1977 Canadian MO Problems/Problem 7

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Problem

A rectangular city is exactly $m$ blocks long and $n$ blocks wide (see diagram). A woman lives on the southwest corner of the city and works in the northeast corner. She walks to work each day but, on any given trip, she makes sure that her path does not include any intersection twice. Show that the number $f(m,n)$ of different paths she can take to work satisfies $f(m,n)\le 2^{mn}$.

$\underbrace{ \left. \begin{array}{|c|c|c|c|c|c|c|c|c|c|c| }      \hline      &&&&&&&&&& \\ \hline      &&&&&&&&&& \\ \hline      &&&&&&&&&& \\ \hline      &&&&&&&&&& \\ \hline      &&&&&&&&&& \\ \hline      &&&&&&&&&& \\ \hline      &&&&&&&&&& \\ \hline \end{array} \right\}n}_m$

Solution

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