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

### Unsolved

### Holeless Variants

### 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

### Unsolved

### Impossible

### Holeless Variants

### Unsolved

### Impossible

### Unsolved

### Impossible

### Holeless Variants

### Unsolved

### Impossible

### Unsolved

### Impossible

### Unsolved

### Impossible

Last revised 2019-12-13.

Back to Polyform Oddities
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Polyform Curiosities

Col. George Sicherman
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