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.

Here are minimal figures for all triamond-tetriamond pairs.

3:4 3:4 3:4


	3:4	3:4	3:4

Horizontal Mirror Variant

Holeless Horizontal Mirror Variant

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

Horizontal Mirror Variants

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

Horizontal Mirror Variants

Holeless Horizontal Mirror Variants

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

Horizontal Mirror Variants

Holeless Horizontal Mirror 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 54:63 ? 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	54:63	?	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

7 Hexiamonds, 6 Heptiamonds

14 Hexiamonds, 12 Heptiamonds

More Hexiamonds and Heptiamonds

Holeless Variants

Last revised 2024-11-10.

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Col. George Sicherman [ HOME | MAIL ]