Related papers: Magic rectangles with empty cells

A chessboard has the property that every row and every column has as many white squares as black squares. In this mostly methodological note, we address the problem of counting such rectangular arrays with a fixed (numeric) number of rows,…

Let $\mathcal{R}$ denote the set of integers $n$ that can be represented as the sum $n = x^2 + y^2$ with $(x,y) = 1$. Let $a$ and $b$ be integers with $a>0$, $a \nmid b$. We show that for sufficiently large positive integer $N$ there are…

A more sums than differences (MSTD) set is a finite subset S of the integers such |S+S| > |S-S|. We show that the probability that a uniform random subset of {0, 1, ..., n} is an MSTD set approaches some limit rho > 4.28 x 10^{-4}. This…

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,…

Let A be an m \times n matrix in which the entries of each row are all distinct. Drisko showed that, if m \ge 2n-1, then A has a transversal: a set of n distinct entries with no two in the same row or column. We generalize this to matrices…

We study the function M(t,n) = Floor[ 1 / {t^(1/n)} ], where t is a positive real number, Floor[.] and {.} are the floor and fractional part functions, respectively. In a recent article in the Monthly, Nathanson proved that if log(t) is…

This paper attacks the following problem. We are given a large number $N$ of rectangles in the plane, each with horizontal and vertical sides, and also a number $r<N$. The given list of $N$ rectangles may contain duplicates. The problem is…

Completing Aronov et al.'s study on zero-discrepancy matrices for digital halftoning, we determine all (m, n, k, l) for which it is possible to put mn consecutive integers on an m-by-n board (with wrap-around) so that each k-by-l region…

Let $M$ be a non-zero binary matrix with distinct rows where the rows are closed under certain logical operators. In this article, we investigate the existence of columns containing an equal or greater number of ones than zeros.…

A Heffter array $H(n;k)$ is an $n\times n$ matrix such that each row and column contains $k$ filled cells, each row and column sum is divisible by $2nk+1$ and either $x$ or $-x$ appears in the array for each integer $1\leq x\leq nk$.…

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…

In this article we consider questions related to the behavior of the moments $M_{m}\left( \left\{ z_{j}\right\} \right) $ when the indices are restricted to specific subsequences of integers, such as the even or odd moments. If $n\geq2$ we…

We use a greedy probabilistic method to prove that for every $\epsilon > 0$, every $m\times n$ Latin rectangle on $n$ symbols has an orthogonal mate, where $m=(1-\epsilon)n$. That is, we show the existence of a second Latin rectangle such…

We study $m \times n$ matrices whose columns are of the form \[\{(a_{1j},\ldots, a_{nj}): \quad a_{1j} = \lambda_j,\ a_{ij} = \pm\lambda_j\ , \ \lambda_j >0 ,\ j=1,2,\ldots,n\}.\] We explicitly construct for all $a = (a_1,\ldots,…

I found a novel class of magic square analogue, magic 24-cell. The problem is to assign the consecutive numbers 1 through 24 to the vertices in a graph, which is composed of 24 octahedra and 24 vertices, to make the sum of the numbers of…

Let m and n be integers, $2 \leq m \leq n$. An m by n array consists of mn cells, arranged in m rows and n columns, and each cell contains exactly one symbol. A transversal of an array consists of m cells, one from each row and no two from…

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.

We give a complete classification of modular categories of dimension $p^3m$ where $p$ is prime and $m$ is a square-free integer. When $p$ is odd, all such categories are pointed. For $p=2$ one encounters modular categories with the same…

A set of arithmetical sequences $$ a_1\, (\bmod{ \,\, m_1}) \quad, \quad a_2 \, (\bmod{\,\, m_2}) \quad, \quad \dots \quad , \quad a_k \, (\bmod{\,\,m_k}) \quad \quad , $$ with $$ m_1 \leq m_2 \leq \dots \leq m_k \quad \quad , $$ is called…

We study the singularity probability of random integer matrices. Concretely, the probability that a random $n \times n$ matrix, with integer entries chosen uniformly from $\{-m,\ldots,m\}$, is singular. This problem has been well studied in…