Tiling a Triangle Polyhex with a Polyhex

Introduction

There are two kinds of equilateral triangular polyhexes:

Straight	Ragged

Here I show the smallest triangular polyhex of each type that various small polyhexes can tile, and identify what larger triangles they can tile. Polyhexes not shown cannot tile a triangular polyhex, so far as is known. If you find a smaller solution, or a solution for another polyhex, please write.

Most of these results for straight triangles are taken from Erich Friedman's Math Magic for 2003-03. There Erich presents results by Berend Jan van der Zwaag, Brendan Owen, Claudio Baiocchi, Jeremy Galvagni, Andrew Clarke, and Mike Reid.

See also Tiling a Hexagon Polyhex with a Heptahex.

Carl Schwenke and Johann Schwenke identified missing information and provided new solutions.

Straight Triangles

Monohex

Tile	Smallest Triangle	Sizes
		all

Dihex

Tile	Smallest Triangle	Sizes
		n ≡ 0 or 3 (mod 4)

Trihexes

Tile	Smallest Triangle	Sizes
		n ≡ 0, 2, 9, or 11 (mod 12)

Tetrahexes

Tile	Smallest Triangle	Sizes
		n ≡ 0 or 7 (mod 8)

Pentahexes

Tile	Smallest Triangle	Sizes
		n ≡ 0 or 4 (mod 5), except 4 and 10
		n = 5, …, 54, …, 59, 60, 60k−1, …, 65, …, 119, 120, …, ? (Carl Schwenke and Johann Schwenke)
		n ≡ 0 or 4 (mod 5), except 4, 9, 10, 14, and 19 (Andrew Clarke)
		n ≡ 0 or 4 (mod 5), except 4 and 10

Hexahexes

Tile	Smallest Triangle	Sizes
		n ≡ 0, 3, 20, or 23 (mod 24), except 3
		n ≡ 0, 3, 8, or 11 (mod 12), except 3, 8, 11, and 12
		n = 3 … ?

Heptahexes

Tile	Smallest Triangle	Sizes
		n = 27, 28, 28k−1, 28k, 35, 41, 42, … ?
		n ≡ 0 or 6 (mod 7), except 7, 13, 14, 20
		n = 6, … ?
		n = 6, … ?
		n = 6, … ?
		n = 6, … ?
		n = 6, … ?
		n ≡ 0 or 6 (mod 7), except 7, 13, 14, 20
		n = 6, … ?
		n = 6, 27, 35, 41, 42, 48, 56, 62, 63, 69, … ?
		n = 6, … ?

Ragged Triangles

Monohex

Tile	Smallest Triangle	Sizes
		all

Dihex

Tile	Smallest Triangle	Sizes
		n ≡ 2 or 3 (mod 4), except 2 and 3

Tetrahexes

Tile	Smallest Triangle	Sizes
		n ≡ 2 or 7 (mod 8), except 2 and 7
		n = 2

Pentahexes

Tile	Smallest Triangle	Sizes
		n ≡ 3 (mod 5), except 3 and 8
Tile	Smallest Triangle	Sizes
		n = 18, 23, 28, 33, … ?

Last revised 2025-09-08.