Hexahex Oddities

A hexahex oddity is a figure with binary symmetry formed by an odd number of copies of a hexahex. Here are the minimal known oddities for hexahexes. Please write if you find a smaller solution or solve an unsolved case. Mike Reid contributed improvements to some of these solutions.

For other orders of polyhexes, see Polyhex Oddities.

[ Rowwise Bilateral | Columnwise Bilateral | Birotary on Edge | Birotary on Cell | Double Bilateral on Edge | Double Bilateral on Cell | Sextuple Rotary | Full ]

Rowwise Bilateral

Unsolved

Holeless Variants


 

Columnwise Bilateral

Unsolved

Impossible

Proof.

Color the cells of the plane as shown. Assume without loss of generality that the median of the oddity passes through a white column. Each tile changes the balance of red and green cells by ±2. Thus an odd number of tiles must have unequal numbers of red and green cells. But an oddity with columnwise symmetry must have equal numbers of red and green cells.

Holeless Variants


 

Birotary on Edge

Unsolved

Impossible

Holeless Variants


 

Birotary on Cell

Unsolved

Impossible

Double Bilateral on Edge

Unsolved

Impossible

Holeless Variants


 

Double Bilateral on Cell

Unsolved

Impossible

Sextuple Rotary

Unsolved

Impossible

Full

Unsolved

Impossible

Last revised 2016-02-05.


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Col. George Sicherman [ HOME | MAIL ]