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\newcommand{\titre}{Le car de ramassage}
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    \small Un car de ramassage doit s'arrêter à chaque carrefour d'un quartier, une fois et une seule. Selon le plan c'est possible ou non — et un simple coloriage permet souvent de le dire d'avance, sans essayer.
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\end{center}
\noindent \textbf{Le car part de l'école, au carrefour K, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑨ rue de l'Église & Ⓐ Place aux Herbes & Ⓙ Rond-point des Vaches \\ \hline
① impasse Verte & ⑩ sentier des Prés & Ⓑ Coin du Lavoir & Ⓚ Rond-point du Stade \\ \hline
② rue Verte & ⑪ allée des Tilleuls & Ⓒ Place Sainte-Croix & Ⓛ Carrefour du Moulin \\ \hline
③ avenue de la Gare & ⑫ rue Basse & Ⓓ Carrefour de la Mairie & Ⓜ Place de la Fontaine \\ \hline
④ rue des Lilas & ⑬ rue Neuve & Ⓔ Carrefour du Lavoir & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
⑤ rue des Écoles & ⑭ rue Haute & Ⓕ Rond-point de la Poste & \textbf{a} l'école \\ \hline
⑥ rue du Pont & ⑮ chemin des Vignes & Ⓖ Rond-point de la Gare & \textbf{b} la boulangerie \\ \hline
⑦ chemin Creux & ⑯ impasse du Puits & Ⓗ Place de l'Église & \textbf{c} le marché couvert \\ \hline
⑧ rue du Moulin & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Place du Vieux-Puits & \textbf{d} la bibliothèque \\ \hline
\end{tabular}
\end{center}
\noindent{\small\itshape Légende du plan}\par\addvspace{0.4em}
\begin{enumerate}
  \item Colorier les carrefours du plan avec \textbf{deux couleurs}, de sorte que deux carrefours \textbf{reliés par une rue} ne soient jamais de la même couleur. Combien y en a-t-il de chaque couleur ?
  \item Le car roule d'un carrefour au suivant. Que peut-on dire de la \textbf{couleur} des arrêts successifs ? Qu'en déduire sur les couleurs des arrêts d'une tournée qui les dessert tous ?
  \item \textbf{Le car peut-il desservir tous les carrefours, une fois chacun, et revenir à l'école ?} Si oui, décrire un trajet ; sinon, démontrer que c'est impossible.
  \item \textbf{Et s'il n'est pas obligé de revenir à l'école ?} Même consigne.
\end{enumerate}
\newpage
\section*{\textcolor{red}{Correction}}
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{Le car de ramassage — Correction}}
\needspace{3\baselineskip}
\begin{center}
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\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & A, B, C, F, H, I, J, M & 8 \\ \hline
clairs & D, E, G, K, L & 5 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{8} carrefours sombres et \textbf{5} clairs. Il y en a donc \textbf{3 de plus} du côté des sombres.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 13 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Un circuit revient à son point de départ : il compte autant d'arrêts sombres que de clairs. Or ce quartier a 8 carrefours sombres et 5 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non plus.} Une tournée n'a qu'un départ et qu'une arrivée : elle ne peut absorber qu'\textbf{un} carrefour sombre en trop. Il y en a 3.}\par\addvspace{0.4em}
\end{document}
