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 \begin{center} \textbf{Math 475 Homework 10 (Bonus)} \\
 \textsc{Due May 14th, 2010} \end{center}

\begin{enumerate}

	\item \textbf{Mystery Clock}\begin{footnote}{Peter Gallin, ``What Time Is It?'', via Charles S. Thompson and William S. Bush, ``Improving Middle School Teachers' Reasoning about Proportional Reasoning'', \emph{Mathematics Teaching In The Middle School}, April 2003, Volume 8, Issue 8, Page 398-403. }\end{footnote}.  This clock has fallen off the wall and I don't know which way is 12 o'clock. Figure out the exact time it is, and explain why your answer is the only possible one.
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	\item  Prove $\log_2 5$ is irrational. (\emph{Hint}: what if it were equal to $p/q$? )
	\item Find the triangular numbers in Pascal's Triangle. Explain why the pattern continues forever.
	\item Identify the skew-diagonals of Pascal's Triangle which sum to the Fibonacci numbers. Explain why this pattern continues forever.


\item \textbf{Chameleon Cube}\begin{footnote}{From Dick Stanley.}\end{footnote}. Twenty-seven small cubes (identical in size) can be used to form a large cube. Is it possible to paint each face of each small cube with one of the three colors (red, green, and yellow) in such a way that (for the same paint scheme):
\begin{itemize}

	\item in one way of assembling the large cube, all six faces are red,

	\item in another way of assembling the large cube, all six faces are green,

	\item in still another way of assembling the large cube, all six faces are yellow?

\end{itemize}
You get part credit for proving it's possible. You get the rest of the credit for giving me an explicit paint scheme and rearrangement scheme. 


\end{enumerate}



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