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 heptahex oddities, see Heptahex Oddities. For other orders of polyhexes, see Polyhex Oddities.

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.