# Tiling a Cubic Polycube with Two Pentacubes

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

A cubic polycube is a polycube whose cells form the shape of a cube.

A cube with side n has n3 cells. The smallest cubic polycube whose volume is a multiple of 5 is the cube with side 5, shown above. It has 125 cells, so it can be tiled with 25 pentacubes.

Here I show which pairs of pentacubes can tile the a 5×5×5 cube, using at least one copy of each pentacube. 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. Yellow cells indicate that the tiling is unique.

The E, I, J (and J′), L, N, P, and Y pentacubes can each tile the 5×5×5 cube alone. To see such tilings, click on the corresponding index link in the table.

If you 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 @ @ @ × – @ @ @ @ @ @ @ – – @ @ @ @ @ × @ @ @ @ × @ @ @ @ @ × @ @ @ @ @ @ @ @ @ × @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ × @ @ × × × × @ @ @ @ × @ @ – × × × @ × @ @ × × @ × × @ @ @ @ @ @ @ @ – – – @ @ @ @ @ @ × @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ – @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ × @ @ @ × @ × @ × @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ – @ @ @ @ @ @ @ @ × @ @ @ @ @ @ × @ @ × @ @ @ @ × @ @ @ @ × × @ @ @ @ @ @ @ @ × @ @ × @ × @ × @

Last revised 2024-03-22.

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Col. George Sicherman [ HOME | MAIL ]