# Pentacube Pair Boxes

## Introduction

A pentacube is a solid made of 5 equal cubes joined face to face. There are 29 pentacubes, distinguishing mirror images.

I indicate mirror images with a prime mark (). For example, S′ denotes the mirror image of pentacube S.

All but two of the pentacubes, G and X, can tile a rectangular prism, or box. All but one pair of pentacubes, G-G′, can jointly tile a box.

Here I show the smallest box known to be tilable with each pair of pentacubes, using at least one copy of each pentacube in the pair. If you find a solution smaller than one shown, or solve an unsolved case, please write.

## Tile Counts

Click on an entry in the table to see the corresponding tiling.

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 6 6 12 24 8 12 6 6 8 6 4 6 4 6 8 6 6 12 12 40 4 6 6 12 14 6 8 4 8 8 10 8 6 6 10 8 15 6 8 12 10 6 12 4 6 6 16 4 4 6 6 6 6 4 6 8 4 4 8 6 8 8 12 6 9 8 18 8 6 18 8 9 8 8 4 12 8 6 6 12 8 16 6 4 12 40 5 9 × 8 8 24 12 4 12 8 48 8 6 6 54 40 24 8 18 12 8 18 200 8 32 10 6 8 8 6 4 6 8 4 4 6 8 6 8 8 8 8 6 14 8 6 10 12 3 12 8 3 6 12 12 9 6 6 9 9 6 9 6 8 4 10 4 6 4 4 6 6 8 6 6 6 8 6 8 6 8 4 4 6 4 4 12 16 6 12 6 8 4 6 8 4 3 4 8 8 9 4 4 4 5 8 8 12 6 6 30 8 8 6 16 12 10 6 8 4 4 8 6 8 4 4 8 16 6 8 4 6 6 3 4 3 4 5 3 3 6 4 6 4 6 8 6 4 6 28 6 8 8 6 6 12 10 6 12 18 12 6 12 6 15 8 6 6 12 12 63 4 18 8 16 3 6 9 12 24 12 3 36 8 12 5 63 5

## Impossible Case

G and G′ cannot tile even one edge of a box, much less a whole box.

Last revised 2024-02-12.

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Col. George Sicherman [ HOME | MAIL ]