Enumerating Magic Distinct Labellings of the Cube

TL;DR

This paper classifies all magic labellings of a cube into eight types using MacMahon's partition analysis, providing a polynomial formula for their count and enumerating distinct labellings with symmetry considerations.

Contribution

It introduces a complete classification of cube magic labellings and derives explicit formulas for their enumeration, including symmetry-based simplifications.

Findings

Eight types of magic cube labellings identified

Number of labellings expressed as a polynomial in the magic sum

Enumeration of distinct labellings is a quasi-polynomial of period 720720

Abstract

We find by applying MacMahon's partition analysis that all magic labellings of the cube are of eight types, each generated by six basis elements. A combinatorial proof of this fact is given. The number of magic labellings of the cube is thus reobtained as a polynomial in the magic sum of degree $5$. Then we enumerate magic distinct labellings, the number of which turns out to be a quasi-polynomial of period 720720. We also find the group of symmetry can be used to significantly simplify the computation.