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.

Triamonds and Tetriamonds
Triamonds and Pentiamonds
Tetriamonds and Pentiamonds
Triamonds and Hexiamonds
Tetriamonds and Hexiamonds
Pentiamonds and Hexiamonds
Triamonds and Heptiamonds
Tetriamonds and Heptiamonds
Pentiamonds and Heptiamonds
Hexiamonds and Heptiamonds
Triamonds and Octiamonds
Tetriamonds and Octiamonds

Triamonds and Tetriamonds.

Here are minimal figures for all triamond-tetriamond pairs.

3:4 3:4 3:4


	3:4	3:4	3:4

Triamonds and Pentiamonds.

Here are minimal figures for all triamond-pentiamond pairs.

3:5 3:5 3:5 3:5


	3:5	3:5	3:5	3:5

Tetriamonds and Pentiamonds.

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

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

Triamonds and Hexiamonds.

Here are minimal figures for all triamond-hexiamond pairs.

2:4 1:2 2:4 1:2 1:2 1:2 1:2 1:2 1:2 1:2 1:2 3:6


	2:4	1:2	2:4	1:2	1:2	1:2	1:2	1:2	1:2	1:2	1:2	3:6

Holeless Variant

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 84:126 96:144 — —

2:3 2:3 2:3 2:3 2:3 2:3 — 2:3 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	84:126	96:144	—	—
2:3	2:3	2:3	2:3	2:3	2:3	—	2:3	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 90:108 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	90:108	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

Holeless Variants

Triamonds and Heptiamonds.

Here are minimal figures for all triamond-heptiamond pairs.

3:7 3:7 3:7 3:7 3:7 3:7 3:7 3:7 3:7 3:7 3:7 3:7


	3:7	3:7	3:7	3:7	3:7	3:7	3:7	3:7	3:7	3:7	3:7	3:7


	3:7	3:7	3:7	3:7	3:7	3:7	3:7	3:7	3:7	3:7	3:7	3:7

Tetriamonds and Heptiamonds.

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

72:126 ? 8:14 4:7 36:63 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


72:126	?	8:14	4:7	36:63	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

Holeless Variants

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 ? 30:42 5:7 40:56 10:14 30:42 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	?	30:42	5:7	40:56	10:14	30:42	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

? 60:84 5:7 10:14 10:14 10:14 20:28 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
?	60:84	5:7	10:14	10:14	10:14	20:28	15:21	?	?	5:7	5:7

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.

24:28 6:7 12:14 12:14 12:14 12:14 6:7 12:14 12:14 24:28 36:42 12:14

12:14 12:14 6:7 12:14 12:14 12:14 12:14 6:7 6:7 12:14 36:42 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 72:84 6:7 6:7 12:14 12:14 ? ? 6:7 36:42


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

24:28 12:14 24:28 12:14 12:14 12:14 12:14 ? ? ? 12:14 12:14

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

18:21 12:14 12:14 12:14 6:7 6:7 6:7 12:14 48:56 12:14 6:7 6:7

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

— 36:42 6:7 6:7 12:14 6:7 6:7 6:7 ? 6:7 12:14 6:7

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

— ? ? ? ? ? — — — — ? ?

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


24:28	12:14	24:28	12:14	12:14	12:14	12:14	?	?	?	12:14	12:14
12:14	12:14	12:14	36:42	12:14	6:7	12:14	12:14	?	?	12:14	12:14
18:21	12:14	12:14	12:14	6:7	6:7	6:7	12:14	48:56	12:14	6:7	6:7
6:7	12:14	36:42	12:14	6:7	6:7	6:7	36:42	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	36:42	6:7	6:7	12:14
—	36:42	6:7	6:7	12:14	6:7	6:7	6:7	?	6:7	12:14	6:7
?	6:7	6:7	12:14	6:7	6:7	6:7	6:7	6:7	12:14	12:14	12:14
—	?	?	?	?	?	—	—	—	—	?	?
?	?	12:14	36:42	12:14	12:14	12:14	18:21	?	12:14	6:7	12:14