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)

A computer generation of all pairs of mutually orthogonal Latin squares of order ten and dimension 35 or less is undertaken. All such pairs are successfully generated up to main class equivalence. No pairs of mutually orthogonal Latin squares of order ten exist for dimension 33. Six dimension 34 pairs, which are counterexamples to a conjecture by Moorehouse, are found. Eighty-five pairs can be formed with dimension 35. None of the pairs can be extended to a triple. If a triple of mutually orthogonal Latin squares exists for order ten, the pairs of Latin squares in the triple must be of dimension 36 or 37.

Submitted by Ricardo Pessoa Dos Santos null (ricopessoa@gmail.com) on 2017-12-14T17:35:33Z No. of bitstreams: 1 Dissertação.pdf: 4489608 bytes, checksum: 2c9d751844c4b178546f2154b0718705 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2017-12-14T18:53:30Z (GMT) No. of bitstreams: 1 santos_rp_me_sjrp.pdf: 4489608 bytes, checksum: 2c9d751844c4b178546f2154b0718705 (MD5) / Made available in DSpace on 2017-12-14T18:53:30Z (GMT). No. of bitstreams: 1 santos_rp_me_sjrp.pdf: 4489608 bytes, checksum: 2c9d751844c4b178546f2154b0718705 (MD5) Previous issue date: 2017-11-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Iremos apresentar a um grupo de alunos do Ensino Médio da rede pública de Ensino do Estado de São Paulo, o mundialmente conhecido quebra cabeças Sudoku, e realizar com eles várias atividades buscando apresentá-lo como subsídio didático na aprendizagem de conceitos matemáticos importantes, além de proporcionar oportunidades de aprimorar a concentração e o raciocínio lógico. Iremos explorar conceitos matemáticos ocultos por trás de suas linhas, colunas e blocos, partindo de uma das primeiras perguntas que podem ser feitas: Qual é a quantidade total de jogos válidos existentes? Para responde-la, será proposto a realização de diversas atividades, primeiramente com um Shidoku (matriz 4 × 4), em seguida iremos calcular o total desses jogos. O tamanho reduzido dessa grade, facilita os cálculos manuais, permitindo visualizar e compreender o processo utilizado, aproveitando para introduzir o princípio fundamental da contagem. A discussão principal desse trabalho, concentra-se na exploração de um método para se determinar a quantidade de jogos válidos existentes para um Sudoku, e para isso, utilizaremos as demonstrações de Bertrand Felgenhauer e Frazer Jarvis. Também apresentaremos um método capaz de gerar uma grade completa de Sudoku, partindo de uma matriz quadrada de ordem 3, que em seguida, será utilizada para gerar uma solução de Sudoku ortogonal. Finalizando, iremos apresentar e explorar algumas formas diferenciadas para os quebra cabeças Sudoku, mostrando variações no formato dos blocos, no tamanho das grades e uma variação que utiliza formas geométricas em suas pistas (Shapedoku). Como desafio de leitura, pesquisa e aprofundamento, será proposto o problema ainda em aberto do número mínimo de dados iniciais para se ter um jogo válido. Podemos afirmar que um dos objetivos esperados, é que tal atividade venha interferir na concentração e raciocínio, auxiliando nas atividades propostas nesse trabalho e que possam ser utilizadas em outros problemas do cotidiano. / We will present to a group of high school students of the public Education of Sao Paulo state, the world-known puzzle Sudoku, and perform with them several activities seeking to present it as a didactic subsidy in the learning important mathematical concepts, besides opportunities to enhance concentration and logical reasoning. We will explore hidden mathematical concepts behind their lines, columns and blocks, starting from one of the rst questions that can be asked: What is the total number of valid games in existence? To answer this question, it will be proposed to perform several activities, rst with a Shidoku (4 × 4 matrix), then we will calculate the total of these games. The reduced size of this grid facilitates manual calculations, allowing to visualize and understand the process used, taking advantage to introduce the fundamental principle of counting. The main discussion of this paper focuses on the exploration of a method to determine the amount of valid games existing for a Sudoku, and for that, we will use the demonstrations of Bertrand Felgenhauer and Frazer Jarvis. We will also present a method capable of generating a complete Sudoku grid, starting from a square matrix of order 3, which will then be used to generate an orthogonal Sudoku solution. Finally, we will introduce and explore some di erent shapes for the Sudoku puzzle, showing variations in the shape of the blocks, the size of the grids and a variation that uses geometric forms in their tracks (Shapedoku). As a challenge for reading, searching and deepening, the open problem of the minimum number of initial data to have a valid game will be proposed. We can say that one of the expected objectives is that such activity will interfere in concentration and reasoning, helping in the activities proposed in this paper and that can be used in other daily problems. / 3107510001F5

正交主效應計畫(Orthogonal main-effect plans)因可無相關地估計主效應，故常被應用於一般工業上作為篩選因子之用。然而，實驗通常費時耗財。因此，如何設計一個較經濟且有效的計劃是很重要的。回顧過去相關的研究，Jacroux (1992)提供了最小正交主效應計劃的充份條件及正交主效應計畫之最少實驗次數表(Jacroux 1992)，張純明(1998)針對此表提出修正與補充。在此，我們再次的補足此表。 正交主效應計畫中，如有重複實驗點，則純誤差可被估計，且據此檢定模型之適合度。Jacroux (1993)及張純明(1998)皆曾提出具最多部份重複之正交主效應計畫(Partially replicated orthogonal main-effect plans)。在此，我們討論所有三因子部份重複正交主效應計畫中，可能重複之最大次數，且具體提出建構此最大部份重複之最小正交主效應計畫之方法。 / Orthogonal main-effect plans (OMEP's), being able to estimate the main effects without correlation, are often employed in industrial situations for screening purpose. But experiments are expensive and time consuming. When an economical and efficient design is desired, a minimal orthogonal main-effect plans is a good choice. Jacroux (1992) derived a sufficient condition for OEMP's to have minimal number of runs and provided a table of minimal OMEP run numbers. Chang (1998) corrected and supplemented the table. In this paper, we try to improve the table to its perfection. A minimal OMEP with replicated runs is appreciated even more since then the pure error can be estimated and the goodness-of-fit of the model can be tested. Jacroux (1993) and Chang (1998) gave some partially replicated orthogonal main-effect plans (PROMEP's) with maximal number of replicated points. Here, we discuss minimal PROMEP's with 3 factors in detail. Methods of constructing minimal PROMEP's with replicated runs are provided, and the number of replicated runs are maximal for most cases.

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.