The compatibility problem is to find a figure that can be tiled with each of a set of polyforms. Here I show for each polyomino of orders 1 through 6 the smallest known polyominoes that are not compatible with it.
In most cases, incompatibility is probable but has not been proved by analysis or exhaustion. Proved cases are shown in red. Unproved cases are shown in blue.
Giovanni Resta showed that the L tetromino is compatible with all polyominoes of order 10 or less, and found minimal incompatibilies for many other polyominoes. His site Polipolimini is out and away the best place to find polyomino compatibilities.
Andris Cibulis first studied domino compatibility and found the minimal polyominoes incompatible with the domino. He also eliminated many 11-ominoes for the L tetromino by finding complex compatibility figures for them.
Juris Čerņenoks found complex compatibility figures for two 11-ominoes with the L tetromino.
Thanks to Erich Friedman for suggesting this page.
See also Minimal Incompatibility for Polyiamonds and Minimal Incompatibility for Polyhexes.
| Monomino | |||||
|---|---|---|---|---|---|
 | ∞ | None | |||
| Domino | |||||
 | 13 |  | |||
| Trominoes | |||||
 | 5 | 
|  | 9 |  |
| Tetrominoes | |||||
 | 5 | 
|  | 11 |  |
 | 7 | 
|  | 7 |  |
 | 5 |  | |||
| Pentominoes | |||||
 | 5 | 
|  | 7 |  |
 | 8 | 
|  | 7 |  |
 | 5 | 
|  | 7 |  |
 | 5 | 
|  | 7 |  |
 | 8 | 
|  | 5 |  |
 | 5 | 
|  | 3 |  |
| Hexominoes | |||||
 | 5 | 
|  | 5 |  |
 | 7 | 
|  | 5 |  |
 | 5 | 
|  | 5 |  |
 | 5 | 
|  | 5 |  |
 | 5 | 
|  | 5 |  |
 | 5 | 
|  | 7 |  |
 | 7 | 
|  | 5 |  |
 | 5 | 
|  | 5 |  |
 | 4 | 
|  | 7 |  |
 | 7 | 
|  | 4 |  |
 | 7 | 
|  | 5 |  |
 | 5 | 
|  | 7 |  |
 | 5 | 
|  | 7 |  |
 | 5 | 
|  | 4 |  |
 | 5 | 
|  | 5 |  |
 | 5 | 
|  | 4 |  |
 | 4 | 
|  | 4 |  |
 | 4 |  | |||
Last revised 2015-04-29.