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Suppose I have a collection of switches numbered $0$ through
$n$, all initially off.  Someone comes and flips every switch (so they
are all on).  Another person then flips every second switch, starting
with $0,2,\ldots$.  The next person flips every third switch, and so
on until the last person flips only the flips numbered $0$ and $n$.
Which switches are on at the end?  Prove it.

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