Pentacube Pairs Tiling a Box with All Edges Removed

Introduction

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.

Here I show minimal known tilings of a rectangular box from which all the cells on its edges have been removed. The pentacubes are not reflected in the tilings. 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.

Basic Solutions

	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	Z
A		10	12		10	10		4		12	12		12	16	12	10	10	10	10		12		12	10	16	24	28	16	24
B			10		10	10		12		12	10		12	10	10	10	10	12	10		10		12	10	12	19	21	10	12
E				16	24	12	12	12	12	19	12	16	12	10	12	10	10	12	10	4	28	24	19	12	24	12	32	10	28
F						24		12		19	12		28	12	24	12	12	12	4		24		10	12	24	12	46	16	33
G							10	12	12	28	12	12	12	16	16	12	10	12	4	10	24	16	10	12	12	28	64	16	12
H									4	16	12	10	12	12	12	12	12	12	12	12	16	12	24	12	12	12	12	12	12
I											16		28	19	12	19	12	16	12		28		19	12	19	19	19	16	19
J												12	12	16	19	10	10	12	10	4	12	12	12	12	12	12	12	10	21
K														16	28	12	12	16	12		32		28	12	28	4	32	16	32
L															12	16	10	16	12		16		20	12	20	16	20	16	16
M																10	10	12	4		16		46	10	24	10	68	12	32
N																	12	12	10		12		10	10	12	28	28	12	10
P																		12	10		12		12	10	10	12	10	10	10
Q																			10		16		12	12	12	4	16	16	16
R																				4	10	12	10	10	12	12	10	12	10
S																						32	32	28	32	4	64	19	32
T																								28	32	32	–	24	32
U																									19	4	35	10	12
V																										28	64	19	33
W																											54	16	38
X																												24	–
Y																													28
Z

Odd Solutions

	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	Z
A		19	33		19	33		19		33	19		33	33	33	19	19	35	19		33		33	33	35	–	37	19	33
B			19		19	19		19		19	19		19	19	19	19	19	19	19		19		19	19	19	19	21	19	19
E				33	33	33	33	33	33	19	19	21	33	19	33	19	19	33	19	19	33	33	19	33	33	33	33	19	33
F						33		–		19	19		–	19	33	–	33	33	33		–		35	33	33	–	–	19	33
G							33	33	33	33	21	21	33	19	33	33	19	35	19	19	33	33	33	33	33	–	–	19	35
H									–	19	21	19	33	33	–	–	37	–	–	37	–	–	33	33	33	–	55	19	33
I											19		53	19	19	19	19	19	19		37		19	19	19	19	19	19	19
J												21	21	19	19	21	19	21	19	19	33	33	21	21	21	19	33	19	21
K														33	33	–	55	33	–		–		–	33	33	–	55	19	33
L															19	19	19	33	19		43		37	35	21	43	65	19	53
M																33	33	33	33		33		–	33	33	33	–	33	33
N																	55	–	55		–		33	21	21	–	33	19	33
P																		55	55		33		19	19	19	33	19	19	19
Q																			–		–		33	33	21	37	33	19	33
R																				–	33	37	19	19	19	–	33	19	33
S																						–	–	33	–	–	–	19	33
T																								33	–	33	–	33	–
U																									19	35	35	19	33
V																										33	–	19	33
W																											75	35	33
X																												35	–
Y																													53
Z

Last revised 2026-05-12.

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Col. George Sicherman [ HOME | MAIL ]