M. Ahmed, J. De Loera, R. Hemmecke
Using computational algebraic geometry techniques and Hilbert bases of polyhedral cones we derive explicit formulas and generating functions for the number of magic squares and magic cubes.
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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.
Explicit algorithms are developed for constructing odd order n pandiagonal latin cubes in 3 and 4 dimensions, and these are used to construct pandiagonal magic cubes and 4 dimensional hypercubes, respectively. It is established that these structures exist in 3 dimensions only for odd orders n not less than 11, and in 4 dimensions only for odd orders n not less than 17. Estimates for the number of such structures as functions of n in 3 and 4 dimensions are also presented.
We describe how to construct and enumerate Magic squares, Franklin squares, Magic cubes, and Magic graphs as lattice points inside polyhedral cones using techniques from Algebraic Combinatorics. The main tools of our methods are the Hilbert Poincare series to enumerate lattice points and the Hilbert bases to generate lattice points. We define polytopes of magic labelings of graphs and digraphs, and give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.
We settle the existence of certain "anti-magic" cubes using combinatorial block designs and graph decompositions to align a handful of small examples.
We show that the vertices and edges of a $d$-dimensional grid graph $G=(V,E)$ ($d\geqslant 2$) can be labeled with the integers from $\{1,\ldots,\lvert V\rvert\}$ and $\{1,\ldots,\lvert E\rvert\}$, respectively, in such a way that for every subgraph $H$ isomorphic to a $d$-cube the sum of all the labels of $H$ is the same. As a consequence, for every $d\geqslant 2$, every $d$-dimensional grid graph is $Q_d$-supermagic where $Q_d$ is the $d$-cube.
Critical moments of slices and slabs of the cube (and other polyhedral norms)
提出统一代数-组合框架，计算多面体范数球切片和板的体积与高阶矩的显式分段有理参数公式，并应用于低维立方体，发现新有理函数族和临界点。
Magic Gems: A Polyhedral Framework for Magic Squares
Magic Gems将魔方阵转化为三维多面体，通过统计正交性条件定义其特征，揭示了魔方阵的统计结构和能量特性。
The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes
本文证明阶为2的魔面超立方体在任意维数下存在且唯一（在特定变换下），并应用于推广卡朱拉霍幻方、回答Coxeter问题，证明最完美对象在Weyl群作用下为单轨道。
In pioneering works of Meyer and of McMullen in the early 1970s, the set of Minkowski summands of a polytope was shown to be a polyhedral cone called the type cone. Explicit computations of type cones are in general intractable. Nevertheless, we show that the type cone of the product of simplices is the cone over a simplex. This remarkably simple result derives from insights about rainbow point configurations and the work of McMullen.
This paper aims to address the relation between a magic square of odd order $n$ and a group, and their properties. By the modulo number $n$, we construct entries for each table from initial table of magic square with large number $n^2$. Generalization of the underlying idea is presented, we obtain unique group, and we also prove variants of the main results for magic cubes.
Khajuraho's magic square is an hypercube
该研究指出印度克久拉霍寺庙的泛幻方，其对称性群与四维超立方体的对称性群同构，揭示了该古代幻方的隐藏数学特性。
A knight's tour on a board is a sequence of knight moves that visits each square exactly once. A knight's tour on a square board is called magic knight's tour if the sum of the numbers in each row and column is the same (magic constant). Knight's tour in higher dimensions (n > 3) is a new topic in the age-old world of knight's tours. In this paper, it has been proved that there can't be magic knight's tour or closed knight's tour in an odd order n-dimensional hypercube. 3 \times 4 \times 2n-2 is the smallest cuboid (n \geq 2) and 4 \times 4 \times 4n-2 is the smallest cube in which knight's to…
Weighted sums of left and right hand Kronecker products of Integer Sequence Doubly Affine (ISDA) as well as Generalized Arithmetic Progression Doubly Affine (GAPDA) arrays are used to generate larger ISDA arrays of multiplicative order (compound squares) from pairs of smaller ones. In two dimensions we find general expressions for the eigenvalues (EVs) and singular values (SVs) of the larger arrays in terms of the EVs and SVs of their constituent matrices, leading to a simple result for the rank of these highly singular compound matrices. Since the critical property of the smaller constituent …
Algebraic characterizations of generating and affinely generating $Γ$-magic maps and $Γ$-distance magic labelings on regular graphs
本文通过邻接算子核的嵌入条件刻画正则图上生成与仿射生成幻映射及距离幻标定，构造反例否定Cichacz-Froncek猜想，并提出生成群距离幻猜想。
Tropicalizing Principal Minors of Positive Definite Matrices
本文刻画正定矩阵主子式映射像的热带化，证明其等于仿射热带旗簇与次模锥的交集，并据此发现半正定矩阵主子式的新不等式。
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