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\begin{document}

Name (Last, First): $\underline{\hspace{3in}}$ \hfill Student ID: $\underline{\hspace{2in}}$

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{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1516}{2015-16}:
  \href{\myurl/classes/16-475-ProblemSolving}{MAT 475 Problem Solving Seminar}:
  \hfill\url{http://drorbn.net/16-475}
}

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{\large\bf Quiz 10} on March 31, 2016: ``Argue by Contradiction'' and ``Pursue Parity''. You have 30 minutes to solve the following three problems. Please write on both sides of the page. \hfill {\bf Good Luck!}

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\noindent{\bf Problem 1} (Larson's 1.10.4, reworded). Let $n$ be an odd integer
and let $A$ be a symmetric $n\times n$ "Latin" matrix - every row and every
column in $A$ is a permutation of $\{1,2,\ldots,n\}$. Show that the
diagonal of $A$ is also a permutation of $\{1,2,\ldots,n\}$.

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\picskip{3}\noindent{\bf Problem 2.} Can you pack 125 boxes of size $4\times 2\times 1$
inside one cube of size $10\times 10\times 10$? If you wish to refer to one of the figures on the right, state clearly whether it is figure {\bf A} or {\bf B}.

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\picskip{3}\noindent{\bf Problem 3.} (Larson's 1.10.10, reworded). Show that for every positive integer $a$, the equation $x^2-y^2=a^3$ has solutions with $x,y\in\bbZ$.

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