Diagonal Latin Squares Of Order Research Articles

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.

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.

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.

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.

The existence of orthogonal diagonal Latin squares with 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)) for all v ⩾ 3 n + 2 and v − n even. Further, there exists a magic square of order v with missing subsquare of side n (MS( v, n)) for all v ⩾ 3 n + 2 and v − n even.

Orthogonal diagonal latin squares of order fourteen

AbstractOrthogonal diagonal latin squares of order n, ODLS(n), are orthogonal latin squares of order n with transversals on both the main diagonal and the back diagonal of each square. It has been proven that ODLS(n) exist for all n except n = 2, 3, 6, 10, 14, 15, 18 and 26, in which the first three are impossible. In this note an example of ODLS(14) is given.