# Scaled Three-Pentomino Balanced Rectangles

## 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.

It has long been known that only four pentominoes can tile rectangles:

For other rectangles that these pentominoes tile, see Mike Reid's Rectifiable Polyomino Page.

On Balanced Three-Pentomino Rectangles I study the related problem of tiling some rectangle with three pentominoes, using the same areas of each. Here I study the same problem, using the pentominoes in various sizes. If you find a solution better than one of mine, or solve an unsolved case, please write!

Bryce Herdt improved on some of my solutions.

## Nomenclature

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

## Solutions

5F+5I+5L 65F+5I+5N 165F+5I+5P 95F+5I+5T 135F+5I+5U 6
5F+5I+5V 65F+5I+5W 155F+5I+5X 275F+5I+5Y 125F+5I+5Z 15
5F+5L+5N 65F+5L+5P 65F+5L+5T 65F+5L+5U 65F+5L+5V 6
5F+5L+5W 65F+5L+5X 305F+5L+5Y 95F+5L+5Z 125F+5N+5P 12
5F+5N+5T 185F+5N+5U 65F+5N+5V 65F+5N+5W ×5F+5N+5X ×
5F+5N+5Y 125F+5N+5Z ×5F+5P+5T 125F+5P+5U 35F+5P+5V 6
5F+5P+5W 125F+5P+5X 305F+5P+5Y 65F+5P+5Z 185F+5T+5U 18
5F+5T+5V 245F+5T+5W 245F+5T+5X ×5F+5T+5Y 65F+5T+5Z ×
5F+5U+5V 95F+5U+5W 65F+5U+5X 355F+5U+5Y 125F+5U+5Z 19
5F+5V+5W 185F+5V+5X 575F+5V+5Y 125F+5V+5Z 125F+5W+5X ×
5F+5W+5Y 125F+5W+5Z ×5F+5X+5Y 245F+5X+5Z ×5F+5Y+5Z 12
5I+5L+5N 65I+5L+5P 65I+5L+5T 125I+5L+5U 65I+5L+5V 6
5I+5L+5W 65I+5L+5X 205I+5L+5Y 65I+5L+5Z 95I+5N+5P 6
5I+5N+5T 65I+5N+5U 65I+5N+5V 65I+5N+5W 155I+5N+5X 30
5I+5N+5Y 65I+5N+5Z 125I+5P+5T 65I+5P+5U 65I+5P+5V 6
5I+5P+5W 95I+5P+5X 155I+5P+5Y 65I+5P+5Z 65I+5T+5U 13
5I+5T+5V 95I+5T+5W 125I+5T+5X 205I+5T+5Y 65I+5T+5Z 13
5I+5U+5V 95I+5U+5W 155I+5U+5X 155I+5U+5Y 65I+5U+5Z 15
5I+5V+5W 125I+5V+5X 215I+5V+5Y 65I+5V+5Z 65I+5W+5X 21
5I+5W+5Y 65I+5W+5Z 155I+5X+5Y 185I+5X+5Z 215I+5Y+5Z 12
5L+5N+5P 65L+5N+5T 125L+5N+5U 65L+5N+5V 35L+5N+5W 6
5L+5N+5X 95L+5N+5Y 65L+5N+5Z 65L+5P+5T 65L+5P+5U 6
5L+5P+5V 35L+5P+5W 65L+5P+5X 95L+5P+5Y 65L+5P+5Z 6
5L+5T+5U 125L+5T+5V 65L+5T+5W 175L+5T+5X 65L+5T+5Y 3
5L+5T+5Z 125L+5U+5V 95L+5U+5W 65L+5U+5X 125L+5U+5Y 6
5L+5U+5Z 125L+5V+5W 125L+5V+5X 95L+5V+5Y 95L+5V+5Z 6
5L+5W+5X 365L+5W+5Y 65L+5W+5Z 185L+5X+5Y 185L+5X+5Z 33
5L+5Y+5Z 65N+5P+5T 65N+5P+5U 35N+5P+5V 65N+5P+5W 12
5N+5P+5X 185N+5P+5Y 65N+5P+5Z 65N+5T+5U 125N+5T+5V 9
5N+5T+5W 125N+5T+5X 365N+5T+5Y 65N+5T+5Z 245N+5U+5V 6
5N+5U+5W 245N+5U+5X 185N+5U+5Y 65N+5U+5Z 65N+5V+5W 6
5N+5V+5X 555N+5V+5Y 125N+5V+5Z 65N+5W+5X ×5N+5W+5Y 12
5N+5W+5Z ×5N+5X+5Y 155N+5X+5Z ×5N+5Y+5Z 125P+5T+5U 6
5P+5T+5V 65P+5T+5W 65P+5T+5X 125P+5T+5Y 65P+5T+5Z 12
5P+5U+5V 35P+5U+5W 125P+5U+5X 65P+5U+5Y 35P+5U+5Z 6
5P+5V+5W 65P+5V+5X 185P+5V+5Y 65P+5V+5Z 65P+5W+5X 30
5P+5W+5Y 65P+5W+5Z 65P+5X+5Y 125P+5X+5Z 335P+5Y+5Z 6
5T+5U+5V 125T+5U+5W 185T+5U+5X 485T+5U+5Y 65T+5U+5Z 36
5T+5V+5W 245T+5V+5X 1215T+5V+5Y 125T+5V+5Z 215T+5W+5X 54