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

\begin{document}

\module{Sprouts}{2}{18}{2 periods}

%\prereq{}

\object{\begin{itemize}
        \item Introduce the vocabulary of Graph theory.
	\item Show how a game can be solved by exhaustion.
	\end{itemize}}

\mater{\begin{itemize}
	\item Rules.
	\item {\bf A large room, with not too many tables and chairs, would be better suited for this activity.}
	\end{itemize}}



\sectionblack{Period 1}

\modpart{The rules of the game}{10 mins}

The rules of the game are explained, with a few games played on the board by the teacher and some students. Vocabulary is introduced.

\begin{center}
Graph -- Vertex -- Edge -- Order of a graph -- Degree of a vertex
\end{center}


\modpart{Playtime}{10 mins}

Students play some game of sprouts by pairs.

\modpart{Solving the 1-vertex game}{5 mins}

The 1-vertex game is completely solved and it is shown that it's a second-player win.

\modpart{Solving the 2-vertices game}{30 mins}

Students have to collectively solve the 2-vertices game and find out if there is a winning strategy. Teacher's input should be minimal during this part.

\begin{itemize}
\item The class is divided in four groups.
\item Each group is given some large paper sheets (half clip-board sheets) and some pens.
\item Each group has to build a large tree showing all the possible moves, with one move on each paper. Equivalent moves must not be differentiated.
\item Each group has to devise a winning strategy for the first or second player.
\end{itemize}

\pagebreak

\sectionblack{Period 2}

\modpart{Solving the 2-vertices game (continued)}{10 mins}

Students are given some time to take a look at the tree they did during the previous period and review the winning strategy.

\modpart{Championship}{35 mins}

Each group plays a 2-vertices game against every other group twice (switching first and second players in each game). 

\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
 & G1 & G2 & G3 & G4 \\
\hline
G1 & & & & \\
\hline
G2 & & & & \\
\hline
G3 & & & & \\
\hline
G4 & & & & \\
\hline
\end{tabular}
\end{center}

A mark is then awarded to the group depending on the number of wins :
\begin{description}
\item[1 win :] 12 points ;
\item[2 wins :] 15 points ;
\item[3 wins :] 18 points ;
\item[4 wins :] 19 points ;
\item[5 or 6 wins :] 20 points.  
\end{description}

% \modpart{Solving the $n$-vertices game}{Remaining time}
% 
% Some results about the $n$-vertices game, for $n>3$, are shown by the teacher.

\pagebreak

\moddocdis{Sprouts}{2}{18}{Rules}

\section{Origin and rules}

The game called ``Sprouts'' was created  in 1967 by mathematicians
Michael S. Paterson and John H. Conway (the creator of the game of
Life).

This is a two-players game, played with just pencil and paper. Start
with a number of spots, or vertices. The two players take turns
drawing a line (not forcedly straight) between two vertices or from
a vertex to itself, and placing a new point on the line. A turn is
also called a \emph{move}. A few rules apply :
\begin{itemize}
\itemb A line must not cross any other line.
\itemb At most three lines can start from any vertex. We say that each vertex has exactly three ``lives''.
When exactly three lines start from one vertex, it is considered
\emph{dead} and cannot be used in the game any longer.
\end{itemize}
The last player to be able to play wins.

% \section{Graph theory}
% 
% This game has strong connections with Graph Theory, a branch of
% mathematics whose importance has grown quickly in recent years.
% 
% The vocabulary used to study the game of Sprouts comes from Graph theory. A
% \emph{graph} is a set of points, or \emph{vertices}, connected by
% lines, or \emph{edges}. The \emph{degree} of a vertex is the number
% of edges originated from it. The \emph{size} of a graph is simply
% its number of vertices. The \emph{order} of a graph is its number of
% edges.
% 
% \section{Winning strategies}
% 
% 
% 
% \section{Variants}
% 
% The \emph{Brussels Sprouts} is played like Sprouts, except that vertices are drawn as crosses and at most \emph{four}
%can start from any vertex, one from each arm of the cross.
% 
% This game is easier to solve : ??????????????????
\end{document}
