# Pentomino Compatibility

## Introduction

A *pentomino* is a figure made of five squares joined
edge to edge.
There are 12 such figures, not distinguishing reflections and rotations.
They were first enumerated and studied by Solomon Golomb.
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 1992, when
Pablo Coll first posed and studied the problem for pentominoes.
Two well-known websites, Poly^{2}ominoes by Jorge Mireles and
Polypolyominoes by
Giovanni Resta, present the results of their authors' systematic searches
for compatibility figures.
Mireles's site includes solutions by other researchers, especially Mike Reid.
So far as I know, polyomino compatibility has not been treated in print
since Golomb first raised the issue in 1981,
except in Issue 3 of Rodolfo Marcelo Kurchan's
periodical *Puzzle Fun* (1995), and
in a series of articles called Polyomino Number Theory,

written by Andris Cibulis, Andy Liu, Bob Wainwright,
Uldis Barbans, and Gilbert Lee from 2002 to 2005.

The websites and the articles present a wealth of polyomino compatibilities.
They do not show all current results.
I do so below, for pentomino-pentomino compatibility only.
I am not prepared to maintain a current catalogue of results for
other kinds of pairs of polyominoes, or for larger sets such as
the pentomino triples that Livio Zucca has
collected here.

I also show holeless variants. So far as I know,
these have been covered before only in *Puzzle Fun*.

For compatibility figures with an odd number of tiles, see
Livio Zucca's
Pentomino Odd Pairs.
For Galvagni compatibility (self-compatibility), see
Galvagni Figures & Reid Figures
for Pentominoes.

### Table

This table shows the smallest number of tiles known to suffice
to construct a figure tilable by both pentominoes.
The names used for the pentominoes are
Golomb's original names.

| F | I | L | N | P | T | U | V | W | X | Y | Z |

F | * | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 |

I | 10 | * | 2 | 2 | 2 | 4 | 12 | 4 | 10 | × | 2 | 20 |

L | 2 | 2 | * | 4 | 2 | 2 | 2 | 2 | 2 | 44 | 2 | 2 |

N | 2 | 2 | 4 | * | 2 | 2 | 2 | 2 | 2 | 16 | 2 | 2 |

P | 2 | 2 | 2 | 2 | * | 2 | 2 | 2 | 2 | 4 | 2 | 2 |

T | 2 | 4 | 2 | 2 | 2 | * | 4 | 2 | 14 | 4 | 2 | 2 |

U | 4 | 12 | 2 | 2 | 2 | 4 | * | 2 | 2 | × | 2 | 4 |

V | 4 | 4 | 2 | 2 | 2 | 2 | 2 | * | 6 | × | 2 | 4 |

W | 2 | 10 | 2 | 2 | 2 | 14 | 2 | 6 | * | ? | 2 | 4 |

X | 2 | × | 44 | 16 | 4 | 4 | × | × | ? | * | 2 | ? |

Y | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | * | 2 |

Z | 2 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ? | 2 | * |

### Solutions

So far as I know, these solutions
are minimal. They are not necessarily uniquely minimal.
#### 2 Tiles

#### 4 Tiles

#### 6 Tiles

#### 10 Tiles

#### 12 Tiles

#### 14 Tiles

#### 16 Tiles

#### 20 Tiles

#### 44 Tiles

The thinnest compatibility figure for a pair of pentominoes
is not always that with the smallest area.
Here are the thinnest known solutions for pairs of pentominoes.
### Width 2

### Width 3

### Width 4

### Width 5

### Width 6

### Width 7

### Width 8

### Width 9

### Width 12

### Width 20

If a solution has diagonal symmetry, four copies of it can be arranged
to make a solution with square symmetry.
However, diagonally symmetric solutions are hard to find.

The X pentomino is not compatible with any of the pentominoes
I, U, V, W, and Z.
However, it is compatible with some mixtures of them:

Are there any other hybrid solutions?

### Table

The green figures represent holeless tilings that are
minimal even without the condition of holelessness.

| F | I | L | N | P | T | U | V | W | X | Y | Z |

F | * | 10 | 2 | 2 | 2 | 2 | 4 | 6 | 2 | 2 | 2 | 2 |

I | 10 | * | 2 | 2 | 2 | 32 | ? | 10 | 10 | × | 2 | ? |

L | 2 | 2 | * | 4 | 2 | 2 | 2 | 2 | 2 | × | 2 | 2 |

N | 2 | 2 | 4 | * | 2 | 2 | 2 | 2 | 2 | 16 | 2 | 2 |

P | 2 | 2 | 2 | 2 | * | 2 | 2 | 2 | 2 | 4 | 2 | 2 |

T | 2 | 32 | 2 | 2 | 2 | * | ? | 2 | 16 | 4 | 2 | 30 |

U | 4 | ? | 2 | 2 | 2 | ? | * | ? | 2 | × | 2 | ? |

V | 6 | 10 | 2 | 2 | 2 | 2 | ? | * | 6 | × | 2 | 4 |

W | 2 | 10 | 2 | 2 | 2 | 16 | 2 | 6 | * | ? | 2 | 10 |

X | 2 | × | × | 16 | 4 | 4 | × | × | ? | * | 2 | ? |

Y | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | * | 2 |

Z | 2 | ? | 2 | 2 | 2 | 30 | ? | 4 | 10 | ? | 2 | * |

### Solutions

I omit solutions that are the same as in Basic Solutions above.
So far as I know, the following solutions
are minimal. They are not necessarily uniquely minimal.

#### 6 Tiles

#### 10 Tiles

#### 16 Tiles

#### 30 Tiles

#### 32 Tiles

Last revised 2022-04-17.

Back to Pairwise Compatibility
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Polyform Compatibility
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Polyform Curiosities

Col. George Sicherman
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