# Minimal Incompatibility for Polyhexes

## Introduction

A *polyhex* is a figure made of equal regular hexagons joined
edge to edge.
The *compatibility problem*
is to find a figure that can be tiled with each of a set of polyforms.
Here I show for each polyhex of orders 1 through 5 the smallest known
polyhexes that are *not* compatible with it.

In some cases, incompatibility is probable but has not been proved
by analysis or exhaustion.
Proved cases are shown in red. Unproved cases are shown in blue.

Andris Cibulis first studied dihex compatibility
and found the minimal polyhex incompatible with the dihex.
He also found a very large compatibility for the J tetrahex.

See also Minimal Incompatibility for Polyominoes
and
Minimal Incompatibility for Polyiamonds.

## Solutions

Monohex |

| ∞ | None |

Dihex |
---|

| 16 | |

Trihexes |
---|

| 6 |
| | 10 | |

| 6 | |

Tetrahexes |
---|

| 4 |
| | 10 | |

| 6 |
| | 7 | |

| 7 |
| | 7 | |

| 4 | |

Pentahexes |
---|

| 4 |
| | 6 | |

| 6 |
| | 6 | |

| 6 |
| | 7 | |

| 6 |
| | 6 | |

| 7 |
| | 7 | |

| 6 |
| | 7 | |

| 6 |
| | 7 | |

| 6 |
| | 6 | |

| 6 |
| | 7 | |

| 6 |
| | 7 | |

| 6 |
| | 5 | |

Last revised 2015-07-01.

Back to Pairwise Compatibility
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Polyform Compatibility
<
Polyform Curiosities

Col. George Sicherman
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