Les méthodes de Monte Carlo sont des méthodes probabilistes qui utilisent des ordinateurs pour résoudre de nombreux problèmes de la science à l’aide de nombres aléatoires. Leur principal inconvénient est leur convergence lente. La mise au point de techniques permettant d’accélérer la convergence est un domaine de recherche très actif. C’est l’objectif principal des méthodes déterministes quasi-Monte Carlo qui remplacent les points pseudo-aléatoires de simulation par des points quasi-aléatoires ayant une excellente répartition uniforme. Ces méthodes ne fournissent pas d’intervalles de confiance permettant d’estimer l’erreur. Nous étudions dans ce travail des méthodes stochastiques qui permettent de réduire la variance des estimateurs Monte Carlo : ces techniques de stratification le font en divisant le domaine d’échantillonnageen sous-domaines. Nous examinons l’intérêt de ces méthodes pour l’approximation des chaînes de Markov, la simulation de la diffusion physique et la résolution numérique de la fragmentation.Dans un premier chapitre, nous présentons les méthodes de Monte Carlo pour l’intégration numérique. Nous donnons le cadre général des méthodes de stratification. Nous insistons sur deux techniques : la stratification simple (MCS) et la stratification Sudoku (SS), qui place les points sur des grilles analogues à celle du jeu. Nous pressentons également les méthodesquasi-Monte Carlo qui partagent avec les méthodes de stratification certaines propriétés d'équipartition des points d’échantillonnage.Le second chapitre décrit l’utilisation des méthodes de Monte Carlo stratifiées pour la simulation des chaînes de Markov. Nous considérons des chaînes homogènes uni-dimensionnelles à espace d’états discret ou continu. Dans le premier cas, nous démontrons une réduction de variance par rapport `a la méthode de Monte Carlo classique ; la variance des schémas MCSou SS est d’ordre 3/2, alors que celle du schéma MC est de 1. Les résultats d’expériences numériques, pour des espaces d’états discrets ou continus, uni- ou multi-dimensionnels montrent une réduction de variance liée à la stratification, dont nous estimons l’ordre.Dans le troisième chapitre, nous examinons l’intérêt de la méthode de stratification Sudoku pour la simulation de la diffusion physique. Nous employons une technique de marche aléatoire et nous examinons successivement la résolution d’une équation de la chaleur, d’une équation de convection-diffusion, de problèmes de réaction-diffusion (équations de Kolmogorov et équation de Nagumo) ; enfin nous résolvons numériquement l’équation de Burgers. Dans chacun de ces cas, des tests numériques mettent en évidence une réduction de la variance due à l’emploi de la méthode de stratification Sudoku.Le quatrième chapitre décrit un schéma de Monte Carlo stratifie permettant de simuler un phénomène de fragmentation. La comparaison des performances dans plusieurs cas permet de constater que la technique de stratification Sudoku réduit la variance d’une estimation Monte Carlo. Nous testons enfin un algorithme de résolution d’un problème inverse, permettant d’approcher le noyau de fragmentation, à partir de résultats de l’évolution d’une distribution ;nous utilisons dans ce cas des points quasi-Monte Carlo pour résoudre le problème direct. / Monte Carlo methods are probabilistic schemes that use computers for solving various scientific problems with random numbers. The main disadvantage to this approach is the slow convergence. Many scientists are working hard to find techniques that may accelerate Monte Carlo simulations. This is the aim of some deterministic methods called quasi-Monte Carlo, where random points are replaced with special sets of points with enhanced uniform distribution. These methods do not provide confidence intervals that permit to estimate the errordone. In the present work, we are interested with random methods that reduce the variance of a Monte Carlo estimator : the stratification techniques consist of splitting the sampling area into strata where random samples are chosen. We focus here on applications of stratified methods for approximating Markov chains, simulating diffusion in materials, or solving fragmentationequations.In the first chapter, we present Monte Carlo methods in the framework of numerical quadrature, and we introduce the stratification strategies. We focus on two techniques : the simple stratification (MCS) and the Sudoku stratification (SS), where the points repartitions are similar to Sudoku grids. We also present quasi-Monte Carlo methods, where quasi-random pointsshare common features with stratified points.The second chapter describes the use of stratified algorithms for the simulation of Markov chains. We consider time-homogeneous Markov chains with one-dimensional discrete or continuous state space. We establish theoretical bounds for the variance of some estimator, in the case of a discrete state space, that indicate a variance reduction with respect to usual MonteCarlo. The variance of MCS and SS methods is of order 3/2, instead of 1 for usual MC. The results of numerical experiments, for one-dimensional or multi-dimensional, discrete or continuous state spaces show improved variances ; the order is estimated using linear regression.In the third chapter, we investigate the interest of stratified Monte Carlo methods for simulating diffusion in various non-stationary physical processes. This is done by discretizing time and performing a random walk at every time-step. We propose algorithms for pure diffusion, for convection-diffusion, and reaction-diffusion (Kolmogorov equation or Nagumo equation) ; we finally solve Burgers equation. In each case, the results of numerical tests show an improvement of the variance due to the use of stratified Sudoku sampling.The fourth chapter describes a stratified Monte Carlo scheme for simulating fragmentation phenomena. Through several numerical comparisons, we can see that the stratified Sudoku sampling reduces the variance of Monte Carlo estimates. We finally test a method for solving an inverse problem : knowing the evolution of the mass distribution, it aims to find a fragmentation kernel. In this case quasi-random points are used for solving the direct problem.

This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played.

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

This diploma thesis deals with basic features of magic squares and analyses these features with regard to usability during the teaching at elementary schools. Magic squares are known for hundreds years and since then they have changed due to various modifications, from which other kinds were derived. The first part of the thesis is therefore dedicated to the history. Next chapter deals with the construction of magic squares. The following chapters study similar games as Sudoku, Kakuro and Latin squares. The final part of the thesis is dedicated to the usability of magic squares in teaching mathematics. To practice the given topic, the worksheets which are divided according to their difficulty, were created.

. A Escola Fundamental I tem como objetivo promover em seus alunos o desenvolvimento da capacidade de aprender e de se relacionar através da representação, da comunicação e da resolução de problemas em favor da aprendizagem da língua, da matemática, da representação espacial, temporal e gráfica. No campo da Psicologia e da Educação pesquisas demonstram que o jogo pode ser usado para criar situações-problema e assim contribuir para processos de desenvolvimento e aprendizagem em crianças e adolescentes. Esta pesquisa trata, então, de propor uma forma de intervenção que auxilie a aprendizagem da resolução de problemas, através de um jogo que implica o uso de operações lógicas. O Sudoku foi o jogo escolhido porque os problemas que ele apresenta são da natureza e uso da lógica, no sentido de que para descobrir o número de cada casa vazia o jogador deve coordenar vários aspectos ao mesmo tempo e considerar que há apenas uma resposta para cada caso. O recorte de idade feito para a presente pesquisa, crianças com 6-7 anos, se apoia na teoria piagetiana que afirma que as construções operatórias dão início, em média, aos sete anos e seguem até a vida adulta. Assim, criou-se, através de oficinas com o jogo Sudoku, situações-problema que exercitaram o uso do pensamento lógico favorecendo os processos de desenvolvimento e de aprendizagem requeridos para o alcance das expectativas previstas para sujeitos que cursam o segundo ano da Escola Fundamental I. Foram feitas 17 oficinas, sendo duas por semana e com duração de uma hora cada, em três turmas de segundo ano de uma escola particular do município de Ribeirão Preto, alcançando um total 60 participantes. A fundamentação metodológica utilizada nas oficinas se baseia nos modelos Taos, Paris, Atenas e Eldorado desenvolvidos por Gruber e Vonèche (1995) que têm como objetivo aplicar à Educação as contribuições de Piaget ao explorar o pensamento e a compreensão da criança em relação à natureza dos objetos, ao tempo, espaço, juízo moral e etc. Os dados foram analisados a partir dos seguintes eixos: 1) como os participantes resolveram e compreenderam o jogo Sudoku; 2) qual o valor das situações-problema para o primeiro eixo; 3) como e por que aplicar os Modelos Taos, Paris, Atenas e Eldorado em oficinas sobre o jogo Sudoku; e 4) como praticar uma visão construtivista, apoiada na perspectiva de Piaget, em uma situação de jogo com sujeitos de 6-7 anos. Os resultados apontam que no início do processo de intervenção poucos sujeitos conseguiam resolver o Sudoku da forma correta. No entanto, no desenrolar das oficinas foi possível identificar uma considerável aprendizagem deste jogo por parte dos participantes e, mais do que isto, observou-se um apreciável desenvolvimento dos sujeitos quanto à compreensão do jogo. Atribui-se tal desenvolvimento e aprendizagem às atividades das oficinas em resolver os jogos e às situações-problema. Conclui-se, ainda, que os modelos de Gruber e Vonèche formaram uma excelente base de intervenção para o aprendizado do jogo e de suas implicações lógicas. Assim, confiamos que tal estudo aqui apresentado possa colaborar de alguma maneira com educadores e psicólogos interessados no uso do jogo para o desenvolvimento e a aprendizagem de seus alunos e pacientes a respeito do pensamento lógico / Primary school has as a target the promotion of the development of its students capacity to learn and relate through representation, communication and problem solving in favor of language learning, mathematics and spatial, temporal and graphical representation. In the fields of Psychology and Education researches show that gaming can be used to create a problem situation contributing to the process of development and learning in children and teenagers. This research proposes a form of intervention that helps the learning of problem solving, through a game that implies the use of logical operations. Sudoku was the game/puzzle chosen because the problems that it presents are of logical nature and use, in a sense that for discovering the empty squares right number the player must coordinate several aspects at the same time while taking into consideration that there is only one right answer for each case. Children from 6 to 7 years old were chosen because of the Piagetian theory that states that operatory constructions begin at that age. Therefore, through Sudokus workshops problem situations were created to exercise the use of logical thought favoring the process of development and learning that is required for the achievement of expectations regarding the individuals going through the second year of primary school. Seventeen workshops of one hour each, were made twice a week, with three groups attending the second year of a private primary school in Ribeirão Preto, totaling in 60 participants. The methodological grounding was based on the models Taos, Paris, Athens and Eldorado developed by Gruber and Vonèche (1995) these models had an which was objective to apply Piagets contribution to Education as he explored a childs thought and comprehension of the nature of objects, time, moral judgement, etc. The data was analyzed through the following axes: 1) How participants solved and comprehended the Sudoku puzzle; 2) What is the value of the problem situations for the first axis; 3) How and why to apply the models Taos, Paris, Athens and Eldorado in the Sudoku workshops; and 4) How to practice a constructivist viewpoint, based on Piagets perspective, in game situations with children from 6 to 7 years old. The results demonstrate that at the beginning of the intervention few children were able to correctly solve the Sudoku puzzle. However, through the workshops a considerable amount of learning was identified, and more than that, a development of the game comprehension. The learning and development achieved is attributed to the workshops activities of puzzle solving and problem situation. We conclude that Gruber and Vonèches models formed an excellent intervention base for the learning of the puzzle and its logical implications. Therefore, we trust that the present study may collaborate in some ways with educators and psychologists interested in the use of games for their students and patients development and learning of logical thought

. A Escola Fundamental I tem como objetivo promover em seus alunos o desenvolvimento da capacidade de aprender e de se relacionar através da representação, da comunicação e da resolução de problemas em favor da aprendizagem da língua, da matemática, da representação espacial, temporal e gráfica. No campo da Psicologia e da Educação pesquisas demonstram que o jogo pode ser usado para criar situações-problema e assim contribuir para processos de desenvolvimento e aprendizagem em crianças e adolescentes. Esta pesquisa trata, então, de propor uma forma de intervenção que auxilie a aprendizagem da resolução de problemas, através de um jogo que implica o uso de operações lógicas. O Sudoku foi o jogo escolhido porque os problemas que ele apresenta são da natureza e uso da lógica, no sentido de que para descobrir o número de cada casa vazia o jogador deve coordenar vários aspectos ao mesmo tempo e considerar que há apenas uma resposta para cada caso. O recorte de idade feito para a presente pesquisa, crianças com 6-7 anos, se apoia na teoria piagetiana que afirma que as construções operatórias dão início, em média, aos sete anos e seguem até a vida adulta. Assim, criou-se, através de oficinas com o jogo Sudoku, situações-problema que exercitaram o uso do pensamento lógico favorecendo os processos de desenvolvimento e de aprendizagem requeridos para o alcance das expectativas previstas para sujeitos que cursam o segundo ano da Escola Fundamental I. Foram feitas 17 oficinas, sendo duas por semana e com duração de uma hora cada, em três turmas de segundo ano de uma escola particular do município de Ribeirão Preto, alcançando um total 60 participantes. A fundamentação metodológica utilizada nas oficinas se baseia nos modelos Taos, Paris, Atenas e Eldorado desenvolvidos por Gruber e Vonèche (1995) que têm como objetivo aplicar à Educação as contribuições de Piaget ao explorar o pensamento e a compreensão da criança em relação à natureza dos objetos, ao tempo, espaço, juízo moral e etc. Os dados foram analisados a partir dos seguintes eixos: 1) como os participantes resolveram e compreenderam o jogo Sudoku; 2) qual o valor das situações-problema para o primeiro eixo; 3) como e por que aplicar os Modelos Taos, Paris, Atenas e Eldorado em oficinas sobre o jogo Sudoku; e 4) como praticar uma visão construtivista, apoiada na perspectiva de Piaget, em uma situação de jogo com sujeitos de 6-7 anos. Os resultados apontam que no início do processo de intervenção poucos sujeitos conseguiam resolver o Sudoku da forma correta. No entanto, no desenrolar das oficinas foi possível identificar uma considerável aprendizagem deste jogo por parte dos participantes e, mais do que isto, observou-se um apreciável desenvolvimento dos sujeitos quanto à compreensão do jogo. Atribui-se tal desenvolvimento e aprendizagem às atividades das oficinas em resolver os jogos e às situações-problema. Conclui-se, ainda, que os modelos de Gruber e Vonèche formaram uma excelente base de intervenção para o aprendizado do jogo e de suas implicações lógicas. Assim, confiamos que tal estudo aqui apresentado possa colaborar de alguma maneira com educadores e psicólogos interessados no uso do jogo para o desenvolvimento e a aprendizagem de seus alunos e pacientes a respeito do pensamento lógico / Primary school has as a target the promotion of the development of its students capacity to learn and relate through representation, communication and problem solving in favor of language learning, mathematics and spatial, temporal and graphical representation. In the fields of Psychology and Education researches show that gaming can be used to create a problem situation contributing to the process of development and learning in children and teenagers. This research proposes a form of intervention that helps the learning of problem solving, through a game that implies the use of logical operations. Sudoku was the game/puzzle chosen because the problems that it presents are of logical nature and use, in a sense that for discovering the empty squares right number the player must coordinate several aspects at the same time while taking into consideration that there is only one right answer for each case. Children from 6 to 7 years old were chosen because of the Piagetian theory that states that operatory constructions begin at that age. Therefore, through Sudokus workshops problem situations were created to exercise the use of logical thought favoring the process of development and learning that is required for the achievement of expectations regarding the individuals going through the second year of primary school. Seventeen workshops of one hour each, were made twice a week, with three groups attending the second year of a private primary school in Ribeirão Preto, totaling in 60 participants. The methodological grounding was based on the models Taos, Paris, Athens and Eldorado developed by Gruber and Vonèche (1995) these models had an which was objective to apply Piagets contribution to Education as he explored a childs thought and comprehension of the nature of objects, time, moral judgement, etc. The data was analyzed through the following axes: 1) How participants solved and comprehended the Sudoku puzzle; 2) What is the value of the problem situations for the first axis; 3) How and why to apply the models Taos, Paris, Athens and Eldorado in the Sudoku workshops; and 4) How to practice a constructivist viewpoint, based on Piagets perspective, in game situations with children from 6 to 7 years old. The results demonstrate that at the beginning of the intervention few children were able to correctly solve the Sudoku puzzle. However, through the workshops a considerable amount of learning was identified, and more than that, a development of the game comprehension. The learning and development achieved is attributed to the workshops activities of puzzle solving and problem situation. We conclude that Gruber and Vonèches models formed an excellent intervention base for the learning of the puzzle and its logical implications. Therefore, we trust that the present study may collaborate in some ways with educators and psychologists interested in the use of games for their students and patients development and learning of logical thought

The main objective of this dissertation is to present an algebraic method capable of determining a solution, if any, of a non linear polynomial equation systems using Gröbner basis. In order to accomplish that, we first present some concepts and theorems linked to polynomial rings with several undetermined and monomial ideals where we highlight the division extended algorithm, the Hilbert Basis and the Buchberger´s algorithm. Beyond that, using basics of Elimination and Extension Theorems, we present an algebraic solution to the map coloring that use 3 colors as well as a general solution to the Sudoku puzzle. / O objetivo principal desse trabalho é, usando bases de Gröbner, apresentar um método algébrico capaz de determinar a solução, quando existir, de sistemas de equações polinomiais não necessariamente lineares. Para tanto, necessitamos inicialmente apresentar alguns conceitos e teoremas ligados a anéis de polinômios com várias indeterminadas e de ideais monomiais, dentre os quais destacamos o algoritmo extendido da divisão, o teorema da Base de Hilbert e o algoritmo de Buchberger. Além disso, usando noções básicas da Teoria de eliminação e extensão, apresentamos uma solução algébrica para o problema da coloração de mapas usando três cores, bem como um solução geral para o puzzle Sudoku.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work brings an approach to the basic notions of Graph Theory, presenting historical context, concepts, de nitions and examples in order to provide the reader with previous knowledge of the theory. The main objective is to perform a study of the graphs applied to the staining using the dual graph and the method of the greedy algorithm. To do so, we will present the demonstration attempt of the 4-Color Theorem that had been developed by Kempe and the proof of the 5-Color Theorem, made 11 years later by Heawood. Finally, we will seek the resolution of some situations problems that will be modeled through the coloring of vertices. / Este trabalho traz uma abordagem às noções da Teoria dos Grafos, apresentando contexto histórico, conceitos, definições e exemplos com o intuito de proporcionar ao leitor conhecimentos prévios da teoria. O principal objetivo é efetuar um estudo dos grafos aplicados à coloração mediante o uso do grafo dual e o métdo do algoritmo guloso. Para tanto, executaremos um esboço acerca da tentativa de demonstração do Teorema das 4 Cores que fora desenvolvida por Kempe e da prova do Teorema das 5 Cores, feita 11 anos mais tarde por Heawood. Por fim, buscaremos a resolução de algumas situações problemas que serão modeladas através da coloração de vértices. / São Cristóvão, SE

Signal acquisition, signal reconstruction and analysis of spectrum of the signal are the three most important steps in signal processing and they are found in almost all of the modern day hardware. In most of the signal processing hardware, the signal of interest is sampled at uniform intervals satisfying some conditions like Nyquist rate. However, in some cases the privilege of having uniformly sampled data is lost due to some constraints on the hardware resources. In this thesis an important problem of signal reconstruction and spectral analysis from nonuniformly sampled data is addressed and a variety of methods are presented. The proposed methods are tested via numerical experiments on both artificial and real-life data sets. The thesis starts with a brief review of methods available in the literature for signal reconstruction and spectral analysis from non uniformly sampled data. The methods discussed in the thesis are classified into two broad categories - dense and sparse methods, the classification is based on the kind of spectra for which they are applicable. Under dense spectral methods the main contribution of the thesis is a non-parametric approach named LIMES, which recovers the smooth spectrum from non uniformly sampled data. Apart from recovering the spectrum, LIMES also gives an estimate of the covariance matrix. Under sparse methods the two main contributions are methods named SPICE and LIKES - both of them are user parameter free sparse estimation methods applicable for line spectral estimation. The other important contributions are extensions of SPICE and LIKES to multivariate time series and array processing models, and a solution to the grid selection problem in sparse estimation of spectral-line parameters. The third and final part of the thesis contains applications of the methods discussed in the thesis to the problem of radial velocity data analysis for exoplanet detection. Apart from the exoplanet application, an application based on Sudoku, which is related to sparse parameter estimation, is also discussed.

Στη διπλωματική λύνουμε το πρόβλημα του sudoku με χρήση του εκπαιδευτικού ρομπότ της Lego, το LEGO Mindstorm NXT. Το εκπαιδευτικό ρομπότ αυτό δεν έχει συγκεκριμένη μορφή αλλά αποτελείται από αλληλοσυνδεόμενα μεταξύ τους πλαστικά μέρη. Με χρήση αυτών κατασκευάσαμε ένα όχημα που αποτελεί παραλλαγή οχήματος από άλλη εργασία. Το όχημα αυτό μπορεί να κινείται μόνο μπροστά και πίσω. Διαθέτει έναν βραχίονα που μπορεί να κινεί δεξιά-αριστερά και στον οποίο εφαρμόζεται ένας αισθητήρας φωτεινότητας. Τέλος, στον βραχίονα υπάρχει θέση για στυλό. Το πρόβλημα του sudoku που δίνεται στο ρομπότ είναι εκτυπωμένο σε ένα χαρτί Α4. Το ρομπότ αναλαμβάνει να το αναγνωρίσει με τον αισθητήρα, να το επιλύσει και να το αποτυπώσει με τη χρήση του στυλό. Για την επίτευξη αυτού του στόχου επιστρατεύονται αλγόριθμοι ρομποτικής και αλγόριθμοι τεχνητής νοημοσύνης. Συγκεκριμένα για την πλοήγηση του οχήματος εφαρμόζεται μετρική και τοπολογική πλοήγησης, στη συνέχεια για την αναγνώριση του προβλήματος και την ταυτοποίηση κάθε εικόνας που λαμβάνεται υλοποιήσαμε αλγόριθμους μορφολογικής επεξεργασία και τέλος για την επίλυση του προβλήματος sudoku υλοποιήσαμε και συγκρίναμε δύο αλγόριθμους, την αναζήτησης κατά βάθος και την αναζήτηση κατά βάθος με διάδοση περιορισμών. Οι τελικοί αλγόριθμοι που αναπτύχθηκαν διαπιστώσαμε ότι πετυχαίνουν το σκοπό τους αφού το όχημα αναγνωρίζει τους αριθμούς του δοσμένου προβλήματος με ποσοστό επιτυχίας 95%, λύνει τα περισσότερα προβλήματα σε λιγότερο από ένα δευτερόλεπτο και συμπληρώνει επιτυχώς τα κελιά του sudoku με τους σωστούς αριθμούς. Πέρα από αυτές τη σύγκριση των αλγορίθμων θεωρούμε ότι η μελέτη ενός τέτοιου συστήματος είναι ιδανική για εισαγωγή σε θέματα ρομποτικής και μπορεί να χρησιμοποιηθεί ως εκπαιδευτικό εργαλείο πειραματισμού. Μάλιστα ο κώδικας μας σχολιάζεται επαρκώς σε αυτή την εργασία για να είναι ευκολότερη η κατανόηση του. Εκτός αυτού έχουμε αναπτύξει και πρόγραμμα αλληλεπίδρασης χρήστη-ρομπότ μέσω κονσόλας. / We solve the problem of sudoku using the educational robot LEGO Mindstorm NXT, made by LEGO. This educational robot doesn't have specific form but consists of interlinked plastics. We constructed a vehicle that is a variant from another work. This vehicle can move only forward and back. It has an arm that can move side to side and is equipped with a light sensor and a marker. The problem of sudoku is given to the robot in printed form on a A4 paper. The robot at first recognize the problem with the sensor, then it resolves it and finally writes the solution down by using the pen. To achieve this goal we implemented various algorithms. Specifically, we studied robotic algorithms such as metric and topological navigation. Moreover, to identify the printed problem we processed every captured image morphologically and finally to solve the sudoku instance we implemented and compared two methods, first-depth search and first-depth search with constraint propagation. We should mention that our code is written in Java for the lejOS firmware. The final code is capable of recognizing the numbers of the given problem with a success rate of 95%, solving most problems in less than a second and completing the cells on the paper with the correct numbers. Finally, we have developed an accompanying program that is usable for debugging purposes and for calibrating the robot. Even more, it can be used as education tool.