It is easy to see why polyominoes invariant under a rotation of 90o have MTN ∞. Can you improve any of these results? Is there any other polyomino with MTN of 3 besides the straight triomino? Is there a polyomino with MTN of 5? What is the largest MTN you can find for a polyomino?
Can you prove that every integer greater than 1 is a MTN of some shape? What are the MTNs for other polyforms, such as polyiamonds and polyhexes?
POLYIAMONDS
George Sicherman improved one of the hexiamond solutions. Mike Reid improved one of the pentiamonds and one of the hexiamonds, and did all the octiamonds.
The 5-iamonds without holes are here. The 6-iamonds without holes are here. The 7-iamonds without holes are here. The 8-iamonds are here, and those without holes are here. The 9-iamonds are here, and those without holes are here. Generalized 3-iamonds are here.
POLYABOLOES
Mike Reid improved 1 of the polyaboloes, and solved five others by showing how to do all the non-square parallelograms. George Sicherman did the pentaboloes and hexaboloes.
The 6-aboloes are here.
POLYHOUSES
POLYOMINOES
George Sicherman first solved the T pentomino, the S hexomino, and the vast majority of the heptominoes and octominoes! He asked whether there is a shape whose MTN yields an unsymmetric shape.
The smallest solution for the T pentomino is due to Corey Plover. Corey also found better solutions for 6 hexominoes and 3 heptominoes, including one with an asymmetrical minimal configuration.
George Sicherman, Corey Plover and Steve Butler pointed out that there are many polyominoes with MTN 3: scaled up copies of straight triominoes! Corey also found two hexominoes that also have MTN 3.
Joseph DeVincentis and Mike Reid proved that all numbers larger than 1 are the MTN of some polyomino. Mike's construction uses unions of 2x2 squares like this shape with MTN 5:
Mike Reid used the same idea (with 3x3 squares) to generate a shape whose only minimal configuration is asymmetrical:
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:
Mike Reid points out that there are shapes with infinitely many minimal solutions:
Here are the best known solutions for the small polyominoes:
The 5-ominoes without holes can be found here. The 6-ominoes without holes can be found here. The 7-ominoes can be found here, and those without holes can be found here. The 8-ominoes can be found here. Some 9-ominoes can be found here, and here, and here. Most of these are due to George Sicherman.
POLYKINGS
George Sicherman and I managed to solve all the tetrakings except the obvious 2 that won't work. Corey Plover used his program to find most of the hard solutions for the pentakings. George Sicherman solved the last one, and made some pictures.
The 6-kings are here.
POLYPENTS
Scott Reynolds improved the yellow tetrapent, and George Sicherman improved the gray, red, and orange ones. Scott Reynolds managed to do most of the pentapents and George Sicherman solved the last one and improved several others. Then George Sicherman tackled hexapents and heptapents.
The 6-pents are here, here, and here. The 7-pents are here, here, and here.
POLYHEXES
Mike Reid solved all of the pentahexes and most of the hexahexes. George Sicherman improved one of the pentahexes, and did most of the heptahexes.
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The 3-hexes without holes are here. The 4-hexes without holes are here. The 5-hexes without holes are here. The 6-hexes are here, and those without holes are here. The 7-hexes are here and here, and those without holes are here. The 8-hexes are here, here, here, here, here, here, and here.
POLYHEPTS
George Sicherman considered polyhepts:
POLYOCTS
George Sicherman also considered polyocts:
POLYNONS
George Sicherman also considered polynons:
OTHER VARIANTS
Corey Plover thought that MTN's should not allow flipping of the shape. Richard Sabey thought that two tilings should be considered "similar" if one can take some tiles of the first tiling and rotate them all by the same angle to get the second tiling. Corey Plover wrote a computer program to determine the MTN's of polyominoes, with flipping and without, with similarity and not. His results are here for tetrominoes, pentominoes, and hexominoes. The yellow figures are the original problem, the orange is Corey's variant, the blue are Richard's variant, and the purple is both sets of rules.
Mike Reid and George Sicherman also considered pentacubes.
Multiple tiling numbers for tetrominoids are here.
If you can extend any of these results, please
e-mail me.
Click here to go back to Math Magic. Last updated 6/29/07.
