Intercalates and the maximum number of intercalates are presented. We introduced partially intercalate complete Latin squares and results on the existence of some infinite families as well as Latin squares of smaller size is given. The second part of our work summarizes the main results on orthogonal Latin squares. Special type of Latin squares, gerechte designs, are introduced and proof for the existence of such orthogonal Latin squares of prime orders is also presented.

Methods of constructing orthogonal Latin of squares and incomplete balanced block designs are developed. The analysis of these designs is then derived. Particular care is taken in the determination of the number of degrees of freedom involved, a point which is usually neglected in other sources. The principle source of material for this thesis has been H.B. Mann's book, Analysis and Design of Experiments. / Thesis / Master of Arts (MA)

No description available.

This thesis is concerned with quasigroups with a small number of associative triples. The minimum number of associative triples among quasigroups of orders up to seven has already been determined. The goal of this thesis is to determine the minimum for orders eight and nine. This thesis reports that the minimum number of associative triples among quasigroups of order eight is sixteen and among quasigroups of order nine is nine. The latter finding is rather significant and we present a construction of an infinite series of quasigroups with the number of associative triples equal to their order. Findings of this thesis have been a result of a computer search which used improved algorithm presented in this thesis. The first part of the thesis is devoted to the theory that shows how to reduce the search space. The second part deals with the development of the algorithm and the last part analyzes the findings and shows a comparison of the new algorithm to the previous work. It shows that new search program is up to four orders of magnitude faster than the one used to determine the minimum number of associative triples among quasigroups of order seven.

No description available.

No description available.

The intersection of two Latin squares of the same order is the set of cells that contain the same entries in both Latin squares. Determining the order of this set can be asked for any type of Latin square and has been solved for most. Generalizing this to Latin squares of different orders leads to a conjecture of Dukes and Mendelsohn, which will be shown to be true. Results on the intersection of Latin squares, idempotent Latin squares, and idempotent symmetric Latin squares are explored. The relationship between the intersection problem for Latin squares and the intersection problem for Steiner triple systems will also be investigated. In addition to new results, past results are included presenting a common and clear notation. The proofs of some new results are able to replace proofs of past results as well as present a straightforward proof structure to new and past results. Two Latin squares of the same order are said to be r-orthogonal if the set of pairs occurring in corresponding cells has size r. Using this notation, two orthogonal Latin squares of order n are n2-orthogonal. The idea of r-orthogonality is generalized to Latin squares of different orders. The set of possible values is established for r and it is shown that this possible set can be obtained for pairs of Latin squares with certain orders.

The set PLS(a, b; n) is the set of all partial latin squares of order n with a completed rows, b completed columns and all other cells empty. We identify reductions of partial latin squares in PLS(2, 3; n) by using permutations described by filled rows and intersections of filled rows and columns. We find that all partial latin squares in PLS(2, 3;n), where n is sufficiently large, can be completed if such a reduction can be completed. We also show that all partial latin squares in PLS(2, 3; n) where the intersection of filled rows and columns form a latin rectangle have completions for n ≥ 8.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-09-08T12:22:35Z No. of bitstreams: 1 QUADRADOS LATINOS E QUADRADOS MÁGICOS UMA PROPOSTA DIDÁTICA.pdf: 24072473 bytes, checksum: 1d47842f904bd89accec69224c2a3c26 (MD5) / Approved for entry into archive by Fernando Souza (fernandoafsou@gmail.com) on 2017-09-08T13:27:24Z (GMT) No. of bitstreams: 1 QUADRADOS LATINOS E QUADRADOS MÁGICOS UMA PROPOSTA DIDÁTICA.pdf: 24072473 bytes, checksum: 1d47842f904bd89accec69224c2a3c26 (MD5) / Made available in DSpace on 2017-09-08T13:27:24Z (GMT). No. of bitstreams: 1 QUADRADOS LATINOS E QUADRADOS MÁGICOS UMA PROPOSTA DIDÁTICA.pdf: 24072473 bytes, checksum: 1d47842f904bd89accec69224c2a3c26 (MD5) Previous issue date: 2017-03-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study the Latin Squares and the Magic Squares. We explore the mathematical teory and, above all, we study the link between theses objects. We bring the necessary information to support the teacher in the usage of Latin Squares and Magic Squares as content. Our goal is to discuss the usage of games and challenges like didatic tools, and to find a proposal to applicate him in the classroom. / Neste trabalho fizemos uma pesquisa bibliográfica sobre os Quadrados Latinos e os Quadrados Magicos. Mostramos a teoria matematica envolvida e, sobretudo, estudamos a ligação entre esses objetos. Trouxemos as informações necessárias para subsidiar o professor a usar Quadrados Mágicos e Quadrados Latinos como con- teúdos. Nosso objetivo é discutir o uso de jogos e passatempos como ferramenta didática e chegar a uma proposta para utilização desses objetos em sala de aula.

This thesis describes methods of secret communication based on latin squares and their close relative, quasigroups. Different types of cryptosystems are described, including ciphers, public-key cryptosystems, and cryptographic hash functions. There is also a chapter devoted to different secret sharing schemes based on latin squares. The primary objective is to present previously described cryptosystems and secret sharing schemes in a more accessible manner, but this text also defines two new ciphers based on isotopic latin squares and reconstructs a lost proof related to row-latin squares. / Denna uppsats beskriver kryptosystem och metoder för hemlighetsdelning baserade på latinska kvadrater och det närliggande konceptet kvasigrupper. Olika sorters chiffer, både symmetriska och asymmetriska, behandlas. Dessutom finns ett kapitel tillägnat kryptografiska hashfunktioner och ett tillägnat metoder för hemlighetsdelning. Huvudsyftet är att beskriva redan existerande metoder för hemlig kommunikation på ett mer lättillgängligt sätt och med nya exempel, men dessutom återskapas ett, till synes, förlorat bevis relaterat till rad-latinska kvadrater samt beskrivs två nya chiffer baserade på isotopa latinska kvadrater.