# Pentomino Pairs Tiling a Rectangle with One Corner Cell Removed

• Introduction
• Enumeration
• Table
• Solutions
• ## 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 problem of arranging copies of a polyomino to form a rectangle has been studied for a long time. Here I study the problem of arranging copies of two pentominoes to form a rectangle with one corner cell removed.

• Pentomino Pairs Tiling a Rectangle with Two Opposite Corner Cells Removed
• Pentomino Pairs Tiling a Rectangle with Two Neighboring Corner Cells Removed
• Pentomino Pairs Tiling a Rectangle with Three Corner Cells Removed
• Pentomino Pairs Tiling a Rectangle with the Four Corner Cells Removed
• Hexomino Pairs Tiling a Rectangle with One Corner Cell Removed
• ## Enumeration

I use Solomon W. Golomb's original names for the pentominoes:

## Table of Results

This table shows the smallest total number of copies of two pentominoes known to be able to tile a rectangle with three of its corner cells removed, using at least one of each pentomino.

FILNPTUVWXYZ
F * 18 7 × 7 × 4 16 × × 7 ×
I 18 * 7 7 3 24 46 7 35 × 7 35
L 7 7 * 3 3 13 3 4 3 16 7 16
N × 7 3 * 3 16 18 11 × × 11 ×
P 7 3 3 3 * 4 3 3 7 7 3 4
T × 24 13 16 4 * 61 × 24 × 24 ×
U 4 46 3 18 3 61 * 81 × × 4 ×
V 16 7 4 11 3 × 81 * 43 × 11 13
W × 35 3 × 7 24 × 43 * × 11 ×
X × × 16 × 7 × × × × * 31 ×
Y 7 7 7 11 3 24 4 11 11 31 * 7
Z × 35 16 × 4 × × 13 × × 7 *

## Solutions

So far as I know, these solutions have minimal area. They are not necessarily uniquely minimal.

### 81 Tiles

Last revised 2023-06-20.

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Col. George Sicherman [ HOME | MAIL ]