# Pentacube Pair Odd 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. A box is odd if it has an odd number of cells. Such a box must have all its dimensions odd.

Here I show the smallest odd box known to be tilable with each pair of pentacubes, using at least one copy of each of the two pentacubes. 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 25 9 15 ? 45 21 15 15 15 25 15 9 45 ? 15 15 15 15 21 ? 15 15 15 21 49 45 15 9 15 15 35 15 9 15 25 21 15 9 15 21 49 15 21 15 15 15 21 15 15 9 9 9 9 9 9 15 9 9 9 15 9 15 15 9 9 9 21 15 15 21 15 9 15 15 7 21 15 9 15 21 15 21 9 9 21 63 5 9 × 45 ? 27 15 9 21 15 ? 21 9 45 ? ? 33 21 27 15 15 33 ? 15 39 45 9 9 9 9 9 21 15 9 45 45 45 15 15 15 15 9 9 35 15 15 15 21 3 15 15 3 9 15 21 9 9 7 9 9 7 9 9 9 9 15 9 9 9 9 15 9 9 9 9 9 9 21 9 9 15 21 15 9 9 15 27 21 9 15 9 15 15 15 15 7 3 9 9 15 9 9 9 9 5 9 9 21 9 15 ? 15 21 9 21 21 × 15 21 9 15 21 15 15 9 15 21 25 21 15 9 9 9 3 9 3 9 5 3 3 45 9 9 15 9 9 15 15 9 45 21 21 15 9 15 21 117 15 15 27 21 15 21 21 15 15 15 9 15 21 55 9 25 9 21 3 11 9 21 49 15 3 57 15 15 5 63 5

## Impossible Cases

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

M and X, colored like a 3D checkerboard, have four cells of one color and one of the other. Thus an odd number of M and X pentacubes have a net color imbalance that is an odd multiple of 3. Every odd box has a color imbalance of 1.

Last revised 2024-02-05.

Back to Polyform Tiling < Polyform Curiosities
Col. George Sicherman [ HOME | MAIL ]