# Tiling Triangular Prism Polycubes with Two Pentacubes

A pentacube is a solid made of 5 equal cubes joined face to face.

The blue pentacubes have distinct mirror images. The symbols in green are Kate Jones's names for the pentacubes. The symbols in red are Donald Knuth's names for the pentacubes.

A triangle polyomino is a polyomino in the form of an isosceles right triangle. It has two straight edges and one zigzag edge that forms its hypotenuse. A triangular prism polycube is a polycube prism whose base is a triangle polyomino.

Here I show the smallest known triangular prism polycubes that can be tiled with copies of of two different pentacubes, using at least one of each. A prime mark () after a letter denotes a mirror image. For example, S′ is the mirror image of S. To see a tiling, click on the corresponding entry in the table below. Missing entries indicate unsolved cases.

If you find a smaller solution, or solve an unsolved case, please write.

A B E E′ F G G′ H H′ I J J′ K L M N P Q R R′ S S′ T U V W X Y 4 8 18 18 6 20 15 6 4 3 10 4 6 4 12 18 8 15 8 – 12 18 8 18 42 8 10 9 12 6 39 8 6 3 12 8 12 6 8 18 42 4 18 42 8 42 12 4 8 6 8 8 – 10 6 6 4 4 6 12 44 – 20 6 14 12 – 8 – – 12 10 12 – 12 – 42 4 4 – – – 44 8 – – 12 – – 12 18 18 9 8 44 4 – 4 6 3 – – – – – 12 6 – – 4 44 6 6 8 8 6 4 18 15 4 6 18 18 18 6 18 4 12 6 21 6 18 12 12 3 15 10 3 6 15 12 10 10 6 6 10 10 10 8 8 4 12 8 6 4 8 8 3 9 6 8 6 12 42 8 12 10 – – 8 4 42 – – 12 8 42 – 8 – 6 6 4 4 4 4 12 8 8 2 15 10 12 – 8 4 – – 44 9 42 – – 12 – 12 8 4 – 32 63 12 4 – 15 – 4 9 4 8 9 6 3 9 2 12 4 6 8 6 6 6 4 4 8 – 6 – 42 12 21 – – 4 12 – – 44 18 36 – 8 – 6 16 24 – 12 – 12 18 – 6 126 42 – 8 8 – 4 – 30 – 12

Last revised 2024-01-08.

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Col. George Sicherman [ HOME | MAIL ]