominoes in frames, squares missing from the centers of squares. We also investigate tiling equal polyominoes in circles, a totally symmetric set of lattice points whose distance from the center is smaller than a constant. Can you find examples of the smallest frames or circles made from various polyominoes? What other shapes can be tiled well with equal polyominoes?

Frame Tilings:

Small Polyominoes

Pentominoes

Hexominoes

Heptominoes

Octominoes

Nonominoes
 (George Sicherman) (George Sicherman)

Decominoes
 (George Sicherman) (George Sicherman)

Undecominoes
 (George Sicherman) (George Sicherman)
 (George Sicherman)

Mike Reid gave these non-trivial frame tilings for some larger L polyominoes:

Off-Center Frame Tilings:

Small Polyominoes

Pentominoes
 (George Sicherman)

Hexominoes
 (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman)

Heptominoes
 (George Sicherman) (George Sicherman)

Octominoes
 (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)

Nonominoes
 (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman)

George Sicherman investigated triangular frames made from polyiamonds...

...and hexagonal frames made from polyhexes:

Circular Tilings:

Small Polyominoes

Pentominoes

Hexominoes

Heptominoes

Octominoes

Nonominoes
 (George Sicherman)

Decominoes

Undecominoes

I also investigated tilings of donuts, circles missing from the centers of circles. I even allowed disconnected regions. My results are here.

Claudio Baiocchi suggested that we look for full symmetry configurations. We call these Baiocchi figures.

Baiocchi Figures:

Small Polyominoes

Pentominoes
 (GeorgeSicherman) (GeorgeSicherman) (GeorgeSicherman)

Hexominoes
 (GeorgeSicherman) (GeorgeSicherman) (GeorgeSicherman)
 (GeorgeSicherman) (GeorgeSicherman)
 (GeorgeSicherman) (Corey Plover) (GeorgeSicherman) (GeorgeSicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)

George Sicherman also investigated Baiocchi figures made from other polyforms:

Polyiamonds

The 9-iamonds are here.

Polypents
 (Károly Hajba) (George Sicherman) (George Sicherman)

 (George Sicherman)

The pentapents can be found here.

Polyhexes

The hexahexes can be found here.

Polyhepts

The pentahepts can be found here.

Polyocts
 (Károly Hajba) (Károly Hajba)

 (Károly Hajba)
 (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman)

The pentaocts can be found here.

The Baiocchi figures for polycubes can be found here.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 1/1/09.