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.

See also Hexiamond Compatibility and Heptiamond Compatibility.


Triamonds and Tetriamonds.

Here are minimal figures for all triamond-tetriamond pairs.

 4A4I4V
3I3:43:43:4

Triamonds and Pentiamonds.

Here are minimal figures for all triamond-pentiamond pairs.

 5I5J5Q5U
3I3:53:53:53:5

Tetriamonds and Pentiamonds.

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

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

Triamonds and Hexiamonds.

Here are minimal figures for all triamond-hexiamond pairs.

 6A6C6H6I6J6M6O6P6S6V6X6Y
3I2:41:22:41:21:21:21:21:2 1:21:21:23:6

Holeless Variant

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 84:12696:144
4I2:32:32:32:32:32:32:3 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 5:65:65:65:65:65:690:108 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

Holeless Variants

Triamonds and Heptiamonds.

Here are minimal figures for all triamond-heptiamond pairs.

 7A7B7C7D7E7F7G7H7I7J7K7L
3i 3:73:73:73:73:73:73:73:73:73:73:73:7

 7M7N7P7Q7R7S7T7U7V7X7Y7Z
3I 3:73:73:73:73:73:73:73:73:73:73:73:7

Tetriamonds and Heptiamonds.

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

 7A7B7C7D7E7F7G7H7I7J7K7L
4A 72:126?8:144:736:6324:428:1424:42 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 ?8:144:7 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

Holeless Variants

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?30:425:740:5610:1430:425:7??

 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 ?60:845:710:1410:1410:1420:2815:21??5:75:7

7 Pentiamonds, 5 Heptiamonds

More Pentiamonds and Heptiamonds

Holeless Variants

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 24:286:712:1412:1412:1412:146:712:1412:1424:2836:4212:14
6C 12:1412:146:712:1412:1412:1412:146:76:712:1436:426:7
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 18:2124:2812:1472:846:76:712:1412:14??6:736:42

 7M 7N 7P 7Q 7R 7S 7T 7U 7V 7X 7Y 7Z
6A 24:2812:1424:2812:1412:1412:1412:14???12:1412:14
6C 12:1412:1412:1436:4212:146:712:1412:14??12:1412:14
6H 18:2112:1412:1412:146:76:76:712:1448:5612:146:76:7
6I 6:712:1436:4212:146:76:76:736:426:7?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 6:712:1412:146:76:76:76:712:1436:426:76:712:14
6S 36:426:76:712:146:76:76:7?6:712:146:7
6V ?6:76:712:146:76:76:76:76:712:1412:1412:14
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

Holeless Variants

Triamonds and Octiamonds.

Here are minimal known figures for all triamond-octiamond pairs known to be compatible.

Tetriamonds and Octiamonds.

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

2 Tetriamonds, 1 Octiamond

4 Tetriamonds, 2 Octiamonds

6 Tetriamonds, 3 Octiamonds

More Tetriamonds and Octiamonds

Holeless Variants

Last revised 2015-10-14.


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Col. George Sicherman [ HOME | MAIL ]