        \problemdata       % Takes 5 *mandatory* arguments
        {tut34}             % latex-friendly label for the prob.
        {induction}  % The topic(s) of the problem content.
        {}               % Source (if known)
        {F01 PS2-4} % Usage (list ones you are aware of).
        {}        % Last revision info (author, date).

\begin{problem}
The 6.042 course information sheet states:

\begin{quotation}\noindent
Tutorials will be devoted to solving interesting problems, with students
working in teams of three or four.
\end{quotation}

Using strong induction, prove that if a recitation contains at least 6
students, then the class can be divided into teams, each consisting of
either 3 or 4 students.

\solution{\begin{proof}
The proof is by strong induction.  Let $P(n)$ be the proposition that
a recitation with $n$ students can be divided into teams of 3 or 4.
In the base cases, $P(6)$ is true because there could be two teams of
3, $P(7)$ is true because there could be a team of 3 and a team of 4,
and $P(8)$ is true because there could be two teams of 4.  In the
inductive step, for $n \geq 8$, assume $P(6), \ldots, P(n)$ to prove
$P(n+1)$.  If a recitation has $n+1$ students, then three students can
form a team of 3, and the remaining $n-2$ students can be divided into
teams of 3 or 4 by induction.  This shows that $P(6), \ldots, P(n)$
imply $P(n+1)$, and the claim is proved by induction.
\end{proof}}

\end{problem}