# Hexiamond Compatibility

## Introduction

A *hexiamond* is a plane figure made of six equilateral triangles joined
edge to edge.
There are 12 such figures, not distinguishing reflections and rotations.
They were first enumerated by T. H. O'Beirne.
The *compatibility problem*
is to find a figure that can be tiled with each of a set of polyforms.
Polyomino compatibility has been widely studied since the early 1990s.
Polyiamond compatibility was first studied systematically
by Margarita Lukjanska and Andris Cibulis,
who published a paper about it with Andy Liu in 2004 in the *Journal
of Recreational Mathematics.*
While the journal was preparing the paper for print,
I corresponded with the authors,
supplying the missing hexiamond compatibility and some other solutions.
My contributions arrived too late to appear in print,
but I was listed as a joint author.

This web page and my other page, Mixed
Polyiamond Compatibility, extend and correct the solutions in
the JRM article.
For sets of three hexiamonds, see Triple
Hexiamonds.

### Table

This table shows the smallest number of tiles needed
to construct a figure tilable by both hexiamonds.

| A | E | F | H | I | L | O | P | S | U | V | X |

A | * | 3 | 2 | 3 | 2 | 3 | × | 2 | 18 | 3 | 2 | × |

E | 3 | * | 6 | 2 | 3 | 3 | × | 6 | 4 | 3 | 6 | 3 |

F | 2 | 6 | * | 2 | 2 | 2 | 6 | 2 | 2 | 2 | 2 | 6 |

H | 3 | 2 | 2 | * | 3 | 2 | 6 | 2 | 2 | 2 | 3 | 3 |

I | 2 | 3 | 2 | 3 | * | 3 | × | 2 | 3 | 3 | 2 | × |

L | 3 | 3 | 2 | 2 | 3 | * | 6 | 2 | 3 | 2 | 3 | 3 |

O | × | × | 6 | 6 | × | 6 | * | 6 | 3 | 3 | 3 | × |

P | 2 | 6 | 2 | 2 | 2 | 2 | 6 | * | 2 | 2 | 2 | 2 |

S | 18 | 4 | 2 | 2 | 3 | 3 | 3 | 2 | * | 2 | 3 | × |

U | 3 | 3 | 2 | 2 | 3 | 2 | 3 | 2 | 2 | * | 3 | × |

V | 2 | 6 | 2 | 3 | 2 | 3 | 3 | 2 | 3 | 3 | * | × |

X | × | 3 | 6 | 3 | × | 3 | × | 2 | × | × | × | * |

### Solutions

These solutions are minimal. They are not necessarily uniquely minimal.
### 2 Tiles

### 3 Tiles

### 4 Tiles

### 6 Tiles

### 18 Tiles

| A | E | F | H | I | L | O | P | S | U | V | X |

A | * | 3 | × | 3 | 5 | 3 | × | × | 33 | 3 | 5 | × |

E | 3 | * | × | 3 | 3 | 3 | × | × | 5 | 3 | 15 | 3 |

F | × | × | * | × | × | × | × | 3 | × | × | × | × |

H | 3 | 3 | × | * | 3 | 3 | 33 | × | 3 | 5 | 3 | 3 |

I | 5 | 3 | × | 3 | * | 3 | × | × | 3 | 3 | 9 | × |

L | 3 | 3 | × | 3 | 3 | * | 15 | × | 3 | 3 | 3 | 3 |

O | × | × | × | 33 | × | 15 | * | × | 3 | 3 | 3 | × |

P | × | × | 3 | × | × | × | × | * | × | × | × | × |

S | 33 | 5 | × | 3 | 3 | 3 | 3 | × | * | 3 | 3 | × |

U | 3 | 3 | × | 5 | 3 | 3 | 3 | × | 3 | * | 3 | × |

V | 5 | 15 | × | 3 | 9 | 3 | 3 | × | 3 | 3 | * | × |

X | × | 3 | × | 3 | × | 3 | × | × | × | × | × | * |

### Table

This table shows the smallest number of tiles needed
to construct a holeless figure tilable by both hexiamonds.
Cells shaded in green need the same number of tiles if holes are allowed.

| A | E | F | H | I | L | O | P | S | U | V | X |

A | * | × | 2 | 3 | 2 | 4 | × | 2 | ? | 9 | 2 | × |

E | × | * | 32 | 2 | ? | 6 | × | 6 | 6 | ? | ? | × |

F | 2 | 32 | * | 2 | 2 | 2 | 42 | 2 | 2 | 2 | 2 | × |

H | 3 | 2 | 2 | * | 4 | 2 | 12 | 2 | 2 | 2 | 9 | × |

I | 2 | ? | 2 | 4 | * | 3 | × | 2 | 3 | 4 | 2 | × |

L | 4 | 6 | 2 | 2 | 3 | * | 10 | 2 | 3 | 2 | 4 | × |

O | × | × | 42 | 12 | × | 10 | * | 6 | 3 | 3 | 3 | × |

P | 2 | 6 | 2 | 2 | 2 | 2 | 6 | * | 2 | 2 | 2 | 2 |

S | ? | 6 | 2 | 2 | 3 | 3 | 3 | 2 | * | 2 | 3 | × |

U | 9 | ? | 2 | 2 | 4 | 2 | 3 | 2 | 2 | * | 3 | × |

V | 2 | ? | 2 | 9 | 2 | 4 | 3 | 2 | 3 | 3 | * | × |

X | × | × | × | × | × | × | × | 2 | × | × | × | * |

### Solutions

These solutions are the smallest known.
They are not necessarily uniquely minimal.
I show only solutions that differ from the solutions given above.

### 4 Tiles

### 6 Tiles

### 9 Tiles

### 10 Tiles

### 12 Tiles

### 32 Tiles

### 42 Tiles

Solutions with horizontal symmetry can be hard to find.

### Table

| A | E | F | H | I | L | O | P | S | U | V | X |

A | * | ? | 6 | 6 | 6 | 6 | × | 6 | ? | ? | 6 | × |

E | ? | * | ? | 4 | ? | ? | × | 6 | 6 | ? | ? | × |

F | 6 | ? | * | 6 | 6 | 6 | 6 | 6 | 30 | 12 | 6 | × |

H | 6 | 4 | 6 | * | 12 | 6 | 6 | 6 | 12 | 18 | 18 | × |

I | 6 | ? | 6 | 12 | * | 3 | × | 6 | ? | 3 | 6 | × |

L | 6 | ? | 6 | 6 | 3 | * | 6 | 6 | 18 | 3 | 6 | × |

O | × | × | 6 | 6 | × | 6 | * | 6 | 18 | 18 | 18 | × |

P | 6 | 6 | 6 | 6 | 6 | 6 | 6 | * | 18 | 12 | 6 | 2 |

S | ? | 6 | 30 | 12 | ? | 18 | 18 | 18 | * | 16 | 18 | × |

U | ? | ? | 12 | 18 | 3 | 3 | 18 | 12 | 16 | * | 18 | × |

V | 6 | ? | 6 | 18 | 6 | 6 | 18 | 6 | 18 | 18 | * | × |

X | × | × | × | × | × | × | × | 2 | × | × | × | * |

### Solutions

Last revised 2022-04-11.

Back to Pairwise Compatibility
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Polyform Compatibility
<
Polyform Curiosities

Col. George Sicherman
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