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    \small Un facteur veut parcourir toutes les rues de son quartier en n'empruntant chacune qu'une seule fois. Selon le plan, c'est tantôt facile, tantôt impossible — et l'on peut le dire d'avance, sans essayer, en regardant simplement combien de rues aboutissent à chaque carrefour.
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\subsection*{\textcolor{red}{La tournée du facteur}}
\noindent Voici le plan d'un quartier.\par\addvspace{0.35em}
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\end{center}
\noindent \textbf{Le facteur part de la poste et doit emprunter chaque rue exactement une fois. Il n'est pas obligé d'y revenir : il termine sa tournée où il veut.}\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Les rues & \cellcolor{blue!7}\bfseries  & \cellcolor{blue!7}\bfseries Les carrefours & \cellcolor{blue!7}\bfseries Les bâtiments \\ \hline
① chemin du Lavoir & ⑩ rue Neuve & Ⓐ Rond-point du Pont & \textbf{a} la poste \\ \hline
② avenue de la Gare & ⑪ allée des Tilleuls & Ⓑ Place de l'Église & \textbf{b} la bibliothèque \\ \hline
③ rue de la Mairie & ⑫ rue Pasteur & Ⓒ Rond-point de la Poste & \textbf{c} le gymnase \\ \hline
④ sentier des Prés & ⑬ chemin Creux & Ⓓ Carrefour de l'Hôpital & \textbf{d} la salle des fêtes \\ \hline
⑤ chemin des Vignes & ⑭ venelle du Chat & Ⓔ Carrefour des Tilleuls & \textbf{e} le stade \\ \hline
⑥ rue du Four & ⑮ rue Haute & Ⓕ Rond-point des Vaches &  \\ \hline
⑦ quai aux Fleurs & ⑯ boulevard du Parc & Ⓖ Carrefour du Lavoir &  \\ \hline
⑧ rue de l'Église & ⑰ passage du Marché & Ⓗ Carrefour du Moulin &  \\ \hline
⑨ rue du Puits & ⑱ rue des Lilas & Ⓘ Carrefour de la Mairie &  \\ \hline
 &  & Ⓙ Place aux Herbes &  \\ \hline
\end{tabular}
\end{center}
\noindent{\small\itshape Légende du plan}\par\addvspace{0.4em}
\noindent Les questions se suivent : la première n'est qu'un comptage, la dernière demande une preuve.\par\addvspace{0.45em}
\begin{enumerate}
  \item Pour \textbf{chaque carrefour}, compter le nombre de rues qui y aboutissent. Présenter les résultats dans un tableau.
  \item Le facteur imagine sa tournée. À chaque fois qu'il \textbf{traverse} un carrefour, combien de rues utilise-t-il ? Qu'en déduire pour les carrefours où aboutit un nombre \textbf{impair} de rues ?
  \item \textbf{Le facteur peut-il faire sa tournée en partant de la poste ?} Si oui, décrire un trajet possible ; sinon, démontrer que c'est impossible.
  \item Si ce n'est pas possible : une tournée existe-t-elle \textbf{malgré tout} dans ce quartier, en partant d'ailleurs ? Si oui, où faudrait-il installer la poste — et le facteur y reviendrait-il ?
\end{enumerate}
\newpage
\section*{\textcolor{red}{Correction}}
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{La tournée du facteur — Correction}}
\needspace{3\baselineskip}
\begin{center}
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\noindent \textbf{1.} On compte les rues aboutissant à chaque carrefour :\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Carrefour & \cellcolor{blue!7}\bfseries A & \cellcolor{blue!7}\bfseries B & \cellcolor{blue!7}\bfseries C & \cellcolor{blue!7}\bfseries D & \cellcolor{blue!7}\bfseries E & \cellcolor{blue!7}\bfseries F & \cellcolor{blue!7}\bfseries G & \cellcolor{blue!7}\bfseries H & \cellcolor{blue!7}\bfseries I & \cellcolor{blue!7}\bfseries J \\ \hline
Nombre de rues & 3 & 2 & 4 & 3 & 4 & 6 & 4 & 4 & 4 & 2 \\ \hline
Parité & \textbf{impair} & pair & pair & \textbf{impair} & pair & pair & pair & pair & pair & pair \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 36 = 2 \(\times\) 18, car chaque rue est comptée à ses \textbf{deux} extrémités.\par\addvspace{0.3em}
\noindent{\color{red!80!black}\bfseries Les carrefours à nombre \textbf{impair} de rues sont : \textbf{A}, \textbf{D} — il y en a 2.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Quand le facteur \textbf{traverse} un carrefour, il y arrive par une rue et en repart par une autre : il consomme les rues \textbf{deux par deux}. Un carrefour traversé 3 fois voit donc 6 de ses rues utilisées — toujours un nombre \textbf{pair}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Un carrefour où aboutit un nombre \textbf{impair} de rues ne peut donc pas être seulement traversé : il doit être le \textbf{départ} ou l'\textbf{arrivée} de la tournée. Là, une rue reste « non appariée ».}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Ici la poste est \textbf{sur un carrefour}, en \textbf{F}, et le facteur n'a pas à y revenir. Ce carrefour est donc le \textbf{départ} — il a droit d'être impair. Mais il n'y a qu'\textbf{une seule arrivée} : tous les autres carrefours doivent être pairs, sauf un au plus.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Deux carrefours sont impairs, \textbf{A} et \textbf{D} : l'un devrait être le départ, l'autre l'arrivée.}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Or la poste est en \textbf{F}, où aboutit un nombre \textbf{pair} de rues. Ce carrefour ne peut être que traversé, jamais servir de départ.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui, une tournée existe} — mais elle va de \textbf{A} à \textbf{D}. En installant la poste à l'un de ces deux carrefours, le facteur pourrait faire toutes les rues une fois. En revanche il \textbf{ne reviendrait pas} à la poste : il finirait à l'autre bout du quartier.}\par\addvspace{0.4em}
\newpage
\noindent Ce parcours, en partant de A :\par\addvspace{0.35em}
\begin{enumerate}
  \item Il part de Rond-point du Pont \textbf{(A)}, prend le \textbf{chemin du Lavoir ①} vers l'est et atteint Rond-point de la Poste \textbf{(C)}.
  \item Il continue tout droit dans le \textbf{sentier des Prés ④}, vers l'est, jusqu'à Place de l'Église \textbf{(B)}.
  \item Il tourne à droite dans le \textbf{chemin des Vignes ⑤}, vers le sud, jusqu'à Carrefour des Tilleuls \textbf{(E)}.
  \item Il prend dans la \textbf{rue du Four ⑥}, vers le nord-ouest, jusqu'à Rond-point de la Poste \textbf{(C)}.
  \item Il tourne à gauche dans le \textbf{quai aux Fleurs ⑦}, vers le sud-ouest, jusqu'à Rond-point des Vaches \textbf{(F)}.
  \item Il tourne à droite dans l'\textbf{avenue de la Gare ②}, vers le nord-ouest, jusqu'à Rond-point du Pont \textbf{(A)}.
  \item Il tourne à gauche dans la \textbf{rue de la Mairie ③}, vers le sud, jusqu'à Carrefour du Moulin \textbf{(H)}.
  \item Il tourne à gauche dans le \textbf{chemin Creux ⑬}, vers le nord-est, jusqu'à Rond-point des Vaches \textbf{(F)}.
  \item Il continue tout droit dans la \textbf{rue du Puits ⑨}, vers l'est, jusqu'à Carrefour de l'Hôpital \textbf{(D)}.
  \item Il continue tout droit dans la \textbf{rue de l'Église ⑧}, vers l'est, jusqu'à Carrefour des Tilleuls \textbf{(E)}.
  \item Il tourne à droite dans l'\textbf{allée des Tilleuls ⑪}, vers le sud-ouest, jusqu'à Carrefour de la Mairie \textbf{(I)}.
  \item Il tourne à droite dans la \textbf{venelle du Chat ⑭}, vers le nord-ouest, jusqu'à Rond-point des Vaches \textbf{(F)}.
  \item Il tourne à gauche dans la \textbf{rue Pasteur ⑫}, vers le sud, jusqu'à Carrefour du Lavoir \textbf{(G)}.
  \item Il tourne à droite dans la \textbf{rue Haute ⑮}, vers l'ouest, jusqu'à Carrefour du Moulin \textbf{(H)}.
  \item Il tourne à gauche dans la \textbf{rue des Lilas ⑱}, vers le sud-est, jusqu'à Place aux Herbes \textbf{(J)}.
  \item Il tourne à gauche dans le \textbf{passage du Marché ⑰}, vers le nord, jusqu'à Carrefour du Lavoir \textbf{(G)}.
  \item Il tourne à droite dans le \textbf{boulevard du Parc ⑯}, vers l'est, jusqu'à Carrefour de la Mairie \textbf{(I)}.
  \item Il tourne à gauche dans la \textbf{rue Neuve ⑩}, vers le nord, jusqu'à Carrefour de l'Hôpital \textbf{(D)}.
\end{enumerate}
\end{document}
