Related papers: Enumerating Magic Distinct Labellings of the Cube
A magic labelling of a set system is a labelling of its points by distinct positive integers so that every set of the system has the same sum, the magic sum. Examples are magic squares (the sets are the rows, columns, and diagonals) and…
We find by applying MacMahon's partition analysis that all magic squares of order three, up to rotations and reflections, are of two types, each generated by three basis elements. A combinatorial proof of this fact is given.
A magic labeling of a graph is a labeling of the edges by nonnegative integers such that the label sum over the edges incident to every vertex is the same. This common label sum is known as the index. We count magic labelings by maximum…
A magic labelling of a graph $G$ with magic sum $s$ is a labelling of the edges of $G$ by nonnegative integers such that for each vertex $v\in V$, the sum of labels of all edges incident to $v$ is equal to the same number $s$. Stanley gave…
Disregarding the identity, the remaining 63 elements of the generalized three-qubit Pauli group are found to contain 12096 distinct copies of Mermin's magic pentagram. Remarkably, 12096 is also the number of automorphisms of the smallest…
We define a magic square to be a square matrix whose entries are nonnegative integers and whose rows, columns, and main diagonals sum up to the same number. We prove structural results for the number of such squares as a function of the…
We give a variety of magic hexagons of Orders from 3 to 7, many of which are extensions of known results. We also give a theorem that their are an infinite number of magic hexagons of Order $n$ for any fixed positive integer $n$ for any…
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…
The 30 MacMahon colored cubes have each face painted with one of six colors and every color appears on at least one face. One puzzle involving these cubes is to create a $2\times2\times2$ model with eight distinct MacMahon cubes to recreate…
A \emph{magic square} is an $n \times n$ array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for $n=4$, as a function of either the magic sum or an…
A distance magic labeling of an $n$-dimensional hypercube is a labeling of its vertices by natural numbers from $\{0, \ldots, 2^n-1\}$, such that for all vertices $v$ the sum of the labels of the neighbors of $v$ is the same. Such a…
The Antimagic Graph Conjecture asserts that every connected graph $G = (V, E)$ except $K_2$ admits an edge labeling such that each label $1, 2, \dots, |E|$ is used exactly once and the sums of the labels on all edges incident to a given…
We settle the existence of certain "anti-magic" cubes using combinatorial block designs and graph decompositions to align a handful of small examples.
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.
It is unknown at present whether a magic square of squared integers exists. Such an object is defined to be a 3 by 3 grid of 9 distinct integer squares, such that the entries of each row, column, and two main diagonals sum to the same…
An additive-multiplicative magic square is a square grid of numbers whose rows, columns, and long diagonals all have the same sum (called the magic sum) and the same product (called the magic product). There are numerous open problems about…
In recreational mathematics, a normal magic square is an $n \times n$ square matrix whose entries are distinctly the integers $1 \ldots n^2$, such that each row, column, and major and minor traces sum to one constant $\mu$. It has been…
Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of…
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…
We study cylindric partitions with two-element profiles using MacMahon's partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition…