Related papers: Signed magic rectangles with three filled cells in…

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…

This paper studies the set of $n\times n$ matrices for which all row and column sums equal zero. By representing these matrices in a lower dimensional space, it is shown that this set is closed under addition and multiplication, and…

Let $n$ be a positive integer and $X = [x_{ij}]_{1 \leq i, j \leq n}$ be an $n \times n$\linebreak \noindent sized matrix of independent random variables having joint uniform distribution $$\hbox{Pr} {x_{ij} = k \hbox{for} 1 \leq k \leq n}…

Let $n>1$ and $k>0$ be fixed integers. A matrix is said to be level if all its column sums are equal. A level matrix with $m$ rows is called reducible if we can delete $j$ rows, $0<j<m$, so that the remaining matrix is level. We ask if…

Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle…

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 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.…

An $r$-matrix is a matrix with symbols in $\{0,1,\dots,r-1\}$. A matrix is simple if it has no repeated columns. Let the support of a matrix $F$, $\text{supp}(F)$ be the largest simple matrix such that every column in $\text{supp}(F)$ is in…

We find the numbers of $3 \times 3$ magic, semimagic, and magilatin squares, as functions either of the magic sum or of an upper bound on the entries in the square. Our results on magic and semimagic squares differ from previous ones in…

This paper characterizes when an $m \times n$ rectangle, where $m$ and $n$ are integers, can be tiled (exactly packed) by squares where each has an integer side length of at least 2. In particular, we prove that tiling is always possible…

A Heffter array $H(m,n;s,t)$ is an $m \times n$ matrix with nonzero entries from $\mathbb{Z}_{2ms+1}$ such that $i)$ each row contains $s$ filled cells and each column contains $t$ filled cells, $ii)$ every row and column sum to 0, and…

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…

A "signed graph" is a graph $\Gamma$ where the edges are assigned sign labels, either "$+$" or "$-$". The sign of a cycle is the product of the signs of its edges. Let $\mathrm{SpecC}(\Gamma)$ denote the list of lengths of cycles in…

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…

A ring $R$ is trinil clean if every element in $R$ is the sum of a tripotent and a nilpotent. If $R$ is a 2-primal strongly 2-nil-clean ring, we prove that $M_n(R)$ is trinil clean for all $n\in {\Bbb N}$. Furthermore, we show that the…

A KMS matrix is one of the form $$J_n(a)=[{array}{ccccc} 0 & a & a^2 &... & a^{n-1} & 0 & a & \ddots & \vdots & & \ddots & \ddots & a^2 & & & \ddots & a 0 & & & & 0{array}]$$ for $n\ge 1$ and $a$ in $\mathbb{C}$. Among other things, we…

For positive integers $1 \leq k \leq n$ let $M_n$ be the algebra of all $n \times n$ complex matrices and $M_n^{\le k}$ its subset consisting of all matrices of rank at most $k$. We first show that whenever $k>\frac{n}{2}$, any continuous…

This paper is concerned with the problem of finding two sets of integers, $\{a_1, a_2, \ldots$, $a_m\}$ and $\{b_1, b_2, \ldots, b_n\}$, such that all the $mn$ sums $a_i+b_j, i=1, \ldots, m, j=1, \ldots, n$, are perfect squares. A method is…

Let a tribonacci sequence be a sequence of integers satisfying $a_k=a_{k-1}+a_{k-2}+a_{k-3}$ for all $k\ge 4$. For any positive integers $k$ and $n$, denote by $f_k(n)$ the number of tribonacci sequences with $a_1, a_2, a_3>0$ and with…

A frequency rectangle of type FR$(m,n;q)$ is an $m \times n$ matrix such that each symbol from a set of size $q$ appears $n/q$ times in each row and $m/q$ times in each column. Two frequency rectangles of the same type are said to be…