Related papers: Magic Polygons and Their Properties

A curious number is a palindromic number whose base ten representation has the form $a \ldots a b \ldots b a \ldots a$. In this paper, we determine all curious numbers that are perfect squares. Our proof involves reducing the search for…

In this short paper we have produced different kinds of upside down magic squares based on a palindromic day 11.02.2011. In this day appear only the algorisms 0, 1 and 2. Some of the magic squares are bimagic and some are palindromic. Magic…

Permutation matrices play an important role in understand the structure of magic squares. In this work, we use a class of symmetric permutation matrices than can be used to categorize magic squares. Many magic squares with a high degree of…

A polygonal complex in euclidean 3-space is a discrete polyhedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number r of faces surround each edge. It is said to be regular…

Fullerenes are an allotrope of carbon having hollow, cage-like structure. Atoms in the molecule are arranged in pentagonal and hexagonal rings, such that each atom is connected to three other atoms. Simple polyhedra having only pentagonal…

We introduce and study arithmetic polygons. We show that these arithmetic polygons are connected to triples of square pyramidal numbers. For every odd $N\geq3$, we prove that there is at least one arithmetic polygon with $N$ sides. We also…

Recently, the classical Freudenthal Magic Square has been extended over fields of characteristic 3 with two more rows and columns filled with (mostly simple) Lie superalgebras specific of this characteristic. This Supermagic Square will be…

We show arithmetic triplets of Gaussian squares are in 3-to-1 correspondence with Pythagorean triples thereof. This correspondence would transform a solution to the Magic Square of Squares puzzle into a larger structure of perfect Gaussian…

It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for…

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…

A construction of the magic square, and hence of exceptional Lie algebras, is carried out using trialities rather than division algebras. By way of preparation, a comprehensive discussion of trialities is given, incorporating a number of…

In this paper we give the first method for constructing n-multimagic squares (and hypercubes) for any n. We give an explicit formula in the case of squares and an effective existence proof in the higher dimensional case. Finally we prove…

Polygon spaces have been studied extensively, and yet missing from the literature is a simple property that every polygon has: dimension. This is distinct (possibly) from the dimension of the ambient space in which the polygon lives. A…

Let $\Gamma$ be a group of order $n^2$ and $SMS_{\Gamma}(n)=(a_{i,j})_{n\times n}$ be an $n\times n$ array whose entries are all distinct elements of $\Gamma$. If there exists an element $\mu\in\Gamma$ such that for every row $i$, there…

In the early part of the paper, various geometrical formulas are derived. Then, at some point in the paper, the concept of a Pythagorean rational is introduced. A Pythagorean rational is a rational number which is the ratio of two integers…

In this paper, we study the concept of "binary color-coded magic squares" by assigning two distinct colors to the even and odd numbers within a magic square. We investigate the uniqueness of patterns within these squares using three…

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…

Magic-square constraints define Diophantine systems whose solutions, in several natural families, exhibit rigid periodic structure. We study this structure in an oracle setting, where a marked set of integers is given by black-box access…

A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to…

A novel kind of self-referential square matrix is introduced. A certain subset of the matrix entries record the frequencies of occurrence of each distinct number appearing within the entire matrix. Such squares are necessarily elusive. Our…