Pentomino Compatibility

Introduction

The compatibility problem is to find a figure that can be tiled with each of a set of polyforms. Polyomino compatibility has been widely studied since the early 1990s, and two well-known websites, Poly²ominoes by Jorge Mireles and Polypolyominoes by Giovanni Resta, present the results of their authors' systematic searches for compatibility figures. Mireles's site includes solutions by other researchers, especially Mike Reid. So far as I know, polyomino compatibility has not been treated in print since Golomb first raised the issue in 1981, except in a series of articles called Polyomino Number Theory, written by Andris Cibulis, Andy Liu, Bob Wainwright, Uldis Barbans, and Gilbert Lee from 2002 to 2005.

The websites and the articles present a wealth of polyomino compatibilities. They do not show all current results. I do so below, for pentomino-pentomino compatibility only. I am not prepared to maintain a current catalogue of results for other kinds of pairs of polyominoes, or for larger sets such as the pentomino triples that Livio Zucca has collected here.

For compatibility figures with an odd number of tiles, see Pentomino Odd Pairs Update. For Galvagni compatibility (self-compatibility), see Galvagni Figures & Reid Figures for Pentominoes.

Table

	F	I	L	N	P	T	U	V	W	X	Y	Z
F	*	10	2	2	2	2	4	4	2	2	2	2
I	10	*	2	2	2	4	12	4	10	×	2	20
L	2	2	*	4	2	2	2	2	2	44	2	2
N	2	2	4	*	2	2	2	2	2	16	2	2
P	2	2	2	2	*	2	2	2	2	4	2	2
T	2	4	2	2	2	*	4	2	14	4	2	2
U	4	12	2	2	2	4	*	2	2	×	2	4
V	4	4	2	2	2	2	2	*	6	×	2	4
W	2	10	2	2	2	14	2	6	*	?	2	4
X	2	×	44	16	4	4	×	×	?	*	2	?
Y	2	2	2	2	2	2	2	2	2	2	*	2
Z	2	20	2	2	2	2	4	4	4	?	2	*

Solutions

Two Tiles

Four Tiles

Six Tiles

Ten Tiles

Twelve Tiles

Fourteen Tiles

Sixteen Tiles

Twenty Tiles

Forty-Four Tiles