\section{Introduction}

\example{}<lockproblem>
A certain electronic lock has two buttons: \texttt{0} and \texttt{1}.
It opens as soon as the correct two-digit code is entered, completely ignoring
previous inputs.\hspace{-0.5ex}\footnotemark{} For example, if the correct code is \text{10}, the lock will open
once the sequence \texttt{010} is entered.

\vspace{2mm}

Naturally, there are $2^2 = 4$ possible combinations that open this lock. \par
If don't know the lock's combination, we could try to guess it by trying all four combinations. \par
This would require eight key presses: \texttt{0001101100}.

\problem{}
There is, of course, a better way. \par
Unlock this lock with only 5 keypresses.

\begin{solution}
	The sequence \texttt{00110} is guaranteed to unlock this lock.
\end{solution}


\problem{}
Consider the same lock, now set with a three-digit binary code.
\begin{itemize}
	\item How many codes are possible?
	\item What is the shortest sequence that is guaranteed to unlock the lock? \par
	\hint{You'll need 10 digits.}
\end{itemize}

\begin{solution}
	\begin{itemize}
		\item $2^3 = 8$
		\item \texttt{0001110100} will do.
	\end{itemize}
\end{solution}


\problem{}
How about a four-digit code? How many digits do we need? \par

\begin{instructornote}
	Don't spend too long on this problem.
	Provide a solution at the board once everyone has had a few
	minutes to think about this.
\end{instructornote}

\begin{solution}
	Interestingly enough, we can only save one digit. \par
	Any optimal sequence has 15 digits, for example \texttt{0000111101100101000}
\end{solution}

\vfill
\pagebreak