Two-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.

Rodoflo Kurchan's online magazine Puzzle Fun studied the problem of tiling some rectangle with two different pentominoes, in Issue 19, and revisited the problem in Issue 21. The August 2010 issue of Erich Friedman's Math Magic broadened this problem to use two polyominoes of any size, not necessarily the same.

Here I study the related problem of tiling some rectangle with two pentominoes, using the same number of copies of each.

Nomenclature

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

Table

This table shows the smallest total number of pentominoes known to be able to tile a rectangle in equal numbers.

FILNPTUVWXYZ
F * 28 4 × 8 × 12 4 × × 8 ×
I 28 * 4 24 4 56 ? 12 56 ? 8 60
L 4 4 * 4 4 24 4 4 4 ? 8 32
N × 24 4 * 4 24 4 4 × × 8 ×
P 8 4 4 4 * 12 4 6 8 40 4 12
T × 56 24 24 12 * 96 × 64 × 4 ×
U 12 ? 4 4 4 96 * 136 × × 12 ×
V 4 12 4 4 6 × 136 * 72 × 12 4
W × 56 4 × 8 64 × 72 * × 14 ×
X × ? ? × 40 × × × × * 20 ×
Y 8 8 8 8 4 4 12 12 14 20 * 8
Z × 60 32 × 12 × × 4 × × 8 *

Solutions

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

4 Tiles

6 Tiles

8 Tiles

12 Tiles

14 Tiles

20 Tiles

24 Tiles

28 Tiles

32 Tiles

40 Tiles

56 Tiles

60 Tiles

64 Tiles

72 Tiles

96 Tiles

136 Tiles

Last revised 2012-05-24.

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Col. George Sicherman [ HOME | MAIL ]