\documentstyle{article}

\begin{document}

\parindent0pt

\begin{enumerate}

\item[A]
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.

\item[B]
A computable function from the naturals to the naturals is a finite
piece of code (in your favourite programming language). The code takes
as input a natural number and outputs a natural number. Prove that
there are many uncomputable functions.

\item[C]
Prove that in the last 1000 years you have had an ancestor A and an
ancestor B such that A is decended from B on both sides, i.e. both of
A's parents are descendants of B.

\end{enumerate}

\end{document}
