In Summer 2021 Prof. Mark Wildon supervised two undergraduate research projects. The plan was to give our students a taste of mathematical research by looking at some fun and accessible problems.
Eilis Gilmore worked on the Hunter and Rabbit Problem: a rabbit is hidden at one vertex of a graph. On each turn, the hunter may shoot at one vertex. If she shoots at the vertex where the rabbit is, the hunter wins. Otherwise the rabbit jumps from its vertex to a neighbouring vertex. Even though the hunter never knows for sure where the rabbit is until she makes a successful shot, she can win on many small graphs. For instance, if the graph is a line of length five then simply shooting each vertex working left to right and then right to left ensures the rabbit meets a gruesome end. Eilis looked at the smallest graphs on which the rabbit can avoid the hunter for ever, and suggested some new results on the generalisation of the problem where there are multiple hunters all working as a team. She said: 'I found my summer project a rewarding and enjoyable experience, it was a good opportunity to develop my understanding outside the pressure of exams.'
Dimitar Mogilarov worked on combinatorial enumeration. He found a new approach to the problem of counting the number of coin-flip sequences of a specified length not containing five consecutive heads, and then generalized this to solve a wide class of related problems. This led to an exploration of 'bifix-free' sequences: sequences of heads or tails in which no prefix is equal to a suffix. For instance HHTTHTT is bifix-free, but HHTTHHT is not, because it starts and ends with HHT. He found a new way to count these sequences and began work on an open generalization of the problem.
Prof. Wildon commented 'I am delighted by the excellent work of my two Summer students. It was a real pleasure to meet face-to-face to talk about interesting mathematics with two such keen students.'