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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 8} on March 17, 2016: ``Divide into Cases'', ``Work Backwards'', and the Pigeonhole Principle. You have 25 minutes to solve the following two problems. Please write on both sides of the page. \hfill {\bf Good Luck!}

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\noindent{\bf Problem 1 (near Larson's 2.5.11).} Let $T_n$ denote the number of ways of placing $n$ nonattacking rooks on an $n\times n$ chessboard so that the resulting arrangement is symmetric about both diagonals. Find a recursive formula for $T_n$.

\noindent{\bf Problem 2 (near Larson's 2.6.10).} Let $\alpha$ be any real number. Prove that among the numbers $\{\alpha,\,2\alpha,\,\ldots,\,(n-1)\alpha\}$ there is one that differs from an integer by at most $1/n$. {\em Hint.} The bins could be $[\frac{1}{n},\frac{2}{n})$, $[\frac{2}{n},\frac{3}{n})$, \ldots, $[\frac{n-2}{n},\frac{n-1}{n})$.

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