Define a graph G by taking the vertices as $\mathbb{R}^2$ and the edges to be any pair of vertices that are distance 1 apart. The Hadwiger-Nelson Problem asks the chromatic number of this graph, written $\chi(\mathbb{R}^2)$. It is known that either $4\leq \chi(\mathbb{R}^2)\leq 7$ or $5\leq \chi(\mathbb{R}^2)\leq 7$. We explore some approaches to solving this problem, encountering along the way the Axiom of Choice (or lack thereof) and other infinite oddities.