Tiling a Trapezium/Trapezoid with a Pentiamond and a Hexiamond

A pentiamond is a plane figure formed by joining 5 equilateral triangles edge to edge.

A hexiamond is a plane figure formed by joining 6 equilateral triangles edge to edge.

A trapezium is a quadrilateral with just two sides parallel. In the U.S. and Canada it is known as a trapezoid, because of a mixup in a popular 18th-century English dictionary of mathematics. For details, see this Wikipedia page.

A polyiamond with the shape of a trapezium is isosceles. That is, its two sides that are not parallel are equal.

Here I show the smallest trapezium with full symmetry that can be tiled by a pentiamond and a hexiamond, using at least one copy of each. The solutions shown are not necessarily uniquely minimal. I admit triangles as a degenerate case of trapezia. If you find a smaller solution or solve an unsolved case, please write.

In the table, the figures indicate the numbers of polyiamond cells in the solutions.

 6A6E6F6H6I6L6O6P6S6U6V6X
5I1614433641132511655636521
5J???????27???60
5Q77?16?125321639?16112?
5U???????32????

Tilings

11 Cells

16 Cells

21 Cells

27 Cells

32 Cells

33 Cells

39 Cells

51 Cells

55 Cells

60 Cells

63 Cells

64 Cells

65 Cells

77 Cells

112 Cells

125 Cells

144 Cells

Non-Triangular Variants

Last revised 2026-08-10.


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Col. George Sicherman [ HOME | MAIL ]