Mixed Polyiamond Compatibility

Introduction.

Two or more polyiamonds are compatible if there is a polyiamond that each of them can tile. Here I show some compatibilities for two polyiamonds of different orders. Many of them were found by Mike Reid or by Margarita Lukjanska, Andris Cibulis, and Andy Liu, whose article in Volume 33 of the Journal of Recreational Mathematics proves that some pairs of polyiamonds are not compatible. Mike has since sent us more incompatibility proofs. If you find a smaller solution or solve an unsolved case, please write.

For hexiamond-hexiamond compatibility, see the September 2004 issue of Erich Friedman's Math Magic. For heptiamond-heptiamond compatibility, see my Heptiamond Compatibility page.


Tetriamonds and Pentiamonds.

Here are minimal known figures for all tetriamond-pentiamond pairs known to be compatible.

 5I5J5Q5U
4A8:108:108:10
4I4:54:54:54:5
4I4:54:54:54:5

Tetriamonds and Hexiamonds.

Here are minimal known figures for all tetriamond-hexiamond pairs known to be compatible.

 6A6C6H6I6J6M6O6P6S6V6X6Y
4A4:64:612:184:64:64:64:6 ??
4I4:64:62:32:32:32:34:6 2:32:312:188:12
4V6:94:62:3 2:32:34:62:32:32:32:36:92:3

Pentiamonds and Hexiamonds.

Here are minimal known figures for all pentiamond-hexiamond pairs known to be compatible.

 6A6C6H6I6J6M6O6P6S6V6X6Y
5I10:1210:125:65:6 5:65:65:65:610:1210:12
5J10:125:65:65:6 5:65:65:65:65:65:6? 5:6
5Q5:65:65:65:6 5:65:65:65:6 10:125:65:6
5U5:65:610:1230:36 5:65:630:3610:12 5:630:36

Tetriamonds and Heptiamonds.

Here are minimal known figures for all tetriamond-heptiamond pairs known to be compatible.

 7A7B7C7D7E7F7G7H7I7J7K7L
4A 80:140?8:144:788:15424:428:1424:42 8:148:14
4I 4:74:74:74:74:74:74:74:7 4:74:74:74:7
4V 4:74:74:74:74:74:74:74:7 4:74:74:74:7

 7M7N7P7Q7R7S7T7U7V7X7Y7Z
4A ?8:14?4:7 16:2812:2112:2124:4224:42
4I 4:74:74:78:14 4:74:74:74:74:74:74:74:7
4V 12:218:144:74:7 4:74:74:74:74:74:74:74:7

Pentiamonds and Heptiamonds.

Here are minimal known figures for all pentiamond-heptiamond pairs known to be compatible.

 7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K 7L
5I 5:710:145:710:1410:145:75:75:75:75:715:215:7
5J 5:75:75:75:75:75:710:145:75:75:75:75:7
5Q 5:75:75:75:75:75:75:75:75:710:145:75:7
5U 10:1415:2120:28??5:7???5:7??

 7M 7N 7P 7Q 7R 7S 7T 7U 7V 7X 7Y 7Z
5I 10:1410:1410:145:75:710:145:75:720:2810:145:710:14
5J 10:145:75:710:145:75:75:75:75:75:75:75:7
5Q 5:75:710:145:75:75:75:710:1415:2110:145:75:7
5U ??5:710:1410:1410:1440:5615:21??5:75:7

7 Pentiamonds, 5 Heptiamonds

More Pentiamonds and Heptiamonds

Hexiamonds and Heptiamonds.

Here are minimal known figures for all hexiamond-heptiamond pairs known to be compatible.

 7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K 7L
6A ?6:712:1412:1412:1412:146:712:1412:1424:28?12:14
6C 12:1412:146:712:1412:1412:1412:146:76:712:14?6:7
6H 6:76:712:1412:146:712:146:76:712:1412:146:712:14
6I 12:1412:146:76:76:76:76:76:76:76:712:146:7
6J 12:146:76:76:76:76:76:712:146:76:712:146:7
6M 6:76:76:76:712:146:712:1412:146:712:1412:1412:14
6O 6:76:7?6:7??
6P 12:146:76:712:146:712:146:76:76:712:1412:146:7
6S 6:712:1412:146:76:712:1412:146:76:76:76:712:14
6V 6:712:146:712:146:712:146:76:712:146:736:4212:14
6X ??????????
6Y 18:2124:2812:14?6:76:712:1412:14??6:7?

 7M 7N 7P 7Q 7R 7S 7T 7U 7V 7X 7Y 7Z
6A ?12:1436:4212:1412:1412:1412:14???12:1412:14
6C 12:1412:1412:14?12:146:712:1412:14??12:1412:14
6H ?12:1412:1412:146:76:76:712:14?12:146:76:7
6I 6:712:1472:8412:146:76:76:7?6:7?6:712:14
6J 6:76:76:76:76:76:76:76:76:712:146:76:7
6M 6:76:712:146:76:76:76:712:1424:2812:1412:146:7
6O 6:7?6:76:76:76:76:7
6P 6:712:1412:146:76:76:76:712:14?6:76:712:14
6S ??6:76:712:146:76:76:7?6:712:1412:14
6V ?12:146:712:146:76:76:76:76:718:2112:1412:14
6X ??????????
6Y ??12:1436:4212:1412:1412:1418:21?12:146:712:14

7 Hexiamonds, 6 Heptiamonds

14 Hexiamonds, 12 Heptiamonds

More Hexiamonds and Heptiamonds

Last revised 2008-05-06.


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Col. George Sicherman [ HOME | MAIL ]