Blue Red Hackenbush Spiders

Fall 2021

Thesis

Degree Name

Master of Arts (MA)

Department

Mathematics and Statistics

Tim Hsu

Mathematics

Abstract

One of the goals of Combinatorial Game Theory is to find provable winning strategiesfor certain games. In this paper, we give winning strategies for certain spider positions played using the rules of Blue Red Hackenbush and a variant. Blue Red Hackenbush and its variants are played on a graph of a bLue and Red edges that are connected to a vertex called the ground. We will represent the ground as a horizontal black line. In this paper, we study spider graphs played under two different variants: Blue Red Hackenbush and Reverse Blue Red Hackenbush. Both variants are played by two players: Left and Right. On Left’s turn, they must choose a blue edge to delete. Any edges no longer connected to the ground are also deleted. Right’s turn is similar, except they must choose a red edge to delete. The first player unable to move loses. Every Blue Red Hackenbush position can be identified as a dyadic rational. This value completely determines who wins the game playing first and who wins the game playing second. It’s been shown that determining this value is NP-hard for certain kinds of BR Hackenbush games, making the study of even certain classes of positions interesting. Reverse BR Hackenbush is played exactly the same as the usual BR Hackenbush, except after a player deletes an edge (and any subsequently unconnected to the ground edges) they reverse the color of any edges that were in the component they played in. In this paper, we begin by introducing the essentials necessary to analyze spiders. In section 2, we analyze spiders in BR Hackenbush. In particular, we give a solution for a certain class of spiders called balanced spiders. In section 3, we turn our attention to Reverse BR Hackenbush. In particular, we give a solution for two legged spiders. Even though Reverse BR Hackenbush seems like it has similar rules to BR Hackenbush, the results of section 2 and section 3 appear remarkably different.

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