# Minimal Incompatibility for Polyiamonds

## Introduction

A *polyiamond* is a figure made of equal equilateral triangles 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 polyiamond of orders 1 through 7 the smallest known
polyiamonds 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 diamond compatibility
and found the minimal polyiamond incompatible with the diamond.

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

## Solutions

Moniamond |

| ∞ | None |

Diamond |
---|

| 14 | |

Triamond |
---|

| 12 | |

Tetriamonds |
---|

| 6 |
| | 10 | |

| 5 | |

Pentiamonds |
---|

| 6 |
| | 10 | |

| 6 |
| | 4 | |

Hexiamonds |
---|

| 6 |
| | 8 | |

| 4 |
| | 6 | |

| 6 |
| | 9 | |

| 6 |
| | 8 | |

| 6 |
| | 4 | |

| 8 |
| | 4 | |

Heptiamonds |
---|

| 6 |
| | 6 | |

| 6 |
| | 5 | |

| 6 |
| | 4 | |

| 6 |
| | 4 | |

| 6 |
| | 6 | |

| 6 |
| | 5 | |

| 5 |
| | 6 | |

| 4 |
| | 6 | |

| 6 |
| | 6 | |

| 4 |
| | 4 | |

| 4 |
| | 4 | |

| 6 |
| | 4 | |

Last revised 2015-03-21.

Back to Pairwise Compatibility
<
Polyform Compatibility
<
Polyform Curiosities

Col. George Sicherman
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