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
Tetriamonds and Hexiamonds
Pentiamonds and Hexiamonds
Tetriamonds and Heptiamonds
Pentiamonds and Heptiamonds
Hexiamonds and Heptiamonds

Tetriamonds and Pentiamonds.

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

8:10 8:10 8:10 —

4:5 4:5 4:5 4:5

4:5 4:5 4:5 4:5


8:10	8:10	8:10	—
4:5	4:5	4:5	4:5
4:5	4:5	4:5	4:5

Tetriamonds and Hexiamonds.

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

4:6 4:6 12:18 4:6 4:6 4:6 — 4:6 ? ? — —

4:6 4:6 2:3 2:3 2:3 2:3 — 4:6 2:3 2:3 12:18 8:12

6:9 4:6 2:3 2:3 2:3 4:6 2:3 2:3 2:3 2:3 6:9 2:3


4:6	4:6	12:18	4:6	4:6	4:6	—	4:6	?	?	—	—
4:6	4:6	2:3	2:3	2:3	2:3	—	4:6	2:3	2:3	12:18	8:12
6:9	4:6	2:3	2:3	2:3	4:6	2:3	2:3	2:3	2:3	6:9	2:3

Pentiamonds and Hexiamonds.

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

10:12 10:12 5:6 5:6 5:6 5:6 — 5:6 5:6 10:12 — 10:12

10:12 5:6 5:6 5:6 5:6 5:6 5:6 5:6 5:6 5:6 ? 5:6

5:6 5:6 5:6 5:6 5:6 5:6 — 5:6 5:6 10:12 5:6 5:6

5:6 5:6 10:12 30:36 5:6 5:6 — 30:36 10:12 5:6 — 30:36


10:12	10:12	5:6	5:6	5:6	5:6	—	5:6	5:6	10:12	—	10:12
10:12	5:6	5:6	5:6	5:6	5:6	5:6	5:6	5:6	5:6	?	5:6
5:6	5:6	5:6	5:6	5:6	5:6	—	5:6	5:6	10:12	5:6	5:6
5:6	5:6	10:12	30:36	5:6	5:6	—	30:36	10:12	5:6	—	30:36

Tetriamonds and Heptiamonds.

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

80:140 ? 8:14 4:7 88:154 24:42 8:14 24:42 8:14 — — 8:14

4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7

4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7


80:140	?	8:14	4:7	88:154	24:42	8:14	24:42	8:14	—	—	8:14
4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7
4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7

? 8:14 ? 4:7 16:28 12:21 12:21 — — — 24:42 24:42

4:7 4:7 4:7 8:14 4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7

12:21 8:14 4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7 4:7


?	8:14	?	4:7	16:28	12:21	12:21	—	—	—	24:42	24:42
4:7	4:7	4:7	8:14	4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7
12:21	8:14	4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7	4:7

Pentiamonds and Heptiamonds.

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

5:7 10:14 5:7 10:14 10:14 5:7 5:7 5:7 5:7 5:7 15:21 5:7

5:7 5:7 5:7 5:7 5:7 5:7 10:14 5:7 5:7 5:7 5:7 5:7

5:7 5:7 5:7 5:7 5:7 5:7 5:7 5:7 5:7 10:14 5:7 5:7

10:14 15:21 20:28 ? ? 5:7 ? ? ? 5:7 ? ?


5:7	10:14	5:7	10:14	10:14	5:7	5:7	5:7	5:7	5:7	15:21	5:7
5:7	5:7	5:7	5:7	5:7	5:7	10:14	5:7	5:7	5:7	5:7	5:7
5:7	5:7	5:7	5:7	5:7	5:7	5:7	5:7	5:7	10:14	5:7	5:7
10:14	15:21	20:28	?	?	5:7	?	?	?	5:7	?	?

10:14 10:14 10:14 5:7 5:7 10:14 5:7 5:7 20:28 10:14 5:7 10:14

10:14 5:7 5:7 10:14 5:7 5:7 5:7 5:7 5:7 5:7 5:7 5:7

5:7 5:7 10:14 5:7 5:7 5:7 5:7 10:14 15:21 10:14 5:7 5:7

? ? 5:7 10:14 10:14 10:14 40:56 15:21 ? ? 5:7 5:7


10:14	10:14	10:14	5:7	5:7	10:14	5:7	5:7	20:28	10:14	5:7	10:14
10:14	5:7	5:7	10:14	5:7	5:7	5:7	5:7	5:7	5:7	5:7	5:7
5:7	5:7	10:14	5:7	5:7	5:7	5:7	10:14	15:21	10:14	5:7	5:7
?	?	5:7	10:14	10:14	10:14	40:56	15:21	?	?	5:7	5:7

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.

? 6:7 12:14 12:14 12:14 12:14 6:7 12:14 12:14 24:28 ? 12:14

12:14 12:14 6:7 12:14 12:14 12:14 12:14 6:7 6:7 12:14 ? 6:7

6:7 6:7 12:14 12:14 6:7 12:14 6:7 6:7 12:14 12:14 6:7 12:14

12:14 12:14 6:7 6:7 6:7 6:7 6:7 6:7 6:7 6:7 12:14 6:7

12:14 6:7 6:7 6:7 6:7 6:7 6:7 12:14 6:7 6:7 12:14 6:7

6:7 6:7 6:7 6:7 12:14 6:7 12:14 12:14 6:7 12:14 12:14 12:14

6:7 — — 6:7 ? — — 6:7 ? — ? —

12:14 6:7 6:7 12:14 6:7 12:14 6:7 6:7 6:7 12:14 12:14 6:7

6:7 12:14 12:14 6:7 6:7 12:14 12:14 6:7 6:7 6:7 6:7 12:14

6:7 12:14 6:7 12:14 6:7 12:14 6:7 6:7 12:14 6:7 36:42 12:14

? ? ? — ? ? — ? ? ? ? ?

18:21 24:28 12:14 ? 6:7 6:7 12:14 12:14 ? ? 6:7 ?


?	6:7	12:14	12:14	12:14	12:14	6:7	12:14	12:14	24:28	?	12:14
12:14	12:14	6:7	12:14	12:14	12:14	12:14	6:7	6:7	12:14	?	6:7
6:7	6:7	12:14	12:14	6:7	12:14	6:7	6:7	12:14	12:14	6:7	12:14
12:14	12:14	6:7	6:7	6:7	6:7	6:7	6:7	6:7	6:7	12:14	6:7
12:14	6:7	6:7	6:7	6:7	6:7	6:7	12:14	6:7	6:7	12:14	6:7
6:7	6:7	6:7	6:7	12:14	6:7	12:14	12:14	6:7	12:14	12:14	12:14
6:7	—	—	6:7	?	—	—	6:7	?	—	?	—
12:14	6:7	6:7	12:14	6:7	12:14	6:7	6:7	6:7	12:14	12:14	6:7
6:7	12:14	12:14	6:7	6:7	12:14	12:14	6:7	6:7	6:7	6:7	12:14
6:7	12:14	6:7	12:14	6:7	12:14	6:7	6:7	12:14	6:7	36:42	12:14
?	?	?	—	?	?	—	?	?	?	?	?
18:21	24:28	12:14	?	6:7	6:7	12:14	12:14	?	?	6:7	?

? 12:14 36:42 12:14 12:14 12:14 12:14 ? ? ? 12:14 12:14

12:14 12:14 12:14 ? 12:14 6:7 12:14 12:14 ? ? 12:14 12:14

? 12:14 12:14 12:14 6:7 6:7 6:7 12:14 ? 12:14 6:7 6:7

6:7 12:14 72:84 12:14 6:7 6:7 6:7 ? 6:7 ? 6:7 12:14

6:7 6:7 6:7 6:7 6:7 6:7 6:7 6:7 6:7 12:14 6:7 6:7

6:7 6:7 12:14 6:7 6:7 6:7 6:7 12:14 24:28 12:14 12:14 6:7

— — 6:7 ? — 6:7 6:7 6:7 — 6:7 — 6:7

6:7 12:14 12:14 6:7 6:7 6:7 6:7 12:14 ? 6:7 6:7 12:14

? ? 6:7 6:7 12:14 6:7 6:7 6:7 ? 6:7 12:14 12:14

? 12:14 6:7 12:14 6:7 6:7 6:7 6:7 6:7 18:21 12:14 12:14

? ? ? ? ? ? ? ? — — ? ?

? ? 12:14 36:42 12:14 12:14 12:14 18:21 ? 12:14 6:7 12:14


?	12:14	36:42	12:14	12:14	12:14	12:14	?	?	?	12:14	12:14
12:14	12:14	12:14	?	12:14	6:7	12:14	12:14	?	?	12:14	12:14
?	12:14	12:14	12:14	6:7	6:7	6:7	12:14	?	12:14	6:7	6:7
6:7	12:14	72:84	12:14	6:7	6:7	6:7	?	6:7	?	6:7	12:14
6:7	6:7	6:7	6:7	6:7	6:7	6:7	6:7	6:7	12:14	6:7	6:7
6:7	6:7	12:14	6:7	6:7	6:7	6:7	12:14	24:28	12:14	12:14	6:7
—	—	6:7	?	—	6:7	6:7	6:7	—	6:7	—	6:7
6:7	12:14	12:14	6:7	6:7	6:7	6:7	12:14	?	6:7	6:7	12:14
?	?	6:7	6:7	12:14	6:7	6:7	6:7	?	6:7	12:14	12:14
?	12:14	6:7	12:14	6:7	6:7	6:7	6:7	6:7	18:21	12:14	12:14
?	?	?	?	?	?	?	?	—	—	?	?
?	?	12:14	36:42	12:14	12:14	12:14	18:21	?	12:14	6:7	12:14

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 ]