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.
Here I show the smallest known solutions for sets of one tetriamond, one pentiamond, and one hexiamond. The numbers tell how many cells the polyiamond has. If you find a smaller solution or solve an unsolved case, please write.
This web page was inspired by Livio Zucca's Pento-Tetro-Trominoes.
| 6A-5I-4A 120 | 6A-5I-4I 60 | 6A-5I-4V 60 | 6A-5J-4A 120 | 6A-5J-4I 60 | 6A-5J-4V 60 |
|---|---|---|---|---|---|
 |  |  |  |  | 
|
| 6A-5Q-4A 120 | 6A-5Q-4I 60 | 6A-5Q-4V 60 | 6A-5U-4A — | 6A-5U-4I 60 | 6A-5U-4V 120 |
 |  |  |  |  | 
|
| 6E-5I-4A — | 6E-5I-4I ? | 6E-5I-4V 120 | 6E-5J-4A — | 6E-5J-4I 720 | 6E-5J-4V 60 |
|  |  | |  | 
|
| 6E-5Q-4A — | 6E-5Q-4I 120 | 6E-5Q-4V 60 | 6E-5U-4A — | 6E-5U-4I ? | 6E-5U-4V ? |
|  |  | | |
|
| 6F-5I-4A 120 | 6F-5I-4I 60 | 6F-5I-4V 60 | 6F-5J-4A 120 | 6F-5J-4I 60 | 6F-5J-4V 60 |
 |  |  |  |  | 
|
| 6F-5Q-4A 120 | 6F-5Q-4I 60 | 6F-5Q-4V 60 | 6F-5U-4A — | 6F-5U-4I 120 | 6F-5U-4V 60 |
 |  |  | |  | 
|
| 6H-5I-4A 360 | 6H-5I-4I 60 | 6H-5I-4V 60 | 6H-5J-4A 360 | 6H-5J-4I 60 | 6H-5J-4V 60 |
 |  |  |  |  | 
|
| 6H-5Q-4A 360 | 6H-5Q-4I 60 | 6H-5Q-4V 60 | 6H-5U-4A — | 6H-5U-4I 180 | 6H-5U-4V 120 |
 |  |  | |  | 
|
| 6I-5I-4A 120 | 6I-5I-4I 60 | 6I-5I-4V 60 | 6I-5J-4A 120 | 6I-5J-4I 60 | 6I-5J-4V 60 |
 |  |  |  |  | 
|
| 6I-5Q-4A 120 | 6I-5Q-4I 60 | 6I-5Q-4V 60 | 6I-5U-4A — | 6I-5U-4I ? | 6I-5U-4V ? |
 |  |  | | |
|
| 6L-5I-4A 120 | 6L-5I-4I 60 | 6L-5I-4V 60 | 6L-5J-4A 120 | 6L-5J-4I 60 | 6L-5J-4V 60 |
 |  |  |  |  | 
|
| 6L-5Q-4A 120 | 6L-5Q-4I 60 | 6L-5Q-4V 60 | 6L-5U-4A — | 6L-5U-4I 60 | 6L-5U-4V 60 |
 |  |  | |  | 
|
| 6O-5I-4A — | 6O-5I-4I — | 6O-5I-4V — | 6O-5J-4A — | 6O-5J-4I — | 6O-5J-4V 60 |
| | | | | 
|
| 6O-5Q-4A — | 6O-5Q-4I — | 6O-5Q-4V — | 6O-5U-4A — | 6O-5U-4I — | 6O-5U-4V — |
| | | | |
|
| 6P-5I-4A 120 | 6P-5I-4I 60 | 6P-5I-4V 60 | 6P-5J-4A 120 | 6P-5J-4I 60 | 6P-5J-4V 60 |
 |  |  |  |  | 
|
| 6P-5Q-4A 120 | 6P-5Q-4I 60 | 6P-5Q-4V 60 | 6P-5U-4A — | 6P-5U-4I ? | 6P-5U-4V 360 |
 |  |  | | | 
|
| 6S-5I-4A ? | 6S-5I-4I 60 | 6S-5I-4V 60 | 6S-5J-4A ? | 6S-5J-4I 60 | 6S-5J-4V 60 |
|  |  | |  | 
|
| 6S-5Q-4A ? | 6S-5Q-4I 60 | 6S-5Q-4V 60 | 6S-5U-4A — | 6S-5U-4I 60 | 6S-5U-4V 60 |
|  |  | |  | 
|
| 6U-5I-4A ? | 6U-5I-4I 60 | 6U-5I-4V 60 | 6U-5J-4A ? | 6U-5J-4I 60 | 6U-5J-4V 60 |
|  |  | |  | 
|
| 6U-5Q-4A ? | 6U-5Q-4I 60 | 6U-5Q-4V 60 | 6U-5U-4A — | 6U-5U-4I ? | 6U-5U-4V 60 |
|  |  | | | 
|
| 6V-5I-4A 120 | 6V-5I-4I 60 | 6V-5I-4V 60 | 6V-5J-4A 120 | 6V-5J-4I 60 | 6V-5J-4V 60 |
 |  |  |  |  | 
|
| 6V-5Q-4A 120 | 6V-5Q-4I 60 | 6V-5Q-4V 60 | 6V-5U-4A — | 6V-5U-4I ? | 6V-5U-4V ? |
 |  |  | | |
|
| 6X-5I-4A — | 6X-5I-4I — | 6X-5I-4V — | 6X-5J-4A — | 6X-5J-4I ? | 6X-5J-4V ? |
| | | | |
|
| 6X-5Q-4A — | 6X-5Q-4I ? | 6X-5Q-4V ? | 6X-5U-4A — | 6X-5U-4I — | 6X-5U-4V — |
| | | | |
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Last revised 2024-05-05.