Related papers: Magic squares with all subsquares of possible orde…

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…

Let $(\Gamma,+)$ be an Abelian group of order $n^2$. A $\Gamma$-magic square of order $n$ is an $n\times n$ array whose entries are pairwise distinct elements of $\Gamma$ such that all row sums, column sums, and the two main diagonal sums…

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…

Magic squares have been an enthralling topic in mathematics for centuries. They are formed by filling in all the cells of a square matrix with the numbers starting from one so that the sum of all rows, columns, and diagonals is the same.…

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.

Quantum permutation matrices and quantum magic squares are generalizations of permutation matrices and magic squares, where the entries are no longer numbers but elements from arbitrary (non-commutative) algebras. The famous Birkhoff--von…

We review a known method of compounding two magic square matrices of order m and n with the all-ones matrix to form two magic square matrices of order mn. We show that these compounded matrices commute. Simple formulas are derived for their…

A {\em signed magic rectangle} $SMR(m,n;k, s)$ is an $m \times n$ array with entries from $X$, where $X=\{0,\pm1,\pm2,\ldots, $ $\pm (mk-1)/2\}$ if $mk$ is odd and $X = \{\pm1,\pm2,\ldots,\pm mk/2\}$ if $mk$ is even, such that precisely $k$…

In this paper, we introduce a newly defined algebraic invariant for square matrices termed the \emph{Alternating Power Difference (APD)}. The APD is defined as the signed sum of the powers of diagonal sums along permutations of the…

We explore some of the properties of consecutive, equally-summed arithmetic progressions of odd numbers, particularly their offsets and sums, before using them to prove that no $3\times3$ magic squares of distinct square integers exist.

The aim of this note is to introduce fastest new general methods for the construction of double and single even order magic squares. As in [5], the method for double even order magic squares is fairly straight-forward but some adjustments…

Let $\Gamma$ be a group of order $mnk$ and $MRS_{\Gamma}(m,n;k)=(a_{i,j}^s)_{m\times n}$ be a collection of $k$ arrays $m\times n$ whose entries are all distinct elements of $\Gamma$. If there exist elements $\rho,\sigma\in\Gamma$ such that…

For any $d \geq 2$, we prove that there exists an integer $n_0(d)$ such that there exists an $n \times n$ magic square of $d^\text{th}$ powers for all $n \geq n_0(d)$. In particular, we establish the existence of an $n \times n$ magic…

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…

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…

A $q$-ary maximum distance separable (MDS) code $C$ with length $n$, dimension $k$ over an alphabet $\mathcal{A}$ of size $q$ is a set of $q^k$ codewords that are elements of $\mathcal{A}^n$, such that the Hamming distance between two…

In this paper, we present the problem of counting magic squares and we focus on the case of multiplicative magic squares of order 4. We give the exact number of normal multiplicative magic squares of order 4 with an original and complete…

In this short note we have given an equation based on the date 11.09.2001 and presented some magic squares. The magic squares presented are of order 3x3, 4x4, 5x5, 9x9, 16x16 and 25x25. While the magic square of higher order 9x9, 16x16 and…

A signed magic rectangle $SMR(m,n;r, s)$ is an $m \times n$ array with entries from $X$, where $X=\{0,\pm1,\pm2,\ldots, $ $\pm (ms-1)/2\}$ if $mr$ is odd and $X = \{\pm1,\pm2,\ldots,\pm mr/2\}$ if $mr$ is even, such that precisely $r$ cells…

We construct and classify all possible Magic Squares (MS's) related to Euclidean or Lorentzian rank-3 simple Jordan algebras, both on normed division algebras and split composition algebras. Besides the known Freudenthal-Rozenfeld-Tits MS,…