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<p>The authors find by applying MacMahon’s partition analysis that all magic labellings of the cube are of eight types, each generated by six basic 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 the authors enumerate magic distinct labellings, the number of which turns out to be a quasi-polynomial of period 720720. The authors also find the symmetry group can be used to significantly simplify the computation.</p>

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Enumerating Magic Distinct Labellings of the Cube

  • 期刊论文

  • 发表日期：

    2025年12月13日

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  • Guoce Xin,
  • Yingrui Zhang,
  • Zihao Zhang

摘要

The authors find by applying MacMahon’s partition analysis that all magic labellings of the cube are of eight types, each generated by six basic 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 the authors enumerate magic distinct labellings, the number of which turns out to be a quasi-polynomial of period 720720. The authors also find the symmetry group can be used to significantly simplify the computation.