Polyhex Variations

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Polyhex Variations

   Inspired by an entry regarding polyomino variations in the Puzzle Zapper blog, by Alexandre Muņiz, I decided to look at the same phenomenon for the polyhexes. A hexagon has 12 symmetries - 6 rotations and 6 reflections. By restricting these symmetries, we can create derivative sets of pieces with different properties. The diagrams below show the possible positions of a red "peg" starting at the top-left. Physical pieces could be made in this way, with holes drilled at the positions of any red circles, and pegs set at the upper-left of each hex on the board. Pieces can also be made by adding a suitably shaped single hole, which must be aligned the same way in any solution, or by varying the geometry of the edges. These pieces can be made to completely tile the plane, with the exception of pieces with ternary and vertical symmetry, which require holes, gaps or a surface design.

 

 

Symmetry-Restricted Hexagons

Symmetry Full Sextarotary Ternary & Horizontal Ternary & Vertical Trirotary Double Bilateral Horizontal Vertical Birotary None
Diagram
Name Free Polyhexes One-sided Polyhexes ? ? ? Polybricks ? ? One-sided Polybricks Fixed Polyhexes
Hole
Solid

 

Set Sizes

The pieces for each set consist of a number of copies of the standard "free" set of polyhexes. The number of copies of each piece varies from 1 to 12, depending on the symmetries of the piece and the restrictions imposed by the set.    

 

 
1 1 1 1 1 1 1 1 1 1
1 2 1 1 2 1 1 1 2 2
1 1 2 1 2 1 2 1 1 2
1 1 1 2 2 1 1 2 1 2
1 2 2 2 4 1 2 2 2 4
1 1 1 1 1 2 2 2 3 3
1 1 2 1 2 2 4 3 3 6
1 1 1 2 2 2 3 4 3 6
1 2 1 1 2 3 3 3 6 6
1 2 2 2 4 3 6 6 6 12

 

 

Set Full Sextarotary Ternary & Horizontal Ternary & Vertical Trirotary Double Bilateral Horizontal Vertical Birotary None
OEIS Link A000228 A006535 A197551 A197552 A197549 A057973 A197553 A197554 A197550 A001207
1 1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 1 2 2 2 3 3
3 3 3 4 4 5 5 7 7 7 11
4 7 10 10 10 16 16 24 24 28 44
5 22 33 37 36 62 55 99 98 99 186
6 82 147 147 146 276 225 416 415 433 814
7 333 620 637 631 1222 949 1852 1846 1852 3652
8 1448 2821 2823 2815 5563 4269 8386 8378 8463 16689

 

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