Reversible Peg Solitaire on Graphs

The game of peg solitaire on graphs was introduced by Beeler and Hoilman in 2011. In this game, pegs are initially placed on all but one vertex of a graph G. If xyz forms a path in G and there are pegs on vertices x and y but not z, then a jump places a peg on z and removes the pegs from x and y. A graph is called solvable if, for some configuration of pegs occupying all but one vertex, some sequence of jumps leaves a single peg. We study the game of reversible peg solitaire, where there are again initially pegs on all but one vertex, but now both jumps and unjumps (the reversal of a jump) are allowed. We show that in this game all non-star graphs that contain a vertex of degree at least three are solvable, that cycles and paths on n vertices, where n is divisible by 2 or 3, are solvable, and that all other graphs are not solvable. We also classify the possible starting hole and ending peg positions for solvable graphs

Merging Peg Solitaire in Graphs

Peg solitaire has recently been generalized to graphs. Here, pegs start on all but one of the vertices in a graph. A move takes pegs on adjacent vertices x and y, with y also adjacent to a hole on vertex z, and jumps the peg on x over the peg ony to z, removing the peg on y. The goal of the game is to reduce the number of pegs to one. We introduce the game merging peg solitaire on graphs, where a move takes pegs on vertices x and z (with a hole on y) and merges them to a single peg on y. When can a configuration on a graph, consisting of pegs on all vertices but one, be reduced to a configuration with only a single peg? We give results for a number of graph classes, including stars, paths, cycles, complete bipartite graphs, and some caterpillars

Peg Solitaire on Graphs In Which We Allow Merging and Jumping

Peg solitaire is a game in which pegs are placed in every hole but one and the player jumps over pegs along rows or columns to remove them. Usually, the goal of the player is to leave only one peg. In a 2011 paper, this game is generalized to graphs. In this thesis, we consider a variation of peg solitaire on graphs in which pegs can be removed either by jumping them or merging them together. To motivate this, we survey some of the previous papers in the literature. We then determine the solvability of several classes of graphs including stars and double stars, caterpillars, trees of small diameter, particularly four and five, and articulated caterpillars. We conclude this thesis with several open problems related to this study

Making graphs solvable in peg solitaire

In 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. Since then peg solitaire has been considered on quite a few classes of graphs. Beeler and Gray introduced the natural idea of adding edges to make an unsolvable graph solvable. Recently, the graph invariant ms(G), which is the minimal number of additional edges needed to make G solvable, has been introduced and investigated on banana trees by the authors. In this article, we determine ms(G) for several families of unsolvable graphs. Furthermore, we provide some general results for this number of Hamiltonian graphs and graphs obtained via binary graph operations

Path-Stick Solitaire on Graphs

In 2011, Beeler and Hoilman generalised the game of peg solitaire to arbitrary connected graphs. Since then, peg solitaire and related games have been considered on many graph classes. In this paper, we introduce a variant of the game peg solitaire, called path-stick solitaire, which is played with sticks in edges instead of pegs in vertices. We prove several analogues to peg solitaire results for that game, mainly regarding different graph classes. Furthermore, we characterise, with very few exceptions, path-stick-solvable joins of graphs and provide some possible future research questions

Peg Solitaire on Cartesian Products of Graphs

In 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the "most" unsolvable graphs

Hardness of Games and Graph Sampling

The work presented in this document is divided into two parts. The �rst part presents the hardness of games and the second part presents Graph sampling. Non-deterministic constraint logic[1] is used to prove the hardness of games. The games which are considered in this work is Reversi (2 player bounded game), Peg Solitaire (single player bounded game), Badland (single player bounded game). It also contains a theoretical study of peg solitaire on special graph classes. Reversi is proved to be PSPACE-Complete using Bounded 2CL, Peg Solitaire is proved to be NP-Complete using Bounded NCL. Badland is proved to be NP-Complete by a reduction from 3-SAT. The objective of study of peg solitaire of special graph classes is to �nd the maximum number of marbles we can remove from a fully �lled board, if the player is given the privilege to remove a marble from any cell initially, then following the rules after the initial move. The second part of the work is dedicated to graph sampling. Given a graph G, we try to sample a represen- tative subgraph Gs which is similar to the original graph G. The properties that are being studied are Degree Distribution, Clustering Coefficient, Average Shortest Path Length, Largest Connected Component Size. To measure the similarity between the original graph and sample we use the metrics Kolmogorov - Smirnov test and Kullback - Leibler divergence test. Tightly Induced Edge Sampling performs well on general graphs but it's performance decreases when the graph is a tree. Overall TIBFS and KARGER produces a sample which closely matches the distribution of original graphs.

Razonamiento regresivo en situaciones de resolución de problemas: un modelo multidimensional

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 20-10-20The increasing technological progress has highlighted the importance of problem-solving processes and skills connected to programming methods. Among them, backward reasoning is recognized as a critical issue in advanced mathematics education. This, together with the growing interest in recent years of game-based university education is at the base of this research project. Two objectives are established: on the one hand, to extend the epistemic model of backward reasoning, existing in the mathematical literature, to a cognitive and didactic one; on the other hand, to establish principles for the design of university teaching situations focused on backward reasoning. To reach these objectives, four design experiments using strategy games and mathematical problems are developed. These involved a total of 322 university students, from first year of bachelor to PhD, attending the Universidad Complutense de Madrid (Spain) and the Università di Torino (Italy). They are involved in scientific careers (Mathematics, Mathematics Engineering and Computer Science) and teacher training careers (future mathematics professors in secondary school)...El creciente progreso tecnológico ha puesto de relieve la importancia de los procesos de resolución de problemas y los conocimientos técnicos relacionados con los métodos de programación. Entre ellos, el razonamiento regresivo se reconoce como una cuestión crítica en la enseñanza de las matemáticas avanzada. Esto, junto con el creciente interés en los últimos años de la educación universitaria basada en juegos, es la base de esta investigación. Se establecen dos objetivos: 1) ampliar el modelo epistémico de razonamiento regresivo, existente en la literatura matemática, a uno cognitivo y didáctico, y 2) establecer principios para el diseño de situaciones de enseñanza universitaria centradas en el razonamiento regresivo. Para lograr estos objetivos, se desarrollan cuatro Design experiments utilizando juegos de estrategia y problemas matemáticos. En ellos participaron un total de 322 estudiantes universitarios, desde el primer año de grado hasta el doctorado, procedentes de la Universidad Complutense de Madrid (España) y de la Università di Torino (Italia). Son estudiantes de las ramas científica y de ingeniería (Matemáticas, Ingeniería Matemática e Informática) y en la especialidad de formación de profesores (futuros profesores de matemáticas en la escuela secundaria)...Fac. de Ciencias MatemáticasTRUEunpu

Large scale parallel state space search utilizing graphics processing units and solid state disks

The evolution of science is a double-track process composed of theoretical insights on the one hand and practical inventions on the other one. While in most cases new theoretical insights motivate hardware developers to produce systems following the theory, in some cases the shown hardware solutions force theoretical research to forecast the results to expect. Progress in computer science rely on two aspects, processing information and storing it. Improving one side without touching the other will evidently impose new problems without producing a real alternative solution to the problem. While decreasing the time to solve a challenge may provide a solution to long term problems it will fail in solving problems which require much storage. In contrast, increasing the available amount of space for information storage will definitively allow harder problems to be solved by offering enough time. This work studies two recent developments in the hardware to utilize them in the domain of graph searching. The trend to discontinue information storage on magnetic disks and use electronic media instead and the tendency to parallelize the computation to speed up information processing are analyzed. Storing information on rotating magnetic disk has become the standard way since a couple of years and has reached a point where the storage capacity can be seen as infinite due to the possibility of adding new drives instantly with low costs. However, while the possible storage capacity increases every year, the transferring speed does not. At the beginning of this work, solid state media appeared on the market, slowly suppressing hard disks in speed demanding applications. Today, when finishing this work solid state drives are replacing magnetic disks in mobile computing, and computing centers use them as caching media to increase information retrieving speed. The reason is the huge advantage in random access where the speed does not drop so significantly as with magnetic drives. While storing and retrieving huge amounts of information is one side of the medal, the other one is the processing speed. Here the trend from increasing the clock frequency of single processors stagnated in 2006 and the manufacturers started to combine multiple cores in one processor. While a CPU is a general purpose processor the manufacturers of graphics processing units (GPUs) encounter the challenge to perform the same computation for a large number of image points. Here, a parallelization offers huge advantages, so modern graphics cards have evolved to highly parallel computing instances with several hundreds of cores. The challenge is to utilize these processors in other domains than graphics processing. One of the vastly used tasks in computer science is search. Not only disciplines with an obvious search but also in software testing searching a graph is the crucial aspect. Strategies which enable to examine larger graphs, be it by reducing the number of considered nodes or by increasing the searching speed, have to be developed to battle the rising challenges. This work enhances searching in multiple scientific domains like explicit state Model Checking, Action Planning, Game Solving and Probabilistic Model Checking proposing strategies to find solutions for the search problems. Providing an universal search strategy which can be used in all environments to utilize solid state media and graphics processing units is not possible due to the heterogeneous aspects of the domains. Thus, this work presents a tool kit of strategies tied together in an universal three stage strategy. In the first stage the edges leaving a node are determined, in the second stage the algorithm follows the edges to generate nodes. The duplicate detection in stage three compares all newly generated nodes to existing once and avoids multiple expansions. For each stage at least two strategies are proposed and decision hints are given to simplify the selection of the proper strategy. After describing the strategies the kit is evaluated in four domains explaining the choice for the strategy, evaluating its outcome and giving future clues on the topic