The basic shape of a Polyhextan is a hexagon made up of six right isosceles triangles. This particular shape has not been named in an established polyform classification. Since polyforms made from right isosceles triangles are generally called Polytans, the term Hextan was introduced for this hexagon.

The grid cannot be completely filled with Hextans. Arranging them inevitably leaves gaps, creating a grid with holes.

Connected Hextans form what are known as Polyhextans. The illustration shows all 20 Pentahextans—that is, every combination of five connected Hextans. These shapes fit into a regular grid called Bevelled Squares in the PolySolver puzzle solver.

There is one Dihextan, three Trihextans, seven Tetrahextans, 20 Pentahextans, 60 Hexahextans, and 204 Heptahextans.

Shapes and symmetry

The underlying grid allows shapes with four axes of reflection and fourfold rotational symmetry. Both Polyhextan puzzle forms shown here have this maximum symmetry. Free 3D-printing templates for the HexTan puzzle are available in the “Resources & Downloads” section.

1. HexTan puzzle

The seven Tetrahextans alone cannot form a shape with this symmetry. The HexTan puzzle therefore also includes two Trihextans and one Dihextan.

2. PentaHexTan puzzle

The PentaHexTan puzzle consists of the 20 Pentahextans.

The 3D-printing files include the 10 Polyhextan puzzle pieces and a slider box.