Diagonal Latin Squares Research Articles

USE OF X-BASED DIAGONAL FILLINGS AND ESODLS CMS SCHEMES FOR ENUMERATION OF MAIN CLASSES OF DIAGONAL LATIN SQUARES

The concept of X-based diagonal fillings in diagonal Latin squares (DLS) is introduced, their enumeration is carried out them, isomorphism classes are constructed, and the interconnection of these problems with problems based on card decks is shown. On their basis, a method for generating strongly normalized canonical forms of the DLS using a limited set of ESODLS CMS schemes is proposed. It makes possible to reduce the necessary computational costs by 3-4 orders of magnitude compared to a software implementation based on a Brute Force based enumeration of the DLS. Using the developed software implementations and the Gerasim@ Home volunteer distributed computing project on the BOINC platform, the number of main classes of DLS of order 9 has been calculated. This number was previously unknown and amounted to 3 292 326 155 394.

Использование специализированных вычислителей на узле грид-системы для решения комбинаторных задач.

The paper discusses the use of specialized Intel Xeon Phi devices in a desktop grid system. As an example of the successful use of specialized devices, the combinatorial problem of finding diagonal Latin squares is given. Features of implementation of the computing application are discussed. The results of work of a computer application in multithreaded modes on different processors andspecialized devices are given.

Unified Construction of Normal Bimagic Squares of Doubly Even Orders Based on Quasi Bimagic Pairs

A general magic square of order n is an n × n matrix consisting of integers in such a way that the sum of all elements in each row, each column, main diagonal and back diagonal is the same number called the magic sum of this matrix. A general magic square of order 2n is normal if its entries are 4n 2 consecutive odd integers 1−4n 2, 3−4n 2,…,4n 2 −3, 4n 2 −1. A normal bimagic square of order 4n is a normal magic square such that the sum of squares of all elements in each row, each column, main diagonal and back diagonal is the same number. Using the reflection matrix R and a quasi bimagic pair (A,B) where A and B are two special 2n × 2n matrices consisting of odd integers, we give a unified and very simple construction of normal bimagic square H of order 4n for all n ≥ 2 : . We construct a quasi bimagic pair by means of orthogonal diagonal latin squares for n ≠ 3 and by means of the computer seeking for n = 3.

Using Volunteer Computing to Study Some Features of Diagonal Latin Squares

AbstractIn this research, the study concerns around several features of diagonal Latin squares (DLSs) of small order. Authors of the study suggest an algorithm for computing minimal and maximal numbers of transversals of DLSs. According to this algorithm, all DLSs of a particular order are generated, and for each square all its transversals and diagonal transversals are constructed. The algorithm was implemented and applied to DLSs of order at most 7 on a personal computer. The experiment for order 8 was performed in the volunteer computing project Gerasim@home. In addition, the problem of finding pairs of orthogonal DLSs of order 10 was considered and reduced to Boolean satisfiability problem. The obtained problem turned out to be very hard, therefore it was decomposed into a family of subproblems. In order to solve the problem, the volunteer computing project SAT@home was used. As a result, several dozen pairs of described kind were found.

On the Construction of Triples of Diagonal Latin Squares of Order 10

We provide a triple of diagonal Latin squares of order 10 that is the closest to being a triple of mutually orthogonal diagonal Latin squares found so far. It was obtained by constructing all orthogonal mates for diagonal Latin squares generated according to a specific scheme. We also show that a triple of mutually orthogonal diagonal Latin squares of order 10 cannot be constructed based on diagonal Latin squares from specific families.

Применение высокопроизводительных вычислений для поиска троек взаимно частично ортогональных диагональных латинских квадратов порядка 10

This paper deals with the search for triples of partially orthogonal Latin squares of order 10. For every known pair of orthogonal Latin squares of order 10 we add a third diagonal Latin square in such a way that the orthogonality condition between it and squares from a considered pair coincides in the maximum possible number of cells. Two approaches are used: the first one is based on reducing an original problem to Boolean satisfiability problem; the second one is based on Brute force method. Several triples of the aforementioned kind with high characteristics were constructed. The experiments were held in the volunteer computing project SAT@home and on a computing cluster.

Поиск пар ортогональных диагональных латинских квадратов порядка 10 в проекте добровольных распределенных вычислений SAT@HOME

In this paper we consider the approach to search for systems of orthogonal Latin Squares, that is based on reducing the corresponding problems to Boolean Satisfiability problem. We constructed the SAT encoding for finding orthogonal diagonal Latin Squares of order 10. Using this encoding and the resources provided by the volunteer computing project SAT@home we managed to find 17 previously unknown pairs. Based on these 17 pairs and on 3 previously published pairs we constructed the pseudotriples of diagonal Latin Squares of order 10. To construct pseudotriples we employed the computing cluster. The last step required us to make parallel implementation of the algorithm for generating diagonal Latin Squares of order 10.

Some constructions for [formula omitted] pairwise orthogonal diagonal Latin squares based on difference matrices

A diagonal Latin square is one whose main and back diagonals are both transversals. Difference matrices, quasi-difference matrices and holey quasi-difference matrices are used to construct t pairwise orthogonal diagonal Latin squares of order v (t PODLS(v)) in this paper. For t=3,4,5, known existence results for t PODLS(v) are then updated; for t=3, 4, and 5, there remain respectively 1, 11 and 32 unsolved cases, the largest of which are v=10, 54 and 98 respectively.

Strongly symmetric self-orthogonal diagonal Latin squares and Yang Hui type magic squares

In this paper, a strongly symmetric self-orthogonal diagonal Latin square of order n with a special property (∗SSSODLS(n)) is introduced. It is proved that a ∗SSSODLS(n) exists if and only if n≡0(mod4) and n≠4. As an application, it is shown that there exists a Yang Hui type magic square YMS(n,4) if and only if n≡0(mod4) and n≠4.

Upper bounds on sets of orthogonal colorings of graphs

We generalize the notion of orthogonal latin squares to colorings of simple graphs. Twon-colorings of a graph are said to be orthogonal if whenever two vertices share a color in one coloring they have distinct colors in the other coloring. We show that the usual bounds on the maximum size of a certain set of orthogonal latin structures such as latin squares, row latin squares, equi-n squares, single diagonal latin squares, double diagonal latin squares, or sudoku squares are special cases of bounds on orthogonal colorings of graphs.

Existence of strong symmetric self-orthogonal diagonal Latin squares

A diagonal Latin square is a Latin square whose main diagonal and back diagonal are both transversals. A Latin square is self-orthogonal if it is orthogonal to its transpose. A diagonal Latin square L of order n is strongly symmetric, denoted by SSSODLS ( n ) , if L ( i , j ) + L ( n − 1 − i , n − 1 − j ) = n − 1 for all i , j ∈ N = { 0 , 1 , … , n − 1 } . In this note, we shall prove that an SSSODLS ( n ) exists if and only if n ≡ 0 , 1 , 3 ( mod 4 ) and n ≠ 3 .

Some comments on Latin squares and on Graeco-Latin squares, illustrated with postage stamps and old playing cards

We present some comments on Latin squares and on Graeco-Latin squares, with special emphasis on their use in statistics and in a historical context. We also comment on the Knut Vik square, the knight’s move design and the knight’s tour, as well as the Magic Card Puzzle. We consider the well-known 36 officers problem studied by Euler (Verhandelingen uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen, vol. 9 (Middleburg 1782), pp. 85–239, 1779/1782), and give two examples of diagonal Latin squares of order 6 due, respectively, to Abbe Francois-Guillaume Poignard (Chez Guillaume Fricx, Imprimeur & libraire rue Bergestract, a l’enseigne des quatre Evangelistes, Bruxelles [4] 79 pp. (p. 71 folded), 1704) and Jozsef Denes (J Lond Math Soc Ser 2, 6(4):679–689, 1970). We illustrate our comments with images of postage stamps and old playing cards. An extensive annotated bibliography ends the paper.

Latin and cross Latin squares

Latin squares were first introduced and studied by the famous mathematician Leonhard Euler in the 1700s. Through the years, Latin squares have been used in areas such as statistics, graph theory, coding theory, the generation of random numbers as well as in the design and analysis of experiments. Recently, with the international popularity of Sudoku puzzles, Latin squares are in the spotlight as every solution to a Sudoku puzzle is a Latin square. This article describes Latin, diagonal Latin and cross Latin squares, as well as a method of constructing such squares.

Existence of self-orthogonal diagonal Latin squares with a missing subsquare

We shall refer to a diagonal Latin square which is orthogonal to its transpose as a self-orthogonal diagonal Latin square, briefly SODLS. This article investigates the spectra of SODLS and SODLS with a missing subsquare. It is found that for any positive integer v except 2, 3 or 6, there exists a SODLS of order v, and for any positive integers v and n, a SODLS of order v missing a subsquare of order n exists if and only if v⩾3 n+2 and v− n is even, except ( v, n)=(8,2) and except possibly for (v,n)∈{(3n+2,n) : n=6,8,10} .

Existence of (3, 1, 2)‐conjugate orthogonal diagonal latin squares

We shall refer to a diagonal Latin square which is orthogonal to its (3,1,2)-conjugate, and the latter is also a diagonal Latin square, as a (3,1, 2)-conjugate orthogonal diagonal Latin square, briefly CODLS. This article investigates the spectrum of CODLS and it is found that it contains all positive integers υ except 2, 3, 6, and possibly 10. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 297–308, 2001

Existence of (3, 1, 2)‐conjugate orthogonal diagonal latin squares

AbstractWe shall refer to a diagonal Latin square which is orthogonal to its (3,1,2)‐conjugate, and the latter is also a diagonal Latin square, as a (3,1, 2)‐conjugate orthogonal diagonal Latin square, briefly CODLS. This article investigates the spectrum of CODLS and it is found that it contains all positive integers υ except 2, 3, 6, and possibly 10. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 297–308, 2001

Constructing self-conjugate self-orthogonal diagonal latin squares

In this paper, we give some constructions of self-conjugate self-orthogonal diagonal Latin squares (SCSODLS). As an application of such constructions we disproof the conjecture about SCSODLS and show that there exist SCSODLS of orderV, whenevervэ1 (mod 12), with the possible exception ofve{13, 85, 133}.

Existence of self-conjugate self-orthogonal diagonal latin squares

In this article we give some new constructions of self-conjugate self-orthogonal diagonal Latin squares (SCSODLS). As an application of such constructions, we give a conclusive result regarding the existence of SCSODLS and show that there exists an SCSODLS of order n if and only if n ≡ 0, 1 (mod 4), except for n = 5. This result completely disproves a conjecture of Danhof, Phillips, and Wallis about SCSODLS in Danhof, Philips, and Wallis, JCMCC, 8 (1990), 3–8. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:51–62, 1998

Existence of conjugate orthogonal diagonal Latin squares

We shall refer to a diagonal Latin square which is orthogonal to its (3,1,2)-conjugate, and the latter is also a diagonal Latin square, as a (3,1, 2)-conjugate orthogonal diagonal Latin square, briefly CODLS. This article investigates the spectrum of CODLS and it is found that it contains all positive integers υ except 2, 3, 6, and possibly 10. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 297–308, 2001

Orthogonal diagonal Latin squares with small subsquares

We prove that there exists a pair of orthogonal diagonal Latin squares of order v with missing subsquares of side n(ODLS( v, n)) if and only if v ⩾ 3 n + 2 and v − n even, in the cases n ⩽ 6.