# Three-Hexiamond Balanced Hexagons

## Introduction

A hexiamond is a figure made of six equilateral triangles joined edge to edge. There are 12 such figures, not distinguishing reflections and rotations. They were first enumerated and studied by T. H. O'Beirne.

It has long been known that eight hexiamonds can tile regular hexagons:

Here I study the related problem of tiling some regular hexagon with three hexiamonds, using the same number of copies of each.

For two hexiamonds, see Two-Hexiamond Balanced Hexagons.

## Table

This table shows the smallest total number of hexiamonds known to be able to tile a hexagon in equal numbers.

 A E F 36 A E H 81 A E I ? A E L 9 A E O ? A E P 36 A E S 36 A E U 9 A E V 36 A E X ? A F H 36 A F I 36 A F L 36 A F O ? A F P 36 A F S 36 A F U 36 A F V 36 A F X 144 A H I 9 A H L 9 A H O 36 A H P 36 A H S 81 A H U 36 A H V 36 A H X 324 A I L 36 A I O ? A I P 36 A I S 36 A I U 9 A I V 36 A I X ? A L O 9 A L P 36 A L S 9 A L U 9 A L V 9 A L X 36 A O P 36 A O S 324 A O U 144 A O V 36 A O X 144 A P S 36 A P U 36 A P V 36 A P X 36 A S U 36 A S V 36 A S X ? A U V 9 A U X 36 A V X 81 E F H 36 E F I 36 E F L 36 E F O 144 E F P 9 E F S 36 E F U 36 E F V 36 E F X ? E H I 36 E H L 36 E H O ? E H P 36 E H S 36 E H U 9 E H V 36 E H X ? E I L 9 E I O ? E I P 36 E I S 36 E I U 36 E I V 36 E I X ? E L O 9 E L P 36 E L S 9 E L U 9 E L V 9 E L X ? E O P 36 E O S ? E O U ? E O V 36 E O X ? E P S 36 E P U 36 E P V 36 E P X ? E S U 9 E S V 9 E S X ? E U V 36 E U X 81 E V X ? F H I 36 F H L 36 F H O 36 F H P 36 F H S 36 F H U 36 F H V 36 F H X 36 F I L 36 F I O ? F I P 9 F I S 36 F I U 36 F I V 36 F I X 36 F L O 36 F L P 9 F L S 36 F L U 36 F L V 36 F L X 36 F O P 36 F O S 144 F O U 144 F O V 36 F O X 144 F P S 36 F P U 9 F P V 36 F P X 36 F S U 36 F S V 36 F S X 144 F U V 36 F U X 36 F V X 36 H I L 36 H I O 9 H I P 36 H I S 9 H I U 9 H I V 9 H I X 81 H L O 9 H L P 36 H L S 36 H L U 9 H L V 9 H L X 36 H O P 36 H O S 9 H O U 36 H O V 36 H O X 144 H P S 36 H P U 36 H P V 36 H P X 36 H S U 36 H S V 9 H S X ? H U V 9 H U X 36 H V X 81 I L O 36 I L P 36 I L S 9 I L U 36 I L V 36 I L X 81 I O P 36 I O S ? I O U 81 I O V 9 I O X 144 I P S 36 I P U 36 I P V 36 I P X 36 I S U 36 I S V 9 I S X 144 I U V 9 I U X 9 I V X 9 L O P 36 L O S ? L O U ? L O V 36 L O X 81 L P S 36 L P U 36 L P V 36 L P X 36 L S U 9 L S V 36 L S X 144 L U V 36 L U X 36 L V X 36 O P S 36 O P U 36 O P V 36 O P X 36 O S U ? O S V 9 O S X ? O U V 9 O U X 81 O V X 36 P S U 36 P S V 36 P S X 36 P U V 36 P U X 36 P V X 36 S U V 36 S U X 36 S V X 324 U V X 36

## Solutions

6A+6E+6F6A+6E+6H6A+6E+6I6A+6E+6L6A+6E+6O
6A+6E+6P6A+6E+6S6A+6E+6U6A+6E+6V6A+6E+6X
6A+6F+6H6A+6F+6I6A+6F+6L6A+6F+6O6A+6F+6P
6A+6F+6S6A+6F+6U6A+6F+6V6A+6F+6X6A+6H+6I
6A+6H+6L6A+6H+6O6A+6H+6P6A+6H+6S6A+6H+6U
6A+6H+6V6A+6H+6X6A+6I+6L6A+6I+6O6A+6I+6P
6A+6I+6S6A+6I+6U6A+6I+6V6A+6I+6X6A+6L+6O
6A+6L+6P6A+6L+6S6A+6L+6U6A+6L+6V6A+6L+6X
6A+6O+6P6A+6O+6S6A+6O+6U6A+6O+6V6A+6O+6X
6A+6P+6S6A+6P+6U6A+6P+6V6A+6P+6X6A+6S+6U
6A+6S+6V6A+6S+6X6A+6U+6V6A+6U+6X6A+6V+6X
6E+6F+6H6E+6F+6I6E+6F+6L6E+6F+6O6E+6F+6P
6E+6F+6S6E+6F+6U6E+6F+6V6E+6F+6X6E+6H+6I
6E+6H+6L6E+6H+6O6E+6H+6P6E+6H+6S6E+6H+6U
6E+6H+6V6E+6H+6X6E+6I+6L6E+6I+6O6E+6I+6P
6E+6I+6S6E+6I+6U6E+6I+6V6E+6I+6X6E+6L+6O
6E+6L+6P6E+6L+6S6E+6L+6U6E+6L+6V6E+6L+6X
6E+6O+6P6E+6O+6S6E+6O+6U6E+6O+6V6E+6O+6X
6E+6P+6S6E+6P+6U6E+6P+6V6E+6P+6X6E+6S+6U
6E+6S+6V6E+6S+6X6E+6U+6V6E+6U+6X6E+6V+6X
6F+6H+6I6F+6H+6L6F+6H+6O6F+6H+6P6F+6H+6S
6F+6H+6U6F+6H+6V6F+6H+6X6F+6I+6L6F+6I+6O
6F+6I+6P6F+6I+6S6F+6I+6U6F+6I+6V6F+6I+6X
6F+6L+6O6F+6L+6P6F+6L+6S6F+6L+6U6F+6L+6V
6F+6L+6X6F+6O+6P6F+6O+6S6F+6O+6U6F+6O+6V
6F+6O+6X6F+6P+6S6F+6P+6U6F+6P+6V6F+6P+6X
6F+6S+6U6F+6S+6V6F+6S+6X6F+6U+6V6F+6U+6X
6F+6V+6X6H+6I+6L6H+6I+6O6H+6I+6P6H+6I+6S
6H+6I+6U6H+6I+6V6H+6I+6X6H+6L+6O6H+6L+6P
6H+6L+6S6H+6L+6U6H+6L+6V6H+6L+6X6H+6O+6P
6H+6O+6S6H+6O+6U6H+6O+6V6H+6O+6X6H+6P+6S
6H+6P+6U6H+6P+6V6H+6P+6X6H+6S+6U6H+6S+6V
6H+6S+6X6H+6U+6V6H+6U+6X6H+6V+6X6I+6L+6O
6I+6L+6P6I+6L+6S6I+6L+6U6I+6L+6V6I+6L+6X
6I+6O+6P6I+6O+6S6I+6O+6U6I+6O+6V6I+6O+6X
6I+6P+6S6I+6P+6U6I+6P+6V6I+6P+6X6I+6S+6U
6I+6S+6V6I+6S+6X6I+6U+6V6I+6U+6X6I+6V+6X
6L+6O+6P6L+6O+6S6L+6O+6U6L+6O+6V6L+6O+6X
6L+6P+6S6L+6P+6U6L+6P+6V6L+6P+6X6L+6S+6U
6L+6S+6V6L+6S+6X6L+6U+6V6L+6U+6X6L+6V+6X
6O+6P+6S6O+6P+6U6O+6P+6V6O+6P+6X6O+6S+6U
6O+6S+6V6O+6S+6X6O+6U+6V6O+6U+6X6O+6V+6X
6P+6S+6U6P+6S+6V6P+6S+6X6P+6U+6V6P+6U+6X
6P+6V+6X6S+6U+6V6S+6U+6X6S+6V+6X6U+6V+6X

Last revised 2012-07-04.

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Col. George Sicherman [ HOME | MAIL ]