Tetrahex-Pentahex Oddities

• Introduction
• Basic Solutions
• Holeless Variants

Introduction

Here are the smallest known fully symmetric polyhexes with an odd number of cells, formed by copies of a given tetrahex and pentahex, using at least one of each.

See also Tetromino-Pentomino Oddities, Pentomino Pair Oddities, Hexiamond Pair Oddities, Pentahex Pair Oddities, and Trihex-Pentahex Oddities.

Johann Schwenke improved on some of my solutions.

Chronology of Solutions

Basic Solutions

4I+5A 85	4I+5C 175	4I+5D 37	4I+5E 61	4I+5F 85
4I+5H 109	4I+5I 25	4I+5J 79	4I+5K 85	4I+5L 49
4I+5N 37	4I+5P 37	4I+5Q 85	4I+5R 61	4I+5S 121
4I+5T 109	4I+5U 121	4I+5V 61	4I+5W 145	4I+5X 121
4I+5Y 37	4I+5Z 79	4J+5A 37	4J+5C 37	4J+5D 37
4J+5E 37	4J+5F 37	4J+5H 37	4J+5I 37	4J+5J 37
4J+5K 37	4J+5L 37	4J+5N 37	4J+5P 37	4J+5Q 37
4J+5R 37	4J+5S 37	4J+5T 37	4J+5U 37	4J+5V 37
4J+5W 37	4J+5X 37	4J+5Y 37	4J+5Z 37	4O+5A 91
4O+5C 55	4O+5D 31	4O+5E 61	4O+5F 55	4O+5H 55
4O+5I 61	4O+5J 67	4O+5K 55	4O+5L 37	4O+5N 55
4O+5P 31	4O+5Q 31	4O+5R 61	4O+5S 67	4O+5T 79
4O+5U 127	4O+5V 19	4O+5W 43	4O+5X 67	4O+5Y 55
4O+5Z 55	4Q+5A 31	4Q+5C 25	4Q+5D 31	4Q+5E 25
4Q+5F 31	4Q+5H 31	4Q+5I 37	4Q+5J 37	4Q+5K 25
4Q+5L 37	4Q+5N 19	4Q+5P 19	4Q+5Q 13	4Q+5R 31
4Q+5S 13	4Q+5T 13	4Q+5U 37	4Q+5V 37	4Q+5W 37
4Q+5X 37	4Q+5Y 37	4Q+5Z 37	4S+5A 85	4S+5C 97
4S+5D 31	4S+5E 67	4S+5F 25	4S+5H 55	4S+5I 103
4S+5J 55	4S+5K 25	4S+5L 61	4S+5N 31	4S+5P 25
4S+5Q 67	4S+5R 31	4S+5S 73	4S+5T 73	4S+5U 67
4S+5V 109	4S+5W 55	4S+5X 67	4S+5Y 55	4S+5Z 49
4U+5A 13	4U+5C 55	4U+5D 37	4U+5E 49	4U+5F 49
4U+5H 49	4U+5I 13	4U+5J 25	4U+5K 49	4U+5L 37
4U+5N 49	4U+5P 31	4U+5Q 61	4U+5R 55	4U+5S 49
4U+5T 67	4U+5U 55	4U+5V 61	4U+5W 79	4U+5X 49
4U+5Y 49	4U+5Z 49	4Y+5A 109	4Y+5C 133	4Y+5D 79
4Y+5E 145	4Y+5F 85	4Y+5H 55	4Y+5I 175	4Y+5J 19
4Y+5K 49	4Y+5L 79	4Y+5N 19	4Y+5P 19	4Y+5Q 37
4Y+5R 79	4Y+5S 157	4Y+5T 25	4Y+5U 49	4Y+5V 31
4Y+5W 103	4Y+5X 31	4Y+5Y 91	4Y+5Z 79

Holeless Variants

Solutions shown above that are holeless are not shown here.

4I+5C 217	4I+5F 121	4I+5J 85	4I+5L 61	4I+5S 169
4I+5T 175	4I+5U 259	4I+5W 151	4O+5J 85	4O+5N 61
4O+5U 193	4O+5Y 61	4Q+5A 37	4S+5C —	4S+5E 73
4S+5I 109	4S+5J 85	4S+5K 55	4S+5P 31	4S+5Q —
4S+5T 103	4S+5X 73	4S+5Z 55	4U+5C 115	4U+5E 91
4U+5F 55	4U+5J 85	4U+5K 55	4U+5N 55	4U+5Q 67
4U+5R 73	4U+5S 109	4U+5T 133	4U+5U 73	4U+5V 103
4U+5W 85	4U+5X 73	4U+5Y 61	4U+5Z 73	4Y+5A 205
4Y+5C —	4Y+5E 217	4Y+5F 115	4Y+5I —	4Y+5K 91
4Y+5L 85	4Y+5Q 67	4Y+5R 121	4Y+5S —	4Y+5T 31
4Y+5U 115	4Y+5V 97	4Y+5W 133	4Y+5Z 109

Last revised 2024-06-24.

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