Pentacube Odd Cakes

Introduction

Pentominoes can form oddities with any even symmetry up to full symmetry, the 8-fold symmetry of a square. For example, Mike Reid arranged 13 copies of the P pentomino to make this oddity with square symmetry:

No oddities with square symmetry are known for pentominoes T, U, and V.

A pentacube is a solid made of five cubes joined face to face. There are 23 pentacubes, not distinguishing mirror images:

Define a polycube cake as an arrangement of one or more copies of a polycube to form a prism whose base is a polyomino with full (square) symmetry. Here is an example of a cake for the J pentacube:

It is easy to construct a cake for each pentacube. In the following section, for each pentacube I show a smallest known cake with an odd number of tiles. Even though pentominoes T, U, and V have no known full-symmetry oddities, the corresponding pentacubes have odd cakes.

If you find a smaller odd cake for a pentacube or solve an unsolved case, please write.

See also Polycube Oddities.

Table

A	B	E	F	G	H	I	J	K	L	M	N	P	Q	R	S	T	U	V	W	X	Y	Z
?	33	25	21	?	37	1	15	45	15	?	25	9	21	4655	405	75	27	33	29	1	21	73

Tilings

Pentacubes I and X are odd cakes in their own right.

The minimal known odd cakes for pentacubes F, W, X, and Z are pancakes, flat cakes. They are minimal known full-symmetry oddities for the corresponding pentominoes. See Pentomino Oddities.

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 2026-09-22.