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?
ANSWERS
Frame Tilings:
Small Polyominoes
Pentominoes
Hexominoes
Heptominoes
Octominoes
Nonominoes
![](0108/fGG9.gif) (George Sicherman)
| ![](0108/fVV9.gif) (George Sicherman)
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Decominoes
![](0108/f10-127.gif) (George Sicherman)
| ![](0108/f10-128.gif) (George Sicherman)
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Undecominoes
![](0108/f11-198.gif) (George Sicherman)
| ![](0108/f11-199.gif) (George Sicherman)
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![](0108/f11-200.gif) (George Sicherman)
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Mike Reid gave these non-trivial frame tilings for some larger L polyominoes:
Off-Center Frame Tilings:
Small Polyominoes
Pentominoes
Hexominoes
![](0108/oJ6.gif) (George Sicherman)
| ![](0108/oF6.gif) (George Sicherman)
| ![](0108/oA6.gif) (George Sicherman)
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![](0108/oD6.gif) (George Sicherman)
| ![](0108/oG6.gif) (George Sicherman)
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Heptominoes
![](0108/oL7.gif) (George Sicherman)
| ![](0108/oD7.gif) (George Sicherman)
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Octominoes
| ![](0108/oD8.gif) (George Sicherman)
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![](0108/oL8.gif) (George Sicherman)
| ![](0108/oA8.gif) (George Sicherman)
| ![](0108/oF8.gif) (George Sicherman)
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![](0108/oJ8.gif) (George Sicherman)
| ![](0108/oN8.gif) (George Sicherman)
| ![](0108/oG8.gif) (George Sicherman)
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Nonominoes
![](0108/9-883.png) (George Sicherman)
| ![](0108/9-901.png) (George Sicherman)
| ![](0108/9-931.png) (George Sicherman)
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![](0108/9-1031.png) (George Sicherman)
| ![](0108/9-378.png) (George Sicherman)
| ![](0108/9-354.png) (George Sicherman)
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![](0108/9-115.png) (George Sicherman)
| ![](0108/9-125.png) (George Sicherman)
| ![](0108/9-336.png) (George Sicherman)
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![](0108/9-339.png) (George Sicherman)
| ![](0108/9-71.png) (George Sicherman)
| ![](0108/9-2.png) (George Sicherman)
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![](0108/9-376.png) (George Sicherman)
| ![](0108/9-197.png) (George Sicherman)
| ![](0108/9-562.png) (George Sicherman)
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| ![](0108/9-357.png) (George Sicherman)
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George Sicherman investigated triangular frames made from polyiamonds...
...and hexagonal frames made from polyhexes:
Circular Tilings:
Small Polyominoes
Pentominoes
Hexominoes
Heptominoes
Octominoes
Nonominoes
![](0108/cD9.gif) (George Sicherman)
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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
![](0108/sU5.gif) (George Sicherman)
| ![](0108/sZ5.gif) (George Sicherman)
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| ![](0108/sT5.gif) (George Sicherman)
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Hexominoes
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| ![](0108/sRR6.gif) (George Sicherman)
| ![](0108/fX6.gif) (George Sicherman)
| ![](0108/sLL6.gif) (George Sicherman)
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![](0108/sL6.gif) (George Sicherman)
| ![](0108/sY6.gif) (George Sicherman)
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![](0108/sJJ6.gif) (George Sicherman)
| ![](0108/sNN6.gif) (Corey Plover)
| ![](0108/sS6.gif) (George Sicherman)
| ![](0108/sUU6.gif) (George Sicherman)
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![](0108/sTT6.gif) (George Sicherman)
| ![](0108/sZZ6.gif) (George Sicherman)
| ![](0108/sE6.gif) (George Sicherman)
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George Sicherman also investigated Baiocchi figures made from other polyforms:
Polyiamonds
The 9-iamonds are here.
Polypents
![](0108/spent1.gif) (Károly Hajba)
| ![](0108/3-1.gif) (George Sicherman)
| ![](0108/3-2.gif) (George Sicherman)
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![](0108/spent2.gif) (George Sicherman)
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The pentapents can be found here.
Polyhexes
The hexahexes can be found here.
Polyhepts
The pentahepts can be found here.
Polyocts
![](0108/soct1.gif) (Károly Hajba)
| ![](0108/oct1.gif) (Károly Hajba)
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![](0108/soct2.gif) (Károly Hajba)
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![](0108/oct2.gif) (George Sicherman)
| ![](0108/oct3.gif) (George Sicherman)
| ![](0108/oct4.gif) (George Sicherman)
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![](0108/oct5.gif) (George Sicherman)
| ![](0108/oct6.gif) (George Sicherman)
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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.
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