# Minimal Oddities for the P Pentacube

## Introduction

A pentacube is a solid made of five equal cubes joined face to face. There are 29 different pentacubes, distinguishing mirror images. The P pentacube is shaped like the letter P:

A polyform oddity is a shape with even symmetry formed by joining an odd number of copies of a polyform.

Polycubes can belong to any of 33 symmetry classes, including asymmetry; see Polycube Symmetries. Of these symmetry classes, 31 have even order and can be symmetries of oddities.

Here I show minimal oddities for the P pentacube that belong to every even symmetry class. If you find a smaller example for any symmetry class, please write.

For other pentacubes, see Pentacube Oddities with Full Symmetry and its links to other pentacube oddity pages.

## Solutions

The symmetry codes are those of W. F. Lunnon; see Polycube Symmetries. The order of a symmetry is shown in parentheses next to its code. After that appears the number of copies of the P pentacube. An asterisk means that the figure is unique for its number of tiles.

C4(2) 3B6(2) 5CF6(2) 5F5(2) 3
E4(2) 1*A12(4) 5*J10(4) 7BC10(4) 7
BB10(4) 9CK6(4) 5*BE4(4) 5*CE3(4) 3
BF6(4) 5EE4(4) 3CD10(6) 5*FF4(6) 5*
H12(6) 5*AB16(8) 9EF6(8) 5*BFF8(8) 7
CJ6(8) 9AE8(8) 9EFF7(8) 7EEE6(8)
BD34(12) 17DF6(12) 5*BBC2(16) 9*R56(24) 25*
CCC20(24) 17*DEE25(24) 25*G1(48) 25*

Last revised 2024-05-12.

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Col. George Sicherman [ HOME | MAIL ]