Project Ideas

I am happy to supervise projects in any area of combinatorics or algebra. This document gives some possible subjects. Most could be adapted to be suitable as either a 3rd year, 4th year or MSc project. Combinatorial game theory. In the game of Domineering the two players are called Horizontal and Vertical. In each turn, Vertical places a domino vertically on the board, and then Horizontal places a domino horizontally on the board. The first player who is unable to move loses. For example, the diagram below shows a position with Horizontal to play. I am happy to supervise projects in any area of combinatorics or algebra. This document gives some possible subjects. Most could be adapted to be suitable as either a 3rd year, 4th year or MSc project. Combinatorial game theory. In the game of Domineering the two players are called Horizontal and Vertical. In each turn, Vertical places a domino vertically on the board, and then Horizontal places a domino horizontally on the board. The first player who is unable to move loses. For example, the diagram below shows a position with Horizontal to play. 1 2 3 4 5 6 7 The aim of a project in this area would be to explain the remarkable connection between numbers and games that makes it possible to assign numerical values to Domineering positions, and so to decide optimal moves in even quite complicated positions. For example, of the boards , , , the first three have values 1/2, −1/2 and 2 respectively. The third has a value ï¿¿ that lies outside the real number system. The theory can also be applied to many other games, including Nim, Dots-and-Boxes and go; any of these games could be used as an example in a project. For an undergraduate level introduction see [1]. 1 The aim of a project in this area would be to explain the remarkable connection between numbers and games that makes it possible to assign numerical values to Domineering positions, and so to decide optimal moves in even quite complicated positions. For example, of the boards , , , the first three have values 1/2, −1/2 and 2 respectively. The fourth has a value that lies outside the real number system. The theory can also be applied to many other games, including Nim, Dots-and-Boxes