# Tiling L Shapes with a Pentacube

## 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:

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 pentacube.
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 Pentacube Pairs.

## Tilings

The **G** and **X**
pentacubes cannot tile any L-shaped polycube.
Each of the solutions shown for pentacubes
**T**,
**R**,
**A**,
**Z**,
and
**M**
are formed by joining two rectangular box tilings.
Smaller solutions may exist for these pentacubes.

The solutions shown for pentacubes **I**,
**E**,
and
**B** are formed by joining two
rectangular box tilings.
No smaller solutions exist for these pentacubes.

### 1 Tile

### 2 Tiles

### 6 Tiles

### 8 Tiles

### 16 Tiles

### 20 Tiles

### 33 Tiles

### 42 Tiles

### 120 Tiles

### 128 Tiles

### 216 Tiles

### 240 Tiles

### 396 Tiles

Last revised 2024-02-14.

Back to Polycube Tiling
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Polyform Tiling
<
Polyform Curiosities

Col. George Sicherman
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