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.
The sites include 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 show only minimal solutions with no restriction. Here I show minimal known hexomino compatibility figures without holes. If you find a smaller solution or solve an unsolved case, please let me know.
I am grateful to V. Pocins for improving one of the solutions on this page, and for suggesting improvements to the text.
For pentomino compatibility with or without holes, see Pentomino Compatibility. For hexomino compatibility allowing holes, see Resta's Hexominoes. For pentomino-hexomino compatibility without holes, see Holeless Pentomino-Hexomino Compatibility.
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | * | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | ? | 4 | 4 | 4 | 2 | 2 | 4 | ? | 4 | 10 | 3 | 4 | 4 | × | 3 | ? | 6 | ? | × | 4 | ? | 2 | × | 3 | × | × |
2 | 2 | * | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 2 | 2 | 2 | 4 | 3 | 2 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 18 | ? | × | × |
3 | 2 | 2 | * | 2 | 2 | 2 | 4 | 8 | 2 | ? | 2 | 4 | 2 | 2 | 2 | 2 | ? | 4 | 2 | 6 | 2 | 4 | ? | 6 | 2 | 2 | 2 | 2 | 2 | ? | 8 | 2 | 6 | 2 | 2 |
4 | 2 | 4 | 2 | * | 2 | ? | 6 | ? | 4 | ? | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 4 | ? | 2 | 12 | 8 | 6 | 2 | × | × |
5 | 4 | 2 | 2 | 2 | * | 18 | 2 | ? | 2 | 2 | 2 | 2 | 6 | 4 | 2 | 2 | 8 | 4 | 12 | 3 | 2 | 6 | 6 | 4 | 2 | 2 | 4 | ? | 2 | ? | 4 | ? | ? | × | × |
6 | 2 | 4 | 2 | ? | 18 | * | 2 | 2 | 4 | ? | 4 | 4 | ? | ? | 16 | × | ? | 2 | 12 | × | 2 | ? | 2 | 2 | × | 2 | 2 | ? | 2 | ? | ? | ? | ? | 2 | 2 |
7 | 2 | 2 | 4 | 6 | 2 | 2 | * | 2 | 4 | 2 | 4 | 2 | 2 | 4 | 6 | 2 | 10 | 2 | 2 | 38 | 2 | 2 | 2 | 2 | 4 | 8 | 16 | ? | 2 | 6 | 2 | 2 | ? | ? | 4 |
8 | 2 | 4 | 8 | ? | ? | 2 | 2 | * | 2 | ? | 2 | 2 | ? | 30 | 2 | 4 | 2 | ? | 2 | 3 | 2 | ? | ? | ? | ? | ? | × | ? | 2 | ? | ? | × | × | × | ? |
9 | 4 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | * | 2 | 2 | 6 | 4 | 4 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 2 | ? | 2 | 12 | ? | 4 | ? | 14 | 6 | 6 | × | × |
10 | ? | 2 | ? | ? | 2 | ? | 2 | ? | 2 | * | 2 | 2 | 16 | 2 | ? | 24 | ? | 16 | 6 | × | 2 | ? | 2 | 6 | ? | 2 | 12 | ? | 6 | 2 | ? | ? | × | ? | 2 |
11 | 4 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 2 | 2 | * | 2 | 2 | 4 | 2 | 8 | ? | 4 | 2 | ? | 4 | 2 | 4 | 8 | 8 | 10 | 2 | ? | 4 | ? | ? | 2 | 16 | 2 | 2 |
12 | 4 | 4 | 4 | 2 | 2 | 4 | 2 | 2 | 6 | 2 | 2 | * | 4 | 10 | 8 | 4 | 5 | 2 | 8 | 3 | 2 | 4 | 2 | 2 | 8 | 5 | 2 | 2 | 2 | 2 | 2 | 14 | ? | 10 | ? |
13 | 4 | 2 | 2 | 2 | 6 | ? | 2 | ? | 4 | 16 | 2 | 4 | * | 2 | 2 | 2 | 2 | 10 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 2 |
14 | 2 | 4 | 2 | 2 | 4 | ? | 4 | 30 | 4 | 2 | 4 | 10 | 2 | * | 2 | 2 | ? | 4 | 2 | 2 | 2 | 4 | 32 | 2 | ? | 6 | 4 | ? | 4 | 2 | 4 | 2 | 2 | × | × |
15 | 2 | 8 | 2 | 2 | 2 | 16 | 6 | 2 | 2 | ? | 2 | 8 | 2 | 2 | * | 2 | 2 | 2 | 2 | 6 | 2 | 2 | 36 | 6 | ? | 14 | 4 | ? | 16 | 4 | ? | ? | 6 | × | ? |
16 | 4 | 2 | 2 | 2 | 2 | × | 2 | 4 | 4 | 24 | 8 | 4 | 2 | 2 | 2 | * | 4 | 26 | 2 | 2 | 2 | 4 | ? | 4 | 2 | 8 | 2 | ? | 2 | ? | 2 | 4 | 4 | × | × |
17 | ? | 2 | ? | 2 | 8 | ? | 10 | 2 | 2 | ? | ? | 5 | 2 | ? | 2 | 4 | * | 2 | 2 | 4 | 2 | 2 | 8 | 6 | ? | 2 | 2 | 4 | 2 | ? | ? | ? | ? | ? | ? |
18 | 4 | 2 | 4 | 2 | 4 | 2 | 2 | ? | 2 | 16 | 4 | 2 | 10 | 4 | 2 | 26 | 2 | * | 2 | 3 | 2 | 2 | 4 | 3 | ? | 6 | 2 | 4 | 2 | 4 | 2 | 2 | 6 | 2 | 4 |
19 | 10 | 4 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 6 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | * | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 2 | 4 | 2 | 2 | 2 | 4 |
20 | 3 | 3 | 6 | 2 | 3 | × | 38 | 3 | 2 | × | ? | 3 | 2 | 2 | 6 | 2 | 4 | 3 | 2 | * | 2 | 18 | ? | 10 | × | 4 | 12 | 3 | × | ? | × | 2 | 2 | × | × |
21 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | * | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 14 | 2 | 2 | 4 | 2 | 4 |
22 | 4 | 6 | 4 | 2 | 6 | ? | 2 | ? | 2 | ? | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 18 | 2 | * | 2 | 2 | 2 | 4 | 2 | 28 | 4 | ? | ? | 4 | 2 | 28 | 4 |
23 | × | 2 | ? | 2 | 6 | 2 | 2 | ? | 6 | 2 | 4 | 2 | 4 | 32 | 36 | ? | 8 | 4 | 2 | ? | 2 | 2 | * | 2 | 4 | 4 | 16 | ? | 10 | 2 | 4 | 4 | ? | 12 | 2 |
24 | 3 | 2 | 6 | 2 | 4 | 2 | 2 | ? | 2 | 6 | 8 | 2 | 2 | 2 | 6 | 4 | 6 | 3 | 2 | 10 | 2 | 2 | 2 | * | 4 | 2 | 2 | 4 | 6 | 4 | 4 | 2 | 8 | 4 | 40 |
25 | ? | 2 | 2 | 2 | 2 | × | 4 | ? | ? | ? | 8 | 8 | 4 | ? | ? | 2 | ? | ? | 4 | × | 2 | 2 | 4 | 4 | * | 2 | 8 | ? | 12 | 4 | 4 | 4 | × | × | ? |
26 | 6 | 2 | 2 | 8 | 2 | 2 | 8 | ? | 2 | 2 | 10 | 5 | 4 | 6 | 14 | 8 | 2 | 6 | 4 | 4 | 2 | 4 | 4 | 2 | 2 | * | 2 | 2 | 42 | 4 | 4 | 2 | 2 | ? | 2 |
27 | ? | 2 | 2 | 4 | 4 | 2 | 16 | × | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 12 | 4 | 2 | 16 | 2 | 8 | 2 | * | 2 | 4 | 4 | ? | 4 | ? | 2 | ? |
28 | × | 4 | 2 | ? | ? | ? | ? | ? | ? | ? | ? | 2 | 4 | ? | ? | ? | 4 | 4 | 4 | 3 | 2 | 28 | ? | 4 | ? | 2 | 2 | * | 2 | × | ? | 2 | × | × | × |
29 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 6 | 4 | 2 | 2 | 4 | 16 | 2 | 2 | 2 | 8 | × | 2 | 4 | 10 | 6 | 12 | 42 | 4 | 2 | * | 2 | 2 | 24 | × | × | × |
30 | ? | 2 | ? | 12 | ? | ? | 6 | ? | ? | 2 | ? | 2 | 2 | 2 | 4 | ? | ? | 4 | 2 | ? | 14 | ? | 2 | 4 | 4 | 4 | 4 | × | 2 | * | 4 | 4 | × | × | × |
31 | 2 | 2 | 8 | 8 | 4 | ? | 2 | ? | 14 | ? | ? | 2 | 2 | 4 | ? | 2 | ? | 2 | 4 | × | 2 | ? | 4 | 4 | 4 | 4 | ? | ? | 2 | 4 | * | 4 | × | × | × |
32 | × | 18 | 2 | 6 | ? | ? | 2 | × | 6 | ? | 2 | 14 | 4 | 2 | ? | 4 | ? | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 24 | 4 | 4 | * | 2 | 4 | 2 |
33 | 3 | ? | 6 | 2 | ? | ? | ? | × | 6 | × | 16 | ? | 2 | 2 | 6 | 4 | ? | 6 | 2 | 2 | 4 | 2 | ? | 8 | × | 2 | ? | × | × | × | × | 2 | * | ? | 4 |
34 | × | × | 2 | × | × | 2 | ? | × | × | ? | 2 | 10 | 2 | × | × | × | ? | 2 | 2 | × | 2 | 28 | 12 | 4 | × | ? | 2 | × | × | × | × | 4 | ? | * | 2 |
35 | × | × | 2 | × | × | 2 | 4 | ? | × | 2 | 2 | ? | 2 | × | ? | × | ? | 4 | 4 | × | 4 | 4 | 2 | 40 | ? | 2 | ? | × | × | × | × | 2 | 4 | 2 | * |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 |
Last revised 2020-06-18.