\documentclass[12pt]{article}
\usepackage[Mickael]{ammaths}

\begin{document}

\module{Problems about polyominoes}{2}{14}{4 periods}

\prereq{}

\object{\begin{itemize}
        \item Work on a few classic problems about polyominoes.
	\end{itemize}}

\mater{\begin{itemize}
	\item Hand-out with the solutions.
	\item Cut out tetrominoes and pentominoes.
	\item Different problems. 
	\item Beamer.
	\end{itemize}}

\modpart{Solve some problems}{3 hours}

Students work in teams of 4 or 5.

Some problems about polyominoes are available at the teacher's desk. Each team picks any problem and works on it, with the following rules.
\begin{itemize}
\item Each team can pick any problem, randomly or choosing it on purpose.
\item After a 10 minutes search a team can decide to stop working on this particular problem and pick a new one.
\item Answers must not be passed along from one team to another.
\item Each team must find the answer to at least three problems during the available time.
\item Grades are given according to the following scheme :
	\begin{itemize}
	\item three problems solved : B (15) ;
	\item five problems solved : A (18) ;
	\item eight problems solved : A$^\star$ (20).
	\end{itemize}
\end{itemize}

\modpart{Presentation}{1 hour}

Each team must present to the class the result of one problem.�The quality of the presentation may be awarded a grade.

\pagebreak

\moddocdis{Problems about polyominoes}{2}{14}{Solutions}

\section{Rep-tiles}

\begin{quote}
A polyomino is a \emph{rep-tile} if a larger version of itself can
be tiled with only copies of the initial polyomino. Find four non-trivial polyominoes rep-tiles.
\end{quote}

The easiest way is to find polyominoes that can be used to make a square. Then we just have to reproduce this square
as in the initial polymino.

\begin{multicols}{2}

\subsection*{The L-triomino}

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\subsection*{The L-tetromino}

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\end{multicols}

\subsection*{The T-tetromino}

\begin{center}
\includegraphics[scale=0.8]{images/Ttetrominorep.eps}
\end{center}

\subsection*{The P-pentomino}

\begin{center}
\includegraphics[scale=0.35]{images/Ppentominorep.eps}
\end{center}


\section{Tiling a rectangle with pentominoes}

\begin{quote}
Find a rectangle tileable with the 12 pentominoes used exactly once.
\end{quote}

One possibilty is the $6\times10$ rectangle pictured below. Other possibilities are $5\times 12$, $4\times 15$ or
the $3\times 20$ that is the answer to another problem.
\begin{center}
\includegraphics[scale=0.9]{images/6x10rectangle.eps}
\end{center}

\pagebreak

\section{Tiling a rectangle with the tetrominoes}

\begin{quote}
Prove that it's impossible to tile a rectangle with the 5 tetrominoes used exactly once.
\end{quote}

Any tileable rectangle shouold have an area of $5\times 4=20$. Color the squares of any rectangle of this area black and
white, as on a chessboard. Obviously, there is the same number of black squares and white squares. Now color the squares
of the 5 tetrominoes in the same way :

\begin{center}
\includegraphics[scale=1]{images/tetrominoescolor.eps}
\end{center}

Because of the T-tetromino, the number of black squares and the number of white squares will always be different. So
it's impossible to tile a rectangle with the tetrominoes.

\section{Tiling a special rectangle with pentominoes}

\begin{quote}
Tile a $3\times20$ rectanglewith the 12 pentominoes used exactly once.%p.8
\end{quote}

There are only two solutions, one of them is pictured below. For the other solution, leave the four polyominoes on the
left and the polyomino on the right and rotate the whole middle part.

\begin{center}
\includegraphics[scale=0.7]{images/3x20rectangle.eps}
\end{center}


\section{Three congruent groups of pentominoes}

\begin{quote}
Divide the twelve pentominoes into three groups of four each. Find one 20-square region that \emph{each} of
the three groups will cover.%p.13
\end{quote}

One solution is pictured below. There may be other solutions.
\begin{center}
\includegraphics[scale=0.6]{images/3congruentgroups.eps}
\end{center}

\section{Minimal region for the twelve pentominoes}

\begin{quote}
Find a minimal region made of squares on which each of the 1� pentominoes can fit. %p.14
\end{quote}

Here are two possible answers. Each of the 12 pentominoes fits in any of these regions.
\begin{center}
\includegraphics[scale=1]{images/minimalregions.eps}
\end{center}


\section{A tromino or a tetromino on a chessboard}

\begin{quote}
We've seen that you can tile a chessboard with dominoes. Can you do it with the I-tromino, or with the
L-tromino ?
\end{quote}

A tromino is made of three squares. So any region tileable with a tromino must have an area that is a mutiple of 3.
It's not the case for a chessboard, as the area is $8\times 8=64=3\times 21+1$. So a chessboard is not tileable with
trominoes (I or L). A further question is : what about a chessboard with one square removed ?

\medskip

\begin{quote}
Which tetrominoes can be used to tile a complete chessboard ? %p.6
\end{quote}

The I, L, O and T tetrominoes tile a $4\times 4$ square, to they also tile a chessboard.
\begin{center}
\includegraphics[scale=0.8]{images/tetrotilechess.eps}
\end{center}

\pagebreak

\section{Triplication of pentominoes}

\begin{quote}
Pick a pentomino, then use nine of the other pentominoes to construct a scale model, three times as wide
and three times as high as the given piece.%p.8
\end{quote}

Below are show the triplications of the V pentomino and the X pentomino. Other triplications are possible.
\begin{center}
\includegraphics[scale=0.7]{images/triplication.eps}
\end{center}



\section{Pentominoes on an amputated chessboard}

\begin{quote}
Cover a chessboard with the four corners missing with the 12 pentominoes used exactly once.%p.7
\end{quote}

Here is one solution. Other positions of the four missing squares are possible, including the next one, where the
four missing squares make the square tetromino.
\begin{center}
\includegraphics[scale=0.8]{images/pentoonchess.eps}
\end{center}

\pagebreak

\section{Pentominoes and a tetromino on a chessboard}

\begin{quote}
Cover a chessboard with the twelve pentominoes and the square tetromino.%p.7
\end{quote}

A simple trick is to combine the square tetromino with the V pentomino to make a $3\times 3$ square. Then all we
have to do is tile the remaining portion of the chessboard with the 11 other pentominoes. There are many ways to do so,
below is one example.
\begin{center}
\includegraphics[scale=0.8]{images/pento+tetroonchess.eps}
\end{center}


\section{Three 3$\times$7 rectangles}

\begin{quote}
Divide the twelve pentominoes into three groups of four each. To each group add a monomino and form a
$3\times 7$ rectangle.%p.14
\end{quote}

Here is the only solution. The proof that no other solution is possible starts from the only possibility for the U
and X pentominoes.

\begin{center}
\includegraphics[scale=0.8]{images/3x7rectangles.eps}
\end{center}

\pagebreak

\section{Tetrominoes and a pentomino on a square board}

\begin{quote}
Cover a  5$\times$5 board with the five tetrominoes and one pentomino.%p.17
\end{quote}

Below are shown two possibilities. There may be more.
\begin{center}
\includegraphics[scale=1]{images/5x5square.eps}
\end{center}





\pagebreak

\moddoc{Cut-out tetrominos and pentominoes}

\begin{center}
\includegraphics[scale=1]{images/tetrocutout.eps}

{\sl The 5 free tetrominoes.}

\bigskip

\includegraphics[scale=1]{images/pentocutout.eps}

{\sl The 12 free pentominoes.}


\end{center}

\pagebreak

\moddoc{Problems}

{\Large

\hrulefill
\bigskip

\begin{center}\bf Rep-tiles
\end{center}


A polyomino is a \emph{rep-tile} if a larger version of itself can
be tiled with only copies of the initial polyomino.

Find four non-trivial polyominoes rep-tiles.


\bigskip
\hrulefill
\bigskip

\begin{center}\bf Tiling a rectangle with pentominoes
\end{center}


Find a rectangle tileable with the 12 pentominoes used exactly once.

\bigskip
\hrulefill
\bigskip

\begin{center}\bf Tiling a rectangle with the 5 tetrominoes
\end{center}


Prove that it's impossible to tile a rectangle with the 5 tetrominoes used exactly once.

\bigskip
\hrulefill
\bigskip

\begin{center}\bf Tiling a special rectangle with pentominoes
\end{center}


Tile a $3\times20$ rectanglewith the 12 pentominoes used exactly once.%p.8

\bigskip
\hrulefill

\pagebreak

\hrulefill
\bigskip

\begin{center}\bf Three congruent groups of pentominoes
\end{center}


Divide the twelve pentominoes into three groups of four each. Find one 20-square region that \emph{each} of the three groups will cover.%p.13


\bigskip
\hrulefill
\bigskip

\begin{center}\bf Minimal region for the twelve pentominoes
\end{center}


Find a minimal region made of squares on which each of the 12 pentominoes can fit. %p.14


\bigskip
\hrulefill
\bigskip

\begin{center}\bf A tromino or a tetromino on a checkerboard
\end{center}


We've seen that you can tile a chessboard with dominoes. 

Can you do it with the I-tromino, or with the L-tromino ?

Which tetrominoes can be used to tile a complete chessboard ? %p.6


\bigskip
\hrulefill
\bigskip

\begin{center}\bf Triplication of pentominoes
\end{center}


Pick a pentomino, then use nine of the other pentominoes to construct a scale model, three times as wide and three times as high as the given piece.%p.8


\bigskip
\hrulefill

\pagebreak

\hrulefill
\bigskip

\begin{center}\bf Pentominoes on an amputated chessboard
\end{center}


Cover a chessboard with the four corners missing with the 12 pentominoes used exactly once.%p.7

\bigskip
\hrulefill
\bigskip

\begin{center}\bf Pentominoes and a tetromino on a chessboard
\end{center}


Cover a chessboard with the twelve pentominoes and the square tetromino.%p.7


\bigskip
\hrulefill
\bigskip

\begin{center}\bf Three 3$\times$7 rectangles
\end{center}


Divide the twelve pentominoes into three groups of four each. To each group add a monomino and form a $3\times 7$ rectangle.%p.14


\bigskip
\hrulefill
\bigskip

\begin{center}\bf Tetrominoes and a pentomino on a square board
\end{center}


Cover a  5$\times$5 board with the five tetrominoes and one pentomino.%p.17


\bigskip
\hrulefill





}


\end{document}
