Related papers: Fun with Latin Squares
In this paper, we derive simple closed-form expressions for the $n$-queens problem and three related problems in terms of permanents of $(0,1)$ matrices. These formulas are the first of their kind. Moreover, they provide the first method…
A $k \times n$ partial Latin rectangle is \textit{$C$-sparse} if the number of nonempty entries in each row and column is at most $C$ and each symbol is used at most $C$ times. We prove that the probability a uniformly random $k \times n$…
A {\sf $\mu$-way Latin trade} of volume $s$ is a collection of $\mu$ partial Latin squares $T_1,T_2,...,T_{\mu}$, containing exactly the same $s$ filled cells, such that if cell $(i, j)$ is filled, it contains a different entry in each of…
We give an overview of universal quadratic forms and lattices, focusing on the recent developments over the rings of integers in totally real number fields. In particular, we discuss indecomposable algebraic integers as one of the main…
The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.
A Latin square of order $n$ is an $n$ by $n$ grid filled using $n$ symbols so that each symbol appears exactly once in each row and column. A transversal in a Latin square is a collection of cells which share no symbol, row or column. The…
We combine some known results and techniques with new ones to show that there exists a non-algebraic, multi-linear matroid. This answers an open question by Matus (Discrete Mathematics 1999), and an open question by Pendavingh and van Zwam…
We give three explicit quantum Latin squares of order $6$, with cardinalities $13$, $15$, and $17$. Throughout, vectors differing only by a global phase are counted as identical. The cardinality-$13$ construction is based on an orthogonal…
The sudoku puzzles have a long history, with variations going back more than a hundred years, but its current and perhaps surprising world-wide prominence goes back to certain initiatives and then puzzle-generating computer programmes from…
In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in $\PP ^3$ which are quotients of the bidisc by an irreducible lattice…
We study the tropical lines contained in smooth tropical surfaces in R^3. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however,…
Although Hadamard matrices have been investigated since the nineteenth century, relatively little is known about their higher-dimensional analogues. In this paper, we introduce two constructions of Hadamard hypercubes. The first…
In this article we study some geometric properties of a non-trivial square tile (a non-trivial square tile is a non-constant function on a square). Consider infinitely many copies of this single square tile and cover the plane with them,…
Applying results from partial difference sets, quadratic forms, and recent results of Brouwer and Van Dam, we construct the first known amorphic association scheme with negative Latin square type graphs and whose underlying set is a…
We show that there are uncountably many countable lattices. We give a discussion of which such lattices can be modular or distributive. The method applies to show that certain other classes of structures also have uncountably many…
A latin bitrade $(T^{\diamond}, T^{\otimes})$ is a pair of partial latin squares which defines the difference between two arbitrary latin squares $L^{\diamond} \supseteq T^{\diamond}$ and $L^{\diamond} \supseteq T^{\otimes}$ of the same…
The notion of multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The…
We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci--Lucas summation identities.
We settle the existence of certain "anti-magic" cubes using combinatorial block designs and graph decompositions to align a handful of small examples.
We introduce a condition on arrays in some way maximally distinct from Latin square condition, as well as some other conditions on algebras, graphs and $0,1$-matrices. We show that these are essentially the same structures, generalising a…