Planar graphs : a historical perspective.

The field of graph theory has been indubitably influenced by the study of planar graphs. This thesis, consisting of five chapters, is a historical account of the origins and development of concepts pertaining to planar graphs and their applications. The first chapter serves as an introduction to the history of graph theory, including early studies of graph theory tools such as paths, circuits, and trees. The second chapter pertains to the relationship between polyhedra and planar graphs, specifically the result of Euler concerning the number of vertices, edges, and faces of a polyhedron. Counterexamples and generalizations of Euler\u27s formula are also discussed. Chapter III describes the background in recreational mathematics of the graphs of K5 and K3,3 and their importance to the first characterization of planar graphs by Kuratowski. Further characterizations of planar graphs by Whitney, Wagner, and MacLane are also addressed. The focus of Chapter IV is the history and eventual proof of the four-color theorem, although it also includes a discussion of generalizations involving coloring maps on surfaces of higher genus. The final chapter gives a number of measurements of a graph\u27s closeness to planarity, including the concepts of crossing number, thickness, splitting number, and coarseness. The chapter conclused with a discussion of two other coloring problems - Heawood\u27s empire problem and Ringel\u27s earth-moon problem

Games, puzzles, and computation

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 147-153).There is a fundamental connection between the notions of game and of computation. At its most basic level, this is implied by any game complexity result, but the connection is deeper than this. One example is the concept of alternating nondeterminism, which is intimately connected with two-player games. In the first half of this thesis, I develop the idea of game as computation to a greater degree than has been done previously. I present a general family of games, called Constraint Logic, which is both mathematically simple and ideally suited for reductions to many actual board games. A deterministic version of Constraint Logic corresponds to a novel kind of logic circuit which is monotone and reversible. At the other end of the spectrum, I show that a multiplayer version of Constraint Logic is undecidable. That there are undecidable games using finite physical resources is philosophically important, and raises issues related to the Church-Turing thesis. In the second half of this thesis, I apply the Constraint Logic formalism to many actual games and puzzles, providing new hardness proofs. These applications include sliding-block puzzles, sliding-coin puzzles, plank puzzles, hinged polygon dissections, Amazons, Kohane, Cross Purposes, Tip over, and others.(cont.) Some of these have been well-known open problems for some time. For other games, including Minesweeper, the Warehouseman's Problem, Sokoban, and Rush Hour, I either strengthen existing results, or provide new, simpler hardness proofs than the original proofs.by Robert Aubrey Hearn.Ph.D

Designing a Knowledge Based Puzzle Game - Case: After Now Archeology

Tämä opinnäytetyö tutkii tietoon perustuvia pulmapelejä. Se määrittelee tietoon perustuvien pul-mapelien pienen genren pulmiksi, joiden ratkaiseminen vaatii pelin ulkopuolisen tiedon käyttöä työkaluna. Tutkimus toteutettiin tarkastelemalla opinnäytetyöprojektina toteutettua peliä After Now Ar-cheology. Pelin suunnitteluprosessi käydään läpi, jotta saadaan hyvä kuva niistä suunnittelupäätök-sistä, joita sisältyy pelin ulkoisen tiedon ääreen ohjaavien pulmien suunnittelemiseen. Peliä tutki-taan käyttäen useita pelien ja pelisuunnittelun teorioita ja työkaluja. Täten pyritään ymmärtämään, miten After Now Archeology sijoittuu pelien kentälle, mutta myös sen olemusta yleisemmin pelien ja pelaamisen kulttuurissa. Tietoon perustuvien pulmapelien määritelmä tuotetaan analysoimalla muita pulmapelien genreen kuuluvia pelejä ja vertaamalla niitä tässä projektissa tuotettuun peliin. Tulee ilmi, että tietoon pe-rustuvat pulmapelit on vaikea sijoittaa yleiseen pulmapelien genreen – varsinkaan digitaalisten pe-lien kontekstissa. Pelin pervasiivinen luonne koettelee pelaamisen ja taikapiirin rajoja. Voidaan myös havaita, että pelaajat jakautuvat hyvin voimakkaasti niihin, joilla ei riitä aikaa eikä kärsivälli-syyttä tiedon etsimiseen ja niihin, jotka saattavat käyttää tunteja yhden pulman parissa etsien in-formaatiota ja omaksuen uutta tietoa.This thesis studies knowledge based puzzle games. It defines the small genre of knowledge based puzzles as puzzles, which require using information not available in the game itself as a tool to solve the puzzle. The study is done by examining a game made for the project: After Now Archeology. The design process of the game is explained, to give an idea of what kind of design decisions has been made when making a game that directs the player to exit the game itself and search for the information outside of the game’s magic circle. The game is then examined using various theories and tools of games and game design, to get an understanding of how the game positions itself within the field of games, but also within the context of gaming culture and playing games. A definition of knowledge based puzzles is made by analyzing the games in the puzzle genre and comparing them to the game made in this project. It is found that knowledge based puzzles are difficult to fit into the genre of puzzles – especially among digital games. The pervasive nature of the game bends the concept of gameplay and magic circle. It is also found that the players divide strongly to those who do not have time and patience to search for information, and them who can spend hours on a single puzzle, searching for information and gaining new knowledge

Using the Internet to support Libyan in-service EFL teachers' professional development

Libyan in-service teachers of English with poor INSET provision and low-resourced school environments stand to gain a great deal from Internet-based Continuing Professional Development (I-CPD). The aim of this exploratory and descriptive study was threefold: first, to understand current practices and conditions pertaining to CPD provision for Libyan teachers; second, to explore the potential of Internet-based CPD for Libyan teachers through a bottom-up informal approach; third, to develop an I-CPD model appropriate for the Libyan context. A mixed-method interventionist case study approach formed the methodological framework of the research. Fact Finding (Phase 1) was carried out to scout the field using a teachers' questionnaire and semi-structured interviews at six language schools in Tripoli. In the Case Study (Phase 2) a typical language institution with in-house Internet access was selected to deliver a progressive intervention course designed to meet the needs of teachers in low-resourced school contexts, but with access to public Internet cafés. Eight case teachers were engaged in problem-based learning to enhance their Internet skills, then using instructional, peer and task support teachers were engaged in blended learning via a web-based Yahoo Group. A ten-week long Extended Case Study (Phase 3) merged Case members from Phase 2 with other teachers from Libya and the UK, forming a larger online group (60 participants) facilitated by a web-hosted Virtual Learning Environment (Merlin). The Fact Finding phase revealed an overall intermediate level in Internet skills and encouraging attitudes towards I-CPD. A more organised petroleum sector emerged, where professional development was assigned higher priority than in the public or private sectors. The Case Study data showed moderate teacher participation in blended learning while task responses reflected minimum engagement with tasks, and little critical reflection. The low response in the Extended Case Study phase prompted attention to the possible causes of low online participation. In addition to generic barriers to asynchronous online learning, such as lurking and the lack of time, underlying context-specific causes have emerged which point to what is termed intellectual-error phobia (ie-phobia) within unbonded groups: while teachers readily participated in low-level tasks, when faced with high-order group-based tasks, they admitted fear of posting trivial responses that were archived and perhaps criticised by other teachers. To minimise ie-phobia and encourage online interaction, a blended multi-dimensional support model is proposed in which f2f orientation and social cohesion precede Internet-based learning that adopts progressive online activities, thus gradually fostering teacher independence and promoting sustainable I-CPD that is holistic and optimised

Using the Internet to support Libyan in-service EFL teachers' professional development

Libyan in-service teachers of English with poor INSET provision and low-resourced school environments stand to gain a great deal from Internet-based Continuing Professional Development (I-CPD). The aim of this exploratory and descriptive study was threefold: first, to understand current practices and conditions pertaining to CPD provision for Libyan teachers; second, to explore the potential of Internet-based CPD for Libyan teachers through a bottom-up informal approach; third, to develop an I-CPD model appropriate for the Libyan context. A mixed-method interventionist case study approach formed the methodological framework of the research. Fact Finding (Phase 1) was carried out to scout the field using a teachers' questionnaire and semi-structured interviews at six language schools in Tripoli. In the Case Study (Phase 2) a typical language institution with in-house Internet access was selected to deliver a progressive intervention course designed to meet the needs of teachers in low-resourced school contexts, but with access to public Internet cafés. Eight case teachers were engaged in problem-based learning to enhance their Internet skills, then using instructional, peer and task support teachers were engaged in blended learning via a web-based Yahoo Group. A ten-week long Extended Case Study (Phase 3) merged Case members from Phase 2 with other teachers from Libya and the UK, forming a larger online group (60 participants) facilitated by a web-hosted Virtual Learning Environment (Merlin). The Fact Finding phase revealed an overall intermediate level in Internet skills and encouraging attitudes towards I-CPD. A more organised petroleum sector emerged, where professional development was assigned higher priority than in the public or private sectors. The Case Study data showed moderate teacher participation in blended learning while task responses reflected minimum engagement with tasks, and little critical reflection. The low response in the Extended Case Study phase prompted attention to the possible causes of low online participation. In addition to generic barriers to asynchronous online learning, such as lurking and the lack of time, underlying context-specific causes have emerged which point to what is termed intellectual-error phobia (ie-phobia) within unbonded groups: while teachers readily participated in low-level tasks, when faced with high-order group-based tasks, they admitted fear of posting trivial responses that were archived and perhaps criticised by other teachers. To minimise ie-phobia and encourage online interaction, a blended multi-dimensional support model is proposed in which f2f orientation and social cohesion precede Internet-based learning that adopts progressive online activities, thus gradually fostering teacher independence and promoting sustainable I-CPD that is holistic and optimised