# L Shapes from Pentacube Pairs

## Introduction

A *polycube* is a solid made of equal cubes joined
face to face, and a *pentacube* is a polycube with 5 cells.
There are 29 pentacubes, distinguishing mirror images:

The letters shown in black are my names.
Mirror images of chiral pentacubes are indicated with **′**;
e.g., **G′** is the mirror image of **G**.
The green symbols are Kate Jones's names.
The red symbols are Donald Knuth's names.

I define an *L-shaped polycube*
as a polycube prism whose base is L-shaped; that is, it consists of a rectangle
from one corner of which a smaller rectangle has been excised.

Here I show the smallest known L-shaped polycubes
that can be tiled with a given pair of pentacubes, using at least
one of each.
Chiral pairs of pentacubes are distinguished, and chiral pentacubes may not
be reflected when used in these tilings.

If you find a smaller solution, please write.

See also L Shapes from Two Pentominoes.

## Table of Results

| A | B | E | E′ | F | G | G′ | H | H′ | I | J | J′ | K | L | M | N | P | Q | R | R′ | S | S′ | T | U | V | W | X | Y | Z |

A | | 6 | 6 | 12 | 14 | 6 | 13 | 6 | 6 | 6 | 3 | 6 | 4 | 2 | 6 | 8 | 12 | 6 | 8 | 8 | 80 | 6 | 7 |

B | | | 3 | 12 | 18 | 3 | 7 | 2 | 9 | 6 | 19 | 6 | 5 | 3 | 5 | 12 | 12 | 3 | 8 | 14 | 20 | 6 | 14 |

E | | | | 8 | 9 | 12 | 9 | 6 | 2 | 5 | 6 | 6 | 6 | 6 | 3 | 6 | 3 | 2 | 3 | 8 | 4 | 6 | 4 | 6 | 6 | 9 | 16 | 6 | 6 |

F | | | | | 18 | 9 | 7 | 4 | 9 | 5 | 12 | 8 | 3 | 4 | 9 | 4 | 12 | 2 | 4 | 15 | 42 | 5 | 12 |

G | | | | | | | × | 12 | 8 | 21 | 2 | 6 | 12 | 6 | 56 | 7 | 6 | 3 | 56 | 30 | 12 | 18 | 21 | 6 | 3 | 18 | 440 | 8 | 7 |

H | | | | | | | | | 4 | 5 | 6 | 6 | 6 | 4 | 6 | 6 | 4 | 2 | 8 | 6 | 6 | 4 | 10 | 6 | 4 | 6 | 14 | 6 | 10 |

I | | | | | | | | 8 | 12 | 2 | 11 | 6 | 2 | 4 | 11 | 4 | 7 | 6 | 2 | 7 | 8 | 5 | 7 |

J | | | | | | | | | | | | 3 | 6 | 2 | 7 | 6 | 3 | 3 | 8 | 7 | 6 | 3 | 3 | 4 | 4 | 9 | 12 | 4 | 4 |

K | | | | | | | | | | 8 | 8 | 6 | 3 | 4 | 2 | 9 | 10 | 6 | 8 | 6 | 12 | 6 | 12 |

L | | | | | | | | | | | 8 | 2 | 2 | 4 | 6 | 4 | 6 | 3 | 2 | 3 | 7 | 5 | 7 |

M | | | | | | | | | | | | 10 | 5 | 6 | 19 | 9 | 15 | 3 | 15 | 9 | 20 | 8 | 10 |

N | | | | | | | | | | | | | 2 | 4 | 6 | 7 | 6 | 6 | 7 | 6 | 14 | 6 | 6 |

P | | | | | | | | | | | | | | 2 | 5 | 3 | 2 | 2 | 2 | 4 | 6 | 2 | 2 |

Q | | | | | | | | | | | | | | | 4 | 2 | 3 | 2 | 3 | 4 | 6 | 4 | 4 |

R | | | | | | | | | | | | | | | | | | | | 30 | 12 | 6 | 12 | 6 | 8 | 9 | 18 | 7 | 16 |

S | | | | | | | | | | | | | | | | | | | | | | 4 | 15 | 6 | 3 | 4 | 24 | 8 | 4 |

T | | | | | | | | | | | | | | | | | | 3 | 12 | 9 | 72 | 8 | 21 |

U | | | | | | | | | | | | | | | | | | | 6 | 12 | 3 | 2 | 6 |

V | | | | | | | | | | | | | | | | | | | | 9 | 24 | 8 | 2 |

W | | | | | | | | | | | | | | | | | | | | | 28 | 6 | 16 |

X | | | | | | | | | | | | | | | | | | | | | | 10 | 126 |

Y | | | | | | | | | | | | | | | | | | | | | | | 5 |

Z | | | | | | | | | | | | | | | | | | | | | | | |

## Tilings

The pair **G** and **G′**
cannot tile any L-shaped polycube.
Each of the solutions shown for pairs **A**-**X**,
**C**-**X**, and
**X**-**Z**
was formed by joining two rectangular boxes.
Smaller solutions may well exist for these.
### 2 Tiles

### 3 Tiles

### 4 Tiles

### 5 Tiles

### 6 Tiles

### 7 Tiles

### 8 Tiles

### 9 Tiles

### 10 Tiles

### 11 Tiles

### 12 Tiles

### 13 Tiles

### 14 Tiles

### 15 Tiles

### 16 Tiles

### 18 Tiles

### 19 Tiles

### 20 Tiles

### 21 Tiles

### 24 Tiles

### 28 Tiles

### 30 Tiles

### 42 Tiles

### 56 Tiles

### 72 Tiles

### 80 Tiles

### 126 Tiles

### 440 Tiles

Last revised 2022-10-31.

Back to Polyform Tiling
<
Polyform Curiosities

Col. George Sicherman
[ HOME
| MAIL
]