L Shapes from Pentacube Pairs

Introduction

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

The letters shown in black are my names. Mirror images of chiral pentacubes are indicated with ′; e.g., G′ is the mirror image of G. The green symbols are Kate Jones's names. The red symbols are Donald Knuth's names.

I define an L-shaped polycube as a polycube prism whose base is L-shaped; that is, it consists of a rectangle from one corner of which a smaller rectangle has been excised.

Here I show the smallest known L-shaped polycubes that can be tiled with a given pair of pentacubes, using at least one of each. Chiral pairs of pentacubes are distinguished, and chiral pentacubes may not be reflected when used in these tilings.

If you find a smaller solution, please write.

Table of Results

	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		6	6		12	14		6		13	6		6	6	3	6	4	2	6		8		12	6	8	8	80	6	7
B			3		12	18		3		7	2		9	6	19	6	5	3	5		12		12	3	8	14	20	6	14
E				8	9	9	12	2	6	5	6	6	6	6	3	6	3	2	8	3	6	4	4	6	6	9	16	6	6
F						18		9		7	4		9	5	12	8	3	4	9		4		12	2	4	15	39	5	12
G							×	8	12	21	6	2	12	6	56	7	6	3	30	56	18	12	21	6	3	18	440	8	7
H									4	5	6	6	6	4	6	6	4	2	6	8	4	6	10	6	4	6	14	6	10
I											8		12	2	11	6	2	4	11		4		7	6	2	7	8	5	7
J												3	6	2	7	6	3	3	7	8	3	6	3	4	4	9	12	4	4
K														8	8	6	3	4	2		9		10	6	8	6	12	6	12
L															8	2	2	4	6		4		6	3	2	3	7	5	7
M																10	5	6	19		9		15	3	15	9	20	8	10
N																	2	4	6		7		6	6	7	6	14	6	6
P																		2	5		3		2	2	2	4	6	2	2
Q																			4		2		3	2	3	4	6	4	4
R																				30	6	12	12	6	8	9	18	7	16
S																						4	15	6	3	4	24	8	4
T																								3	12	9	36	8	21
U																									6	12	3	2	6
V																										9	24	8	2
W																											28	6	16
X																												10	126
Y																													5
Z

Tilings

The pair G and G′ cannot tile any L-shaped polycube. Each of the solutions shown for pairs A-X, G-X, and X-Z was formed by joining two rectangular boxes. Smaller solutions may well exist for these.