# Tetromino-Pentomino Compatibility

## Introduction

A *tetromino* is a figure made of four squares joined edge to edge.
A *pentomino* is a figure made of five squares joined edge to edge.
There are 5 tetrominoes and 12 pentominoes,
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 the early 1990s,
and 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.
The sites include 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 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.

Here I show minimal known tetromino-pentomino compatibility
figures, allowing or disallowing holes.
If you find a smaller solution or solve an unsolved case,
please let me know.

The solution shown below for the square tetromino
and the T pentomino is a variant of the minimal solution
found in 2011 by **deepgreen**.
His home page is here.

For compatibility of two pentominoes with or without holes, see
Pentomino Compatibility.
Resta's page of tetromino-pentomino compatibilities was originally
on GeoCities, a defunct web host.
It is now
here.
Mireles's site was also on GeoCities and has not been rebuilt.
The link above is to the Internet Archive.

## Tetromino Names

These are Livio Zucca's names for the tetrominoes:

## Pentomino Names

These are Golomb's names for the pentominoes:

### Table

This table shows the smallest number of tiles known to suffice
to construct a figure tilable by the tetromino and the pentomino.

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

I | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 8 10 | 8 10 | 8 10 | 4 5 | × | 4 5 | 8 10 |

L | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 8 10 | 4 5 | 4 5 |

N | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 8 10 | 4 5 | 4 5 | 4 5 | 8 10 | 4 5 | 4 5 |

Q | 8 10 | 4 5 | 4 5 | 4 5 | 4 5 | 68 85 | 64 80 | 4 5 | 16 20 | × | 8 10 | 4 5 |

T | 4 5 | 8 10 | 4 5 | 8 10 | 4 5 | 4 5 | 8 10 | 8 10 | 8 10 | 4 5 | 4 5 | 8 10 |

### Solutions

### Table

This table shows the smallest number of tiles known to suffice
to construct a holeless figure tilable by the tetromino and the pentomino.
Shaded cells indicate solutions that are minimal even if holes are allowed.

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

I | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 8 10 | 8 10 | 8 10 | 4 5 | × | 4 5 | 8 10 |

L | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 8 10 | 4 5 | 4 5 |

N | 4 5 | 4 5 | 4 5 | 4 5 | 4 5 | 8 10 | 4 5 | 4 5 | 4 5 | 12 15 | 4 5 | 4 5 |

Q | ? | 4 5 | 4 5 | 8 10 | 4 5 | ? | ? | 4 5 | ? | × | 8 10 | 4 5 |

T | 4 5 | 8 10 | 8 10 | 8 10 | 4 5 | 4 5 | 8 10 | 8 10 | 8 10 | 8 10 | 4 5 | 8 10 |

### Solutions

Here I show only solutions that are larger than the
corresponding minimal solutions allowing holes.
So far as I know, these solutions
are minimal. They are not necessarily uniquely minimal.

Solutions with diagonal symmetry are hard to find.
For those that do not have full (square) symmetry,
four copies can be arranged to create full symmetry.
### Table

This table shows the smallest number of tiles known to suffice
to construct a polyomino with diagonal
symmetry, tilable by the tetromino and the pentomino.

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

I | 16 20 | 16 20 | 16 20 | 16 20 | 8 10 | ? | ? | ? | 128 160 | × | 32 40 | ? |

L | 8 10 | 8 10 | 8 10 | 8 10 | 8 10 | 24 30 | 8 10 | 4 5 | 8 10 | 16 20 | 8 10 | 16 20 |

N | 8 10 | ? | 16 20 | 24 30 | 8 10 | ? | ? | ? | 8 10 | 8 10 | 8 10 | ? |

Q | 32 40 | 16 20 | 16 20 | 16 20 | 4 5 | ? | ? | 4 5 | ? | × | 16 20 | 4 5 |

T | 16 20 | 16 20 | 16 20 | 32 40 | 16 20 | ? | ? | ? | 16 20 | ? | 16 20 | ? |

### Solutions

Last revised 2023-12-12.

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Polyform Curiosities

Col. George Sicherman
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