In this thesis, we will give a way to build mechanical metamaterials. We will do this by using a triangular tiling, in which we put spins on the edges of the tiles. These spins have to point either...Show moreIn this thesis, we will give a way to build mechanical metamaterials. We will do this by using a triangular tiling, in which we put spins on the edges of the tiles. These spins have to point either into or out of the triangles and have to satisfy the rule that for every triangle two spins have to point out, and one in, or two spins point in and one out. If we can construct a tiling that is completely filled with these triangles in such a way that all spins on sides of adjacent triangles are pointing in the same direction, we will call this a feasible configuration. Firstly, we derive the number of feasible configurations for the tiling and consider a way to estimate these values. Secondly, we derive the distribution for the number of configurations when there are i spins on the boundary pointing in. Finally, we consider the number of spins in a periodic tiling that can be reversed, independent of all other spins, and derive an upper value and a lower boundary for this.Show less
In this thesis, we will give a way to build mechanical metamaterials. We will do this by using a triangular tiling, in which we put spins on the edges of the tiles. These spins have to point either...Show moreIn this thesis, we will give a way to build mechanical metamaterials. We will do this by using a triangular tiling, in which we put spins on the edges of the tiles. These spins have to point either into or out of the triangles and have to satisfy the rule that for every triangle two spins have to point out, and one in, or two spins point in and one out. If we can construct a tiling that is completely filled with these triangles in such a way that all spins on sides of adjacent triangles are pointing in the same direction, we will call this a feasible configuration. Firstly, we derive the number of feasible configurations for the tiling and consider a way to estimate these values. Secondly, we derive the distribution for the number of configurations when there are i spins on the boundary pointing in. Finally, we consider the number of spins in a periodic tiling that can be reversed, independent of all other spins, and derive an upper value and a lower boundary for this.Show less
Voting is incorporated into our daily lives in more ways that we usually realise. From electing national leaders and parliaments to selecting a group leader of a collaboration project and from...Show moreVoting is incorporated into our daily lives in more ways that we usually realise. From electing national leaders and parliaments to selecting a group leader of a collaboration project and from electing the winner of an Olympic sport event to choosing the best date for a group meeting, we use voting as a solution of many problems, some of which are daily, some of which only turn up every few years. But are the common methods used for voting these days the best and most honest methods? A lot of people probably have heard of the problem with one of the most important elections in the world, the election of the American president. The best-known problem occurred during the 2000 election, where Al Gore got a majority of the votes, but due to the American voting system, George W. Bush jr. won the election. This is just one of the problems the traditional voting systems have. This poses the question: What is honest voting, and what is an honest voting system? In this thesis we will look at the traditional voting systems and their (dis)advantages. Next, we will look at the voting system created by Michel Balinski and Rida Laraki, called the Majority Judgment. Once again, we will look into the system itself, look at its (dis)advantages and compare it to the traditional systems. We will conclude with a small research that was conducted regarding the Dutch political voting system. For writing this thesis, the book “Majority Judgment - Measuring, Ranking and Electing”[1] has been employed. A large part of this thesis consists of transforming this ‘semi-mathematical’ subject into more welldefined and precise mathematical definitions and theorems. If one does not have a background in mathematics, or just wants to have a brief overview of the various voting systems and their uses, an article about this subject can be found in the scientific magazine ’Eureka!’. The specific edition and further information can be found in the bibliography[4].Show less
Hanabi is a co-operative card game for two to five players, in which every player can see the contents of the other players’ hands, but not of their own. By the exchange of hints, a player can...Show moreHanabi is a co-operative card game for two to five players, in which every player can see the contents of the other players’ hands, but not of their own. By the exchange of hints, a player can obtain information about the cards in his or her hand. The thesis consists of two main parts. In the first part, we study the notion of playability. Not every initial configuration of the game can result in a maximum score even if playing perfectly. By employing combinatorics, we derive a formula with which the amount of the initial configurations which can be finished perfectly can be calculated for a simplification of the original game. We also propose an approach using dynamic programming to perform these calculations for slightly more complicated versions of the game. In the second part, we test a variety of strategies in search of good strategies for the original game. We discover that some simple rules give promising results, but that not all strategies which seem good intuitively indeed result in high scores.Show less
Fritzen is een dobbelspel dat vaak als drankspel wordt gespeeld. Er moeten dobbelstenen opzij worden gelegd om een gunstig aantal ogen te behalen. Het totaal aantal ogen dat is afgelegd, bepaalt de...Show moreFritzen is een dobbelspel dat vaak als drankspel wordt gespeeld. Er moeten dobbelstenen opzij worden gelegd om een gunstig aantal ogen te behalen. Het totaal aantal ogen dat is afgelegd, bepaalt de hoogte van de boete die uitgedeeld of verkregen wordt. In dit redelijk eenvoudige spel zijn veel verschillende dobbelsteencombinaties mogelijk. Voor elk van deze combinaties zijn er weer veel verschillende dobbelsteencombinaties die opzijgelegd kunnen worden. Hierdoor is het vaak niet meteen duidelijk welke van deze keuzes het beste resultaat geeft. In deze scriptie zal eerst voor een vereenvoudigde versie van het Fritzen een optimale speelwijze worden bepaald. Dit wordt gedaan aan de hand van een stochastisch dynamisch programmeringsprobleem dat specifiek voor dit spel wordt gedefinieerd. Daarna breiden we dit uit naar het volledige spel, waarvoor ook een optimale speelwijze wordt bepaald. Door een niet eenduidig vastgelegd doel is er niet ´e´en beste speelwijze. Deze speelwijze hangt af van het doel dat wordt gekozen. Onderdeel van deze scriptie is een programma waarmee, per doel, de optimale speelwijze bepaald kan worden.Show less
Within the world of operational research, project scheduling plays a large and important part. Being able to plan a project in such a way that is deemed optimal, by minimizing a given objective, is...Show moreWithin the world of operational research, project scheduling plays a large and important part. Being able to plan a project in such a way that is deemed optimal, by minimizing a given objective, is a challenging mathematical problem. Depending on the constraints placed on the project, there might not even exist any straightforward algorithm to obtain an optimum. For time constrained problems, polynomial time algorithms exists to calculate the most cost effective solution for any provided set of jobs. However, if jobs are additionally required to compete for resources, such a general solution does not exist. However, this thesis studies a method in which the Lagrangian relaxation of a resource constrained project can be efficiently solved by transforming it into an equivalent time constrained problem. This time constrained problem is subsequently solved by computing the minimum cut in a derived directed graphShow less
In this paper we introduce a new kind of game, called a deck building game, of which Dominion is the most prominent example. We focus on the question to what extent traditional game analysis...Show moreIn this paper we introduce a new kind of game, called a deck building game, of which Dominion is the most prominent example. We focus on the question to what extent traditional game analysis techniques can be used to analyze deck building games? To do this, we look at several simple strategies, like Random and Greedy, and some traditional techniques, namely Monte Carlo Tree Search and Dynamic Programming. We compare the strategies for mid (31 turns) to long games (100 turns). We conclude that our implementation of DP seems to be suitable only for games of medium length or shorter because of its space complexity, whereas our implementation of MCTS seems to fall behind other strategies with similar performance in regards of time complexity.Show less
Menigeen heeft wel eens gehoord van het `lights out'-probleem. Over dit probleem is een hoop op internet te vinden, met name met verscheidene roosters, zogenaamde `grids'. Het idee hiervan is dat...Show moreMenigeen heeft wel eens gehoord van het `lights out'-probleem. Over dit probleem is een hoop op internet te vinden, met name met verscheidene roosters, zogenaamde `grids'. Het idee hiervan is dat je een heel rooster, bestaande uit lampjes die ofwel aan ofwel uit staan, vanuit een semi-willekeurige configuratie volledig uit kunt krijgen door goed gekozen lampjes van toestand te laten veranderen, waarbij de volgende eigenschap geldt: Als lampje X van toestand verandert, oftewel van uit naar aan gaat, of van aan naar uit, dan zullen alle aangrenzende lampjes van X tevens van toestand veranderen.Show less
