Pentomino Pair Odd Rectangles

Introduction

It is well known that many pairs of pentominoes can tile rectangles. See for example the August 2010 issue of Erich Friedman's Math Magic.

Here I impose the condition that an odd number of tiles must be used.

Earl S. Kramer was the first to study the problem of arranging copies of two pentominoes to form a rectangle. Mike Reid found sets of prime rectangles for all pairs of non-rectifiable pentominoes. They appear at Pairs of Pentominoes in Rectangles at Andrew Clarke's Poly Pages, and in the January 2001 issue of Math Magic.

Carl Schwenke and Johann Schwenke improved one of my solutions.

Table

This table shows which pairs of pentominoes have solutions, and how many tiles are used in the smallest known solutions. For pairs of pentominoes that cannot tile a rectangle singly, it relies on Mike Reid's results.

FILNPTUVWXYZ
F *337×21×2127××5×
I 33*32132763739×733
L 73*7321219215921
N ×217*21212727××21×
P 213321*321321533
T ×2721213*65×33×9×
U 216321272165*105×311×
V 2779273×105*69×213
W ×3921×2133×69*×17×
X ××5×5×3××*5×
Y 57921391121175*5
Z ×3321×3××3××5*

3 Tiles

5 Tiles

7 Tiles

9 Tiles

11 Tiles

17 Tiles

21 Tiles

27 Tiles

33 Tiles

39 Tiles

55 Tiles

65 Tiles

69 Tiles

105 Tiles

Last revised 2026-04-03.

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Col. George Sicherman [ HOME | MAIL ]