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\lhead{{\bfseries CSC\,236\,H1F}}
\chead{{\bfseries\large Homework Exercise \#\,1}}
\rhead{{\bfseries Fall 2009}}
\lfoot{{Dept.\ of Computer Science,
    University of Toronto (St.~George Campus)}}
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\noindent
\strong{Worth:}  2\%
\hfill
\strong{Due:}  By 6pm on Wednesday 16 September

\vspace{2.5ex}% when not using \begin{enumerate} immediately

\noindent
For each question, please write up detailed answers carefully.
Make sure that you use notation and terminology correctly, and
that you explain and justify what you are doing.
Marks \strong{will} be deducted for
incorrect or ambiguous use of notation and terminology, and for
making incorrect, unjustified, ambiguous, or vague claims
in your solutions.

{}  In particular,
keep in mind that the main point of this exercise is
to get you to practice writing proofs by induction.
This means that you will be marked mostly on
the structure of your proofs and how they were written up---in other words,
having the right idea but writing it up poorly will be worth less than
having the wrong idea but writing it up well\ldots
\begin{enumerate}


\item
    Write a detailed proof that
    for all natural numbers $n$,
    $10^n - 1$ is an integer multiple of $9$.


\item
    Let $P(n)$ be the statement:
      ``\altemph{\strong{any} $n$ squares of various sizes
        can be dissected
        (cut in \strong{a finite number of} straight lines)
        and rearranged to form one large square.}''
    For example, in the picture below,
    each of the three squares on the left can be cut as indicated and
    the pieces can be moved and rotated to form the square on the right.
    \begin{center}
        \includegraphics{squares}
    \end{center}
    Obviously $P(1)$ holds.
    You may use without proof the fact that $P(2)$ holds.

    Write a detailed proof that
    $P(n)$ holds for all natural numbers $n \ge 1$.

    \strong{Bonus:}
    Prove the case $P(2)$.


\end{enumerate}

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