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\begin{document}
\title{Math 101 Homework}
\author{Mary Radcliffe}
\date{due 22 Feb 2015}
\maketitle

Answer the following questions. Fully justify each response.

\begin{enumerate}


\item Prove that Soifer's irregular octagon and square tiling discussed in class has the property that no octagon color realizes distance $\sqrt{5}$, and the square color does not realize distance 1.

\item Consider the following definition. Define $\chi_t$ to be the smallest $k$ such that there is a coloring of $\R^2$ using $k$ colors, having the property that every color set realizes every distance. Is this a meaningful constant? Find its value.

\item Consider the following definition. Define $\chi_f$ to be the maximal $k$ such that there is a coloring of $\R^2$ using $k$ colors, having the property that every color set realizes every distance. Is this a meaningful constant? Find its value.



\end{enumerate}









\end{document}