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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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\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 rue du Four & ⑨ rue de la Mairie & S0 Place de la Fontaine & \textbf{a} la poste \\ \hline
② venelle du Chat & ⑩ chemin du Lavoir & S1 Coin du Lavoir & \textbf{b} le gymnase \\ \hline
③ rue de la Poste & ⑪ passage du Marché & S2 Place aux Herbes & \textbf{c} le stade \\ \hline
④ rue Verte & ⑫ chemin des Vignes & S3 Rond-point des Vaches & \textbf{d} la gendarmerie \\ \hline
⑤ rue Haute & ⑬ rue Basse & S4 Place du Vieux-Puits & \textbf{e} la pharmacie \\ \hline
⑥ sentier des Prés & ⑭ quai aux Fleurs & S5 Place Sainte-Croix &  \\ \hline
⑦ allée des Tilleuls & ⑮ rue du Pont & S6 Rond-point de la Poste &  \\ \hline
⑧ avenue de la Gare &  & S7 Carrefour de l'Hôpital &  \\ \hline
 &  & S8 Place du Marché &  \\ \hline
 &  & S9 Place de l'Église &  \\ \hline
 &  & S10 Placette des Ormes &  \\ \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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\end{center}
\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|c|}
\hline
\cellcolor{blue!7}\bfseries Carrefour & \cellcolor{blue!7}\bfseries S0 & \cellcolor{blue!7}\bfseries S1 & \cellcolor{blue!7}\bfseries S2 & \cellcolor{blue!7}\bfseries S3 & \cellcolor{blue!7}\bfseries S4 & \cellcolor{blue!7}\bfseries S5 & \cellcolor{blue!7}\bfseries S6 & \cellcolor{blue!7}\bfseries S7 & \cellcolor{blue!7}\bfseries S8 & \cellcolor{blue!7}\bfseries S9 & \cellcolor{blue!7}\bfseries S10 \\ \hline
Nombre de rues & 2 & 2 & 2 & 6 & 2 & 2 & 4 & 4 & 2 & 2 & 2 \\ \hline
Parité & pair & pair & pair & pair & pair & pair & pair & pair & pair & pair & pair \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 30 = 2 \(\times\) 15, car chaque rue est comptée à ses \textbf{deux} extrémités.\par\addvspace{0.3em}
\noindent{\color{red!80!black}\bfseries \textbf{Aucun} carrefour n'a un nombre impair de rues : ils sont tous pairs.}\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{S3}, 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{Oui.} Aucun carrefour n'est impair : chacun peut être traversé de part en part. Le facteur revient même à la poste sans l'avoir cherché.}\par\addvspace{0.4em}
\newpage
\noindent Voici une tournée possible — ce n'est pas la seule :\par\addvspace{0.35em}
\begin{enumerate}
  \item Il part de Rond-point des Vaches \textbf{(S3)}, prend la \textbf{venelle du Chat ②} vers le nord et atteint Place de la Fontaine \textbf{(S0)}.
  \item Il tourne à droite dans la \textbf{rue du Four ①}, vers l'est, jusqu'à Coin du Lavoir \textbf{(S1)}.
  \item Il tourne à droite dans la \textbf{rue de la Poste ③}, vers le sud-ouest, jusqu'à Rond-point des Vaches \textbf{(S3)}.
  \item Il tourne à droite dans le \textbf{sentier des Prés ⑥}, vers l'ouest, jusqu'à Place du Vieux-Puits \textbf{(S4)}.
  \item Il tourne à gauche dans le \textbf{chemin du Lavoir ⑩}, vers le sud, jusqu'à Rond-point de la Poste \textbf{(S6)}.
  \item Il tourne à gauche dans l'\textbf{allée des Tilleuls ⑦}, vers le nord-est, jusqu'à Rond-point des Vaches \textbf{(S3)}.
  \item Il tourne à droite dans l'\textbf{avenue de la Gare ⑧}, vers le sud-est, jusqu'à Carrefour de l'Hôpital \textbf{(S7)}.
  \item Il prend dans la \textbf{rue Haute ⑤}, vers le nord, jusqu'à Place aux Herbes \textbf{(S2)}.
  \item Il tourne à droite dans la \textbf{rue Verte ④}, vers l'est, jusqu'à Place Sainte-Croix \textbf{(S5)}.
  \item Il tourne à droite dans le \textbf{passage du Marché ⑪}, vers le sud-ouest, jusqu'à Carrefour de l'Hôpital \textbf{(S7)}.
  \item Il tourne à gauche dans le \textbf{quai aux Fleurs ⑭}, vers le sud-est, jusqu'à Place de l'Église \textbf{(S9)}.
  \item Il tourne à droite dans la \textbf{rue du Pont ⑮}, vers l'ouest, jusqu'à Placette des Ormes \textbf{(S10)}.
  \item Il tourne à droite dans le \textbf{chemin des Vignes ⑫}, vers le nord-ouest, jusqu'à Rond-point de la Poste \textbf{(S6)}.
  \item Il prend dans la \textbf{rue Basse ⑬}, vers l'est, jusqu'à Place du Marché \textbf{(S8)}.
  \item Il tourne à gauche dans la \textbf{rue de la Mairie ⑨}, vers le nord, jusqu'à Rond-point des Vaches \textbf{(S3)}.
\end{enumerate}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries La question ne se pose pas : la tournée est possible au départ de la poste, et elle le resterait où qu'on installe celle-ci — aucun carrefour n'est impair.}\par\addvspace{0.4em}
\end{document}
