\begin{center}
    \textit{"Pivot! PIVOT! \textbf{PIVOT!}"}
\end{center}

\vspace{10pt}

Suppose that you are trying to move your sofa into your new college dorm. The hallway has a $90^\circ$ turn that you need to pivot the sofa through. What is the biggest sofa that you can get into your dorm?

\begin{figure}[h]
    \centering
    \includegraphics[width=0.7\linewidth]{img/sofa.jpg}
    \caption{A very inefficient example of a sofa that can moved through the corner.}
\end{figure}

We'll assume that the sofa is {\em connected}, meaning that it is not broken into multiple pieces.

\begin{problem}{(Open-ended)}
    Try to brainstorm some possible sofas that you can move through the corner.
    \begin{itemize}
        \item Can you find a sofa with area $1$?
        \item How about a sofa with area $\pi/2$?
    \end{itemize}
\end{problem}

\vfill
\pagebreak

\begin{problem}{Lower bound}
    Construct a rectangular sofa with area $1$ that you can move through the corner. \\
    {\em Hint: this sofa doesn't have to rotate, it can just slide.}
\end{problem}
\begin{solution}
    A square with side lengths $1$ works.
\end{solution}
\vfill 
\begin{problem}{Refined lower bounds}
    What if our couch is a semicircle of radius $r$? How large can $r$ be? What area does this give?
\end{problem}
\begin{solution}
    A semicircle with radius $1$ works. This gives an area of $\pi/2$.
\end{solution}
\vfill
\pagebreak

% \definition{Convex} A set $S$ is converge is for every points $x,y$ inside $S$, the line segment $\overline{xy}$ is also inside $S$. 
% INSERT PHOTO HERE

\begin{problem}{Upper bound}
    Consider a sofa with maximum width $w$ and maximum length $\ell$. To find an upper bound for the area of a sofa, we're going to bound $w$ and $\ell$.
    \begin{enumerate}[label={(\alph*)}]
        \item What is the maximum possible width $w$ of the sofa?
\begin{solution}
        $1$, otherwise it can't make it through the hallway
\end{solution}
        \vfill    
        \item Suppose that our sofa is convex. What is the maximum possible length $\ell$ of the sofa?\\
        {\em Hint: consider the situation where the sofa is rotated $45^\circ$. What happens to the base of the sofa? \\
        This gives a} rough {\em upper bound for the maximum area.}
\begin{solution}
        Considering the situation where the sofa is rotated $45^\circ$, the sofa has to fit within the diagonal line of the corner. This gives a maximum length of $2\sqrt{2}$. This gives a rough upper bound of $2 \sqrt{2}$.
\end{solution}
        \vfill
        
        \item 
            (Challenge) Consider the situation where the sofa is rotated $45^\circ$ more closely. If the sofa is convex, what region of the corner must it lie within? What is an upper bound for the area of a convex sofa?\\
            {\em The figure in the following problem may help.}
            
\begin{solution}
            If it is convex, the sofa has to lie within the isoceles right triangle of the corner. (I can't put a figure in a solution.)
            
            Because the sofa can only have width $1$, this reduces to a trapezoid with bases $2\sqrt{2}$ and $2\sqrt{2} - 2$. This gives an upper bound of $2\sqrt{2} - 1$.

            See \url{https://math.stackexchange.com/questions/1847453/whats-the-upper-bound-for-sofa-problem}
\end{solution}  
\vfill
    \end{enumerate}
\end{problem}
\pagebreak



\begin{problem}{(Challenge)}
    Suppose that we no longer force our sofa to be convex. You know that the width of the sofa is $w = 1$ and the sofa is connected. When the sofa is rotated $45^\circ$ (halfway through the corner), what region(s) might the sofa lie in? Use this to find an upper bound on the area of the sofa.\\
    {\em Hint: Using the fact that the maximum width is $1$, draw horizontal strips on the provided figure and look at the overlap with the corner.}
\end{problem}
\begin{figure}[h]
    \centering
    \includegraphics[width=0.5\linewidth]{img/rotated_corner.jpg}
\end{figure}
\begin{solution}
    See \url{https://math.stackexchange.com/questions/1847453/whats-the-upper-bound-for-sofa-problem}
    This gives an upper bound of $2\sqrt{2}$ for a non-convex sofa.
\end{solution}
\vfill





% \begin{problem}{(Huge challenge)}
%     Suppose you want to construct a sofa shaped like a telephone (a cellphone but curved). 
%     \color{red} Detail construction here. \color{black}
%     Convince yourself that this sofa can rotate through the corner.
% \end{problem}

\begin{problem}{(Open-ended)}
    The largest sofa size is $2.219531...$ (only proved in 2024!) with a shape similar to a telephone. Can you get any closer to this optimal value? How might you proceed?
\end{problem}
\vspace{3cm}
\pagebreak