Pentacubes in a Square Box

Introduction

The six blue tiles have distinct mirror images. Kate Jones's systematic names are shown in green. Donald Knuth's names are shown in red. In all three nomenclatures, pentacubes that lie all in one plane are named for the corresponding pentominoes. (Kate Jones uses Solomon Golomb's names; Donald Knuth uses John Conway's names.)

Can a rectangular prism, or box, with at least two dimensions equal, be filled with copies of a pentacube? For each pentacube, here is the smallest known square box that it can fill. Torsten Sillke's polycube box tiling page identifies many boxes besides the smallest that can be tiled by polycubes. Toshihiro Shirakawa's Box Packing Collection has extensive box tiling data for polycubes, edge-polycubes, and polyhypercubes.

In these tilings, the pentacubes are not reflected.

Pentacubes G and X cannot tile a box.

If you find a smaller square box tiling, please write.

See also

Solutions

A

B

E

F

G

H

I

J

K

L

M

N

P

Q

R

S

T

U

V

W

X

Y

Z

Last revised 2026-09-17.