An analogue of Ryser's Theorem for partial Sudoku squares

In 1956 Ryser gave a necessary and sufficient condition for a partial latin rectangle to be completable to a latin square. In 1990 Hilton and Johnson showed that Ryser's condition could be reformulated in terms of Hall's Condition for partial latin squares. Thus Ryser's Theorem can be interpreted as saying that any partial latin rectangle $R$ can be completed if and only if $R$ satisfies Hall's Condition for partial latin squares. We define Hall's Condition for partial Sudoku squares and show that Hall's Condition for partial Sudoku squares gives a criterion for the completion of partial Sudoku rectangles that is both necessary and sufficient. In the particular case where $n=pq$, $p|r$, $q|s$, the result is especially simple, as we show that any $r \times s$ partial $(p,q)$-Sudoku rectangle can be completed (no further condition being necessary).Comment: 19 pages, 10 figure

The Effects of the Interspersal Procedure on Persistence with Computer-Delivered Multiplication Problems

An across-subjects, post-test only design was used in two experiments to assess the impact of interspersing additional math problems (i.e., briefer problems and/or longer problems) among target math problems on students’ persistence when completing computer-delivered math multiplication problems. In Experiment 1, high school students who worked only target problems completed 32% more target problems and worked 22% longer than those who had briefer problems interspersed. Problem completion rates were significantly higher for those who had briefer problems interspersed. These results suggest that altering assignments by interspersing additional, briefer discrete tasks does not always enhance, and in some instances may hinder academic responding. Stimulus preference and within-trial contrast effects provided possible explanations for these results and indicated that interspersing longer problems could, perhaps, cause students to increase persistence. Experiment 2 was designed to replicate Experiment 1 and extend this line of research by investigating the stimulus preference and within-trial contrast hypothesizes. To increase the number of participants and allow for the evaluation of three conditions, college students served as participants for Experiment 2. In Experiment 2, no significant differences among groups (i.e., control group with only target problems, experimental group with brief problems interspersed, and experimental group with long problems interspersed) were found in the amount of time before college students quit working or in their problem completion accuracy levels. Interspersal of the long problems significantly reduced the number of target problems completed. The results failed to support stimulus preference or within-trial contrast theories. Discussion focuses on theoretical and applied implications related to the additive interspersal procedure, the discrete task completion hypothesis, and the delay reduction hypothesis. Applied implications suggest that educators avoid interspersing longer discrete tasks and exercise caution when interspersing brief tasks

Measuring reasoning ability:applications of rule-based item generation

Die Arbeit beschreibt 3 Anwendungen von Automatischer Aufgabengenerierung im Kontext der Erfassung schlussfolgernden Denkens. Studie 1 beschreibt die Validierung eines neuen figuralen Analogietests. Verschiedene erklärende IRT Modelle werden verglichen, unter anderem Modelle zur Prüfung von facetten-spezifischen Geschlechtseffekten. Studie 2 beschreibt die Entwicklung eines Konstruktionsansatzes zur Generierung von Zahlenreihenaufgaben. Die Ergebnisse zeigen, dass parallele Testformen auf Basis des neuen Regelsets konstruiert werden können, und dass Itemschwierigkeiten zu großen Teilen durch die relationale Komplexität von jeweils zwei aufeinanderfolgenden Zahlen erklärt werden können. Studie 3 beschreibt die Überprüfung der interkulturellen Validität eines Tests vom Typ Lateinische Quadrate (ähnlich zu SUDOKU) für russische und deutsche Studierende. Analysen deuten darauf hin, dass DIF eher durch bestimmte Oberflächenmerkmale als durch strukturelle Parameter zustande kommt. This thesis describes three applications of automatic item generation (AIG) for measuring reasoning ability. Study 1 describes the validation of a new figural analogy test. Several explanatory item response theory models were compared, including models to test item facet specific gender effects. Study 2 describes the development of a new item generation model for number series. Results show that parallel test forms can be constructed based on the new rule set, and that item difficulties can be explained to a large extent based on the relational complexity of a series. Study 3 describes an investigation of the cross-cultural validity of the Latin Square Task that shows similarities with the popular SUDOKU puzzles in samples of Russian and German students. Analyses indicate that differential item functioning is caused by certain item surface characteristics rather than by structural item parameters. Implications for automatic item generation and adaptive testing are discussed.<br

Effective player guidance in logic puzzles

Pen & paper puzzle games are an extremely popular pastime, often enjoyed by demographics normally not considered to be ‘gamers’. They are increasingly used as ‘serious games’ and there has been extensive research into computationally generating and efficiently solving them. However, there have been few academic studies that have focused on the players themselves. Presenting an appropriate level of challenge to a player is essential for both player enjoyment and engagement. Providing appropriate assistance is an essential mechanic for making a game accessible to a variety of players. In this thesis, we investigate how players solve Progressive Pen & Paper Puzzle Games (PPPPs) and how to provide meaningful assistance that allows players to recover from being stuck, while not reducing the challenge to trivial levels. This thesis begins with a qualitative in-person study of Sudoku solving. This study demonstrates that, in contrast to all existing assumptions used to model players, players were unsystematic, idiosyncratic and error-prone. We then designed an entirely new approach to providing assistance in PPPPs, which guides players towards easier deductions rather than, as current systems do, completing the next cell for them. We implemented a novel hint system using our design, with the assessment of the challenge being done using Minimal Unsatisfiable Sets (MUSs). We conducted four studies, using two different PPPPs, that evaluated the efficacy of the novel hint system compared to the current hint approach. The studies demonstrated that our novel hint system was as helpful as the existing system while also improving the player experience and feeling less like cheating. Players also chose to use our novel hint system significantly more often. We have provided a new approach to providing assistance to PPPP players and demonstrated that players prefer it over existing approaches

Towards Prescriptive Analytics in Cyber-Physical Systems

More and more of our physical world today is being monitored and controlled by so-called cyber-physical systems (CPSs). These are compositions of networked autonomous cyber and physical agents such as sensors, actuators, computational elements, and humans in the loop. Today, CPSs are still relatively small-scale and very limited compared to CPSs to be witnessed in the future. Future CPSs are expected to be far more complex, large-scale, wide-spread, and mission-critical, and found in a variety of domains such as transportation, medicine, manufacturing, and energy, where they will bring many advantages such as the increased efficiency, sustainability, reliability, and security. To unleash their full potential, CPSs need to be equipped with, among other features, the support for automated planning and control, where computing agents collaboratively and continuously plan and control their actions in an intelligent and well-coordinated manner to secure and optimize a physical process, e.g., electricity flow in the power grid. In today’s CPSs, the control is typically automated, but the planning is solely performed by humans. Unfortunately, it is intractable and infeasible for humans to plan every action in a future CPS due to the complexity, scale, and volatility of a physical process. Due to these properties, the control and planning has to be continuous and automated in future CPSs. Humans may only analyse and tweak the system’s operation using the set of tools supporting prescriptive analytics that allows them (1) to make predictions, (2) to get the suggestions of the most prominent set of actions (decisions) to be taken, and (3) to analyse the implications as if such actions were taken. This thesis considers the planning and control in the context of a large-scale multi-agent CPS. Based on the smart-grid use-case, it presents a so-called PrescriptiveCPS – which is (the conceptual model of) a multi-agent, multi-role, and multi-level CPS automatically and continuously taking and realizing decisions in near real-time and providing (human) users prescriptive analytics tools to analyse and manage the performance of the underlying physical system (or process). Acknowledging the complexity of CPSs, this thesis provides contributions at the following three levels of scale: (1) the level of a (full) PrescriptiveCPS, (2) the level of a single PrescriptiveCPS agent, and (3) the level of a component of a CPS agent software system. At the CPS level, the contributions include the definition of PrescriptiveCPS, according to which it is the system of interacting physical and cyber (sub-)systems. Here, the cyber system consists of hierarchically organized inter-connected agents, collectively managing instances of so-called flexibility, decision, and prescription models, which are short-lived, focus on the future, and represent a capability, an (user’s) intention, and actions to change the behaviour (state) of a physical system, respectively. At the agent level, the contributions include the three-layer architecture of an agent software system, integrating the number of components specially designed or enhanced to support the functionality of PrescriptiveCPS. At the component level, the most of the thesis contribution is provided. The contributions include the description, design, and experimental evaluation of (1) a unified multi-dimensional schema for storing flexibility and prescription models (and related data), (2) techniques to incrementally aggregate flexibility model instances and disaggregate prescription model instances, (3) a database management system (DBMS) with built-in optimization problem solving capability allowing to formulate optimization problems using SQL-like queries and to solve them “inside a database”, (4) a real-time data management architecture for processing instances of flexibility and prescription models under (soft or hard) timing constraints, and (5) a graphical user interface (GUI) to visually analyse the flexibility and prescription model instances. Additionally, the thesis discusses and exemplifies (but provides no evaluations of) (1) domain-specific and in-DBMS generic forecasting techniques allowing to forecast instances of flexibility models based on historical data, and (2) powerful ways to analyse past, current, and future based on so-called hypothetical what-if scenarios and flexibility and prescription model instances stored in a database. Most of the contributions at this level are based on the smart-grid use-case. In summary, the thesis provides (1) the model of a CPS with planning capabilities, (2) the design and experimental evaluation of prescriptive analytics techniques allowing to effectively forecast, aggregate, disaggregate, visualize, and analyse complex models of the physical world, and (3) the use-case from the energy domain, showing how the introduced concepts are applicable in the real world. We believe that all this contribution makes a significant step towards developing planning-capable CPSs in the future.Mehr und mehr wird heute unsere physische Welt überwacht und durch sogenannte Cyber-Physical-Systems (CPS) geregelt. Dies sind Kombinationen von vernetzten autonomen cyber und physischen Agenten wie Sensoren, Aktoren, Rechenelementen und Menschen. Heute sind CPS noch relativ klein und im Vergleich zu CPS der Zukunft sehr begrenzt. Zukünftige CPS werden voraussichtlich weit komplexer, größer, weit verbreiteter und unternehmenskritischer sein sowie in einer Vielzahl von Bereichen wie Transport, Medizin, Fertigung und Energie – in denen sie viele Vorteile wie erhöhte Effizienz, Nachhaltigkeit, Zuverlässigkeit und Sicherheit bringen – anzutreffen sein. Um ihr volles Potenzial entfalten zu können, müssen CPS unter anderem mit der Unterstützung automatisierter Planungs- und Steuerungsfunktionalität ausgestattet sein, so dass Agents ihre Aktionen gemeinsam und kontinuierlich auf intelligente und gut koordinierte Weise planen und kontrollieren können, um einen physischen Prozess wie den Stromfluss im Stromnetz sicherzustellen und zu optimieren. Zwar sind in den heutigen CPS Steuerung und Kontrolle typischerweise automatisiert, aber die Planung wird weiterhin allein von Menschen durchgeführt. Leider ist diese Aufgabe nur schwer zu bewältigen, und es ist für den Menschen schlicht unmöglich, jede Aktion in einem zukünftigen CPS auf Basis der Komplexität, des Umfangs und der Volatilität eines physikalischen Prozesses zu planen. Aufgrund dieser Eigenschaften müssen Steuerung und Planung in CPS der Zukunft kontinuierlich und automatisiert ablaufen. Der Mensch soll sich dabei ganz auf die Analyse und Einflussnahme auf das System mit Hilfe einer Reihe von Werkzeugen konzentrieren können. Derartige Werkzeuge erlauben (1) Vorhersagen, (2) Vorschläge der wichtigsten auszuführenden Aktionen (Entscheidungen) und (3) die Analyse und potentiellen Auswirkungen der zu fällenden Entscheidungen. Diese Arbeit beschäftigt sich mit der Planung und Kontrolle im Rahmen großer Multi-Agent-CPS. Basierend auf dem Smart-Grid als Anwendungsfall wird ein sogenanntes PrescriptiveCPS vorgestellt, welches einem Multi-Agent-, Multi-Role- und Multi-Level-CPS bzw. dessen konzeptionellem Modell entspricht. Diese PrescriptiveCPS treffen und realisieren automatisch und kontinuierlich Entscheidungen in naher Echtzeit und stellen Benutzern (Menschen) Prescriptive-Analytics-Werkzeuge und Verwaltung der Leistung der zugrundeliegenden physischen Systeme bzw. Prozesse zur Verfügung. In Anbetracht der Komplexität von CPS leistet diese Arbeit Beiträge auf folgenden Ebenen: (1) Gesamtsystem eines PrescriptiveCPS, (2) PrescriptiveCPS-Agenten und (3) Komponenten eines CPS-Agent-Software-Systems. Auf CPS-Ebene umfassen die Beiträge die Definition von PrescriptiveCPS als ein System von wechselwirkenden physischen und cyber (Sub-)Systemen. Das Cyber-System besteht hierbei aus hierarchisch organisierten verbundenen Agenten, die zusammen Instanzen sogenannter Flexibility-, Decision- und Prescription-Models verwalten, welche von kurzer Dauer sind, sich auf die Zukunft konzentrieren und Fähigkeiten, Absichten (des Benutzers) und Aktionen darstellen, die das Verhalten des physischen Systems verändern. Auf Agenten-Ebene umfassen die Beiträge die Drei-Ebenen-Architektur eines Agentensoftwaresystems sowie die Integration von Komponenten, die insbesondere zur besseren Unterstützung der Funktionalität von PrescriptiveCPS entwickelt wurden. Der Schwerpunkt dieser Arbeit bilden die Beiträge auf der Komponenten-Ebene, diese umfassen Beschreibung, Design und experimentelle Evaluation (1) eines einheitlichen multidimensionalen Schemas für die Speicherung von Flexibility- and Prescription-Models (und verwandten Daten), (2) der Techniken zur inkrementellen Aggregation von Instanzen eines Flexibilitätsmodells und Disaggregation von Prescription-Models, (3) eines Datenbankmanagementsystem (DBMS) mit integrierter Optimierungskomponente, die es erlaubt, Optimierungsprobleme mit Hilfe von SQL-ähnlichen Anfragen zu formulieren und sie „in einer Datenbank zu lösen“, (4) einer Echtzeit-Datenmanagementarchitektur zur Verarbeitung von Instanzen der Flexibility- and Prescription-Models unter (weichen oder harten) Zeitvorgaben und (5) einer grafische Benutzeroberfläche (GUI) zur Visualisierung und Analyse von Instanzen der Flexibility- and Prescription-Models. Darüber hinaus diskutiert und veranschaulicht diese Arbeit beispielhaft ohne detaillierte Evaluation (1) anwendungsspezifische und im DBMS integrierte Vorhersageverfahren, die die Vorhersage von Instanzen der Flexibility- and Prescription-Models auf Basis historischer Daten ermöglichen, und (2) leistungsfähige Möglichkeiten zur Analyse von Vergangenheit, Gegenwart und Zukunft auf Basis sogenannter hypothetischer „What-if“-Szenarien und der in der Datenbank hinterlegten Instanzen der Flexibility- and Prescription-Models. Die meisten der Beiträge auf dieser Ebene basieren auf dem Smart-Grid-Anwendungsfall. Zusammenfassend befasst sich diese Arbeit mit (1) dem Modell eines CPS mit Planungsfunktionen, (2) dem Design und der experimentellen Evaluierung von Prescriptive-Analytics-Techniken, die eine effektive Vorhersage, Aggregation, Disaggregation, Visualisierung und Analyse komplexer Modelle der physischen Welt ermöglichen und (3) dem Anwendungsfall der Energiedomäne, der zeigt, wie die vorgestellten Konzepte in der Praxis Anwendung finden. Wir glauben, dass diese Beiträge einen wesentlichen Schritt in der zukünftigen Entwicklung planender CPS darstellen.Mere og mere af vores fysiske verden bliver overvåget og kontrolleret af såkaldte cyber-fysiske systemer (CPSer). Disse er sammensætninger af netværksbaserede autonome IT (cyber) og fysiske (physical) agenter, såsom sensorer, aktuatorer, beregningsenheder, og mennesker. I dag er CPSer stadig forholdsvis små og meget begrænsede i forhold til de CPSer vi kan forvente i fremtiden. Fremtidige CPSer forventes at være langt mere komplekse, storstilede, udbredte, og missionskritiske, og vil kunne findes i en række områder såsom transport, medicin, produktion og energi, hvor de vil give mange fordele, såsom øget effektivitet, bæredygtighed, pålidelighed og sikkerhed. For at frigøre CPSernes fulde potentiale, skal de bl.a. udstyres med støtte til automatiseret planlægning og kontrol, hvor beregningsagenter i samspil og løbende planlægger og styrer deres handlinger på en intelligent og velkoordineret måde for at sikre og optimere en fysisk proces, såsom elforsyningen i elnettet. I nuværende CPSer er styringen typisk automatiseret, mens planlægningen udelukkende er foretaget af mennesker. Det er umuligt for mennesker at planlægge hver handling i et fremtidigt CPS på grund af kompleksiteten, skalaen, og omskifteligheden af en fysisk proces. På grund af disse egenskaber, skal kontrol og planlægning være kontinuerlig og automatiseret i fremtidens CPSer. Mennesker kan kun analysere og justere systemets drift ved hjælp af det sæt af værktøjer, der understøtter præskriptive analyser (prescriptive analytics), der giver dem mulighed for (1) at lave forudsigelser, (2) at få forslagene fra de mest fremtrædende sæt handlinger (beslutninger), der skal tages, og (3) at analysere konsekvenserne, hvis sådanne handlinger blev udført. Denne afhandling omhandler planlægning og kontrol i forbindelse med store multi-agent CPSer. Baseret på en smart-grid use case, præsenterer afhandlingen det såkaldte PrescriptiveCPS hvilket er (den konceptuelle model af) et multi-agent, multi-rolle, og multi-level CPS, der automatisk og kontinuerligt tager beslutninger i nær-realtid og leverer (menneskelige) brugere præskriptiveanalyseværktøjer til at analysere og håndtere det underliggende fysiske system (eller proces). I erkendelse af kompleksiteten af CPSer, giver denne afhandling bidrag til følgende tre niveauer: (1) niveauet for et (fuldt) PrescriptiveCPS, (2) niveauet for en enkelt PrescriptiveCPS agent, og (3) niveauet for en komponent af et CPS agent software system. På CPS-niveau, omfatter bidragene definitionen af PrescriptiveCPS, i henhold til hvilken det er det system med interagerende fysiske- og IT- (under-) systemer. Her består IT-systemet af hierarkisk organiserede forbundne agenter der sammen styrer instanser af såkaldte fleksibilitet (flexibility), beslutning (decision) og præskriptive (prescription) modeller, som henholdsvis er kortvarige, fokuserer på fremtiden, og repræsenterer en kapacitet, en (brugers) intention, og måder til at ændre adfærd (tilstand) af et fysisk system. På agentniveau omfatter bidragene en tre-lags arkitektur af et agent software system, der integrerer antallet af komponenter, der er specielt konstrueret eller udbygges til at understøtte funktionaliteten af PrescriptiveCPS. Komponentniveauet er hvor afhandlingen har sit hovedbidrag. Bidragene omfatter beskrivelse, design og eksperimentel evaluering af (1) et samlet multi- dimensionelt skema til at opbevare fleksibilitet og præskriptive modeller (og data), (2) teknikker til trinvis aggregering af fleksibilitet modelinstanser og disaggregering af præskriptive modelinstanser (3) et database management system (DBMS) med indbygget optimeringsproblemløsning (optimization problem solving) der gør det muligt at formulere optimeringsproblemer ved hjælp af SQL-lignende forespørgsler og at løse dem "inde i en database", (4) en realtids data management arkitektur til at behandle instanser af fleksibilitet og præskriptive modeller under (bløde eller hårde) tidsbegrænsninger, og (5) en grafisk brugergrænseflade (GUI) til visuelt at analysere fleksibilitet og præskriptive modelinstanser. Derudover diskuterer og eksemplificerer afhandlingen (men giver ingen evalueringer af) (1) domæne-specifikke og in-DBMS generiske prognosemetoder der gør det muligt at forudsige instanser af fleksibilitet modeller baseret på historiske data, og (2) kraftfulde måder at analysere tidligere-, nutids- og fremtidsbaserede såkaldte hypotetiske hvad-hvis scenarier og fleksibilitet og præskriptive modelinstanser gemt i en database. De fleste af bidragene på dette niveau er baseret på et smart-grid brugsscenarie. Sammenfattende giver afhandlingen (1) modellen for et CPS med planlægningsmulighed, (2) design og eksperimentel evaluering af præskriptive analyse teknikker der gør det muligt effektivt at forudsige, aggregere, disaggregere, visualisere og analysere komplekse modeller af den fysiske verden, og (3) brugsscenariet fra energiområdet, der viser, hvordan de indførte begreber kan anvendes i den virkelige verden. Vi mener, at dette bidrag udgør et betydeligt skridt i retning af at udvikle CPSer til planlægningsbrug i fremtiden

ACMS 18th Biennial Conference Proceedings

Association of Christians in the Mathematical Sciences 18th Biennial Conference Proceedings, June 1-4, 2011, Westmont College, Santa Barbara, CA

Conventionalizing and Axiomatizing in a Community College Mathematics Bridge Course

This dissertation consists of three related papers. The first paper, Rethinking mathematics bridge courses--An inquiry model for community colleges, introduces the activities of conventionalizing and axiomatizing from a practitioner perspective. In the paper, I offer a curricular model that includes both inquiry and traditional instruction for two-year college students interested in mathematics. In particular, I provide both examples and rationales of tasks from the research-based Teaching Abstract Algebra for Understanding (TAAFU) curriculum, which anchors the inquiry-oriented version of the mathematics bridge course. The second paper, the role of past experience in creating a shared representation system for a mathematical operation: A case of conventionalizing, adds to the existing literature on mathematizing (Freudenthal, 1973) by zooming in on the early stages of the classroom enactment of an inquiry-oriented curriculum for reinventing the concept of group (Larsen, 2013). In three case study episodes, I shed light onto How might conventionalizing unfold in a mathematics classroom? and offer an initial framework that relates students\u27 establishment of conventions in light of their past mathematical experiences. The third paper, Collective axiomatizing as a classroom activity, is a detailed case study (Yin, 2009) that examines how students collectively engage in axiomatizing. In the paper, I offer a revision to De Villiers\u27s (1986) model of descriptive axiomatizing. The results of this study emphasize the additions of pre-axiomatic activity and axiomatic creation to the model and elaborate the processes of axiomatic formulation and analysis within the classroom community

PYCSP3: Modeling Combinatorial Constrained Problems in Python

In this document, we introduce PYCSP$3$, a Python library that allows us to write models of combinatorial constrained problems in a simple and declarative way. Currently, with PyCSP$3$, you can write models of constraint satisfaction and optimization problems. More specifically, you can build CSP (Constraint Satisfaction Problem) and COP (Constraint Optimization Problem) models. Importantly, there is a complete separation between modeling and solving phases: you write a model, you compile it (while providing some data) in order to generate an XCSP3 instance (file), and you solve that problem instance by means of a constraint solver. In this document, you will find all that you need to know about PYCSP$3$, with more than 40 illustrative models

Mega - mobile multimodal extended games

Tese de mestrado em Engenharia Informática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2012As aplicações de entretenimento móvel têm hoje em dia um papel importante e significativo no mercado de software, abrangendo um grupo variado de utilizadores. Tudo isto se deve ao repentino sucesso de dispositivos de interacção inovadora, como o Wiimote da Nintendo, o Move da Sony e o Kinect da Microsoft. Por sua vez estas técnicas de interacção multimodal têm sido exploradas para jogos móveis. A recente geração de dispositivos móveis vem equipada com uma grande variedade de sensores, para além dos óbvios como ecrã táctil e microfone. Existem ainda outros componentes interessantes como bússola digital, acelerómetros, sensores ópticos. Os dispositivos móveis são também utilizados como máquina fotográfica digital, agenda pessoal, assim como para ver videos e ouvir música, e claro, para jogar jogos. Olhar para os novos grupos de utilizadores e para as novas formas de jogar e incluir nos jogos formas de interacção novas, usando os atributos e potencialidades de novas plataformas e novas tecnologias é pois um assunto pungente e deveras desafiante. Com este trabalho pretende-se estudar e propor novas dimensões de jogo e interacção com plataformas móveis, sejam smartphones, sejam tablets, que se adequem às mais distintas comunidades de jogadores. Pretende-se sobretudo explorar modalidades alternativas como as baseadas no tacto e vibração, assim como no áudio, combinadas ou não com outras mais tradicionais de foro visual. Almeja-se ainda explorar jogos em grupo, à distância e co-localizados, encontrando e estudando novas formas de expressão em jogos clássicos e jogos inovadores que envolvam pequenos conjuntos de indivíduos. A ubiquidade inerente aos dispositivos móveis faz ainda com que se tenham que encontrar neste jogos de grupo formas de fluxo de jogo que sustentem saídas e entradas rápidas ou menos rápidas sem que ainda assim se perca o interesse e a motivação de jogar. Este trabalho iniciou-se com uma pesquisa intensiva de trabalho relacionado, sobre a área de jogos móveis e suas multimodalidades, passando consequentemente pela acessibilidade inerente, jogos em grupo e suas formas de comunicação e conexão, e por último dando especial atenção a jogos de puzzle, sendo o tipo de jogo focado neste trabalho. Seguidamente, foi efectuado o levantamento de requisitos e exploradas as opções de jogo e de interacção relativas a jogos de puzzle móveis multimodais. No âmbito deste estudo foram criados três pequenos jogos sobre um conceito comum: jogos de puzzle. A primeira aplicação contém três modalidades diferentes de jogo: uma visual, apresentando um jogo de puzzle de imagens baseado nos tradicionais; uma segunda auditiva, que recria o conceito de jogo através de música, tornando as peças em pequenas parcelas sonoras da música de tamanhos equivalentes; e a terceira háptica, criando deste modo um puzzle com peças de padrões vibratórios diferentes. A segunda aplicação recriou o mesmo conceito de jogo, puzzle, no modo audio, mas retirando toda a informação visual, apresentando simples formas de interacção. A terceira aplicação apresenta uma abordagem sobre os jogos em grupo, permitindo jogar puzzles visuais e de audio em dois modos distintos: cooperativo, onde os jogadores têm de jogar em equipa de forma a conseguir completar o puzzle; e competitiva, onde os jogadores são forçados a ser mais rápidos que o adversário de modo a poderem vencer. Todas estas aplicações permitem ao utilizador definir o tamanho do puzzle e o nível de dificuldade, assim como escolher as imagens e músicas que pretendem resolver em forma de puzzle. Foram conduzidos vários testes de utilizador, nomeadamente um para cada aplicação desenvolvida. Sobre a primeira aplicação vinte e quatro participantes jogaram puzzles visuais e auditivos, distribuídos equitativamente pelas modalidades. Deste modo, cada participante resolveu nove puzzles de imagem ou nove puzzles audio distintos. Neste primeiro estudo procurou descobrir-se as estratégias de resolução dos puzzles, procurando principalmente igualdades e diferenças entre os diferentes modos. Para o segundo estudo foi usada a segunda aplicação desenvolvida, e foram abrangidos novamente vinte e quatro utilizadores, doze dos quais sendo cegos. Cada participante resolveu três puzzles audio diferentes. Relativamente a este estudo, foi proposta uma comparação entre os modos estudados anteriormente, especialmente sobre o modo audio, uma vez que foi usado o mesmo procedimento. Para os utilizadores cegos o objectivo foi provar que seria possível criar um jogo divertido, desafiante e sobretudo acessível a partir de um conceito de jogo clássico. Para o último estudo, vinte e quatro participantes, organizados em pares, jogaram puzzles visuais e de audio em modo cooperativo e competitivo. Cada conjunto de participantes resolveu quatro puzzles, um para cada modo de jogo por cada tipo de puzzle, o que significa dois puzzles visuais, um competitivo e outro cooperativo, e dois puzzles audio, sendo também um cooperativo e outro competitivo. O objectivo mais uma vez foi procurar as estratégias de resolução, permitindo também a comparação com outros modos anteriormente estudados. Todos os jogos foram transformados em dados contendo todas as acções que cada jogador tomou durante a resolução do puzzle. Esses dados foram depois transformados em números específicos de forma a poderem ser analisados e discutidos. Os valores obtidos foram divididos em três grupos principais, as tentativas de colocação de peças, o número de ajudas, e o tempo de conclusão do puzzle. Em relação às tentativas de colocação de peças é possível identificar a ordem correspondente segundo três formas distintas, pela classificação do tipo de peças, pela disposição das peças na fita e pela ordem sequencial do puzzle. Os resultados do estudo mostram que uma mesma estratégia de resolução de puzzles é usada através de todos os modos estudados, os jogadores optam por resolver primeiro as zonas mais relevantes do puzzle, deixando as partes mais abstractas e confundíveis para o final. No entanto, parente novas modalidades de jogo, pequenas percentagens de utilizadores mostraram diferentes estratégias de resolução. Através das opiniões dos utilizadores é também possível afirmar que todas as aplicações desenvolvidas são jogáveis, divertidas e desafiantes. No final foi criado um conjunto de componentes reutilizáveis e um conjunto de parâmetros para a criação de novos jogos. Numa linha de trabalho futuro foram propostos vários objectivos interessantes que podem promover e reaproveitar o trabalho desenvolvido. Deste modo foi criado um jogo de puzzle baseado na primeira aplicação desenvolvida, mantendo os modos visual e audio, de forma a poder integrar no mercado de aplicações móveis, permitindo deste modo, um estudo em larga escala sobre os mesmos conceitos estudados neste trabalho. Foi também pensada a criação de um servidor centralizado, permitindo conter os resultados de todos os jogadores de forma a criar um ranking geral, podendo deste modo incentivar os jogadores a melhorar o seu desempenho, e ajudar a promover o próprio jogo. Outra alternativa passa por melhorar e aperfeiçoar o modo háptico, de forma a criar mais uma modalidade viável sobre o mesmo conceito de jogo, de forma a poder ser também estudada de forma equivalente. O puzzle para invisuais pode também ser melhorado e aperfeiçoado de forma a criar mais desafios através da inclusão dum modo háptio. E por fim, não menos importante, criar novas dimensões de jogo em grupo, permitindo jogar os modos cooperativo e competitivo em simultâneo, tendo por exemplo duas equipas de dois jogadores cada, a cooperar entre si para completar o puzzle, e de certa forma a competir contra a outra equipa para terminar primeiro e com melhores resultados. O objectivo seria, mais uma vez, estudar as estratégias usadas.Mobile entertainment applications have an important and significant role in the software market, covering a diverse group of users. All this is due to the sudden success of innovative interaction devices such as Nintendo’s Wiimote, Sony’s Move and Kinect’s Microsoft. On the other hand, these multimodal interaction techniques have been explored for mobile games. The latest generation of mobile devices is equipped with a wide variety of sensors, in addition to the obvious such as touch screen and microphone. There are other interesting components such as digital compass, accelerometers and optical sensors. Mobile devices are also used as a digital camera, personal organizer, to watch videos and listen to music, and of course, to play games. Looking for the new users groups and for the new ways to play the games and include new forms of interaction, using the attributes and capabilities of new platforms and new technologies is an issue as poignant and very challenging. This work aims to study and propose new dimensions of play and interaction with mobile platforms, whether smartphones or tablets, which suit most distinct communities of players. It is intended primarily to explore alternative modalities such as touch-based and vibratory, as well as audio based, combined or not with traditional visual ones. It also aims at exploring group games, spatially distributed and co-located, finding and studying new forms of expression in classic games and innovative games that involve small sets of individuals. The ubiquity inherent to mobile devices leads us to find input and output flows which support rapid or less rapid entry commands, without losing the interest and motivation to play. In addition to the design and implementation of three or four small game applications intended to create a set of reusable components and a set of guidelines for creating new games