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\newcommand{\titre}{La tournée du facteur}
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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 allée des Peupliers & ⑬ allée des Tilleuls & S0 Place aux Herbes & \textbf{a} la poste \\ \hline
② rue des Lilas & ⑭ rue de la Poste & S1 Carrefour de l'Hôpital & \textbf{b} la bibliothèque \\ \hline
③ quai aux Fleurs & ⑮ rue du Four & S2 Rond-point de la Poste & \textbf{c} la salle des fêtes \\ \hline
④ boulevard du Parc & ⑯ rue des Jardins & S3 Carrefour Saint-Roch &  \\ \hline
⑤ chemin de Ronde & ⑰ venelle du Chat & S4 Rond-point du Stade &  \\ \hline
⑥ chemin des Vignes & ⑱ avenue de la Gare & S5 Rond-point des Vaches &  \\ \hline
⑦ rue des Écoles & ⑲ rue du Puits & S6 Carrefour des Tilleuls &  \\ \hline
⑧ rue du Moulin & ⑳ chemin Creux & S7 Coin du Lavoir &  \\ \hline
⑨ sentier des Prés & 21 chemin du Lavoir & S8 Carrefour des Quatre-Chemins &  \\ \hline
⑩ rue Verte & 22 passage du Marché & S9 Carrefour du Moulin &  \\ \hline
⑪ rue de l'Église & 23 rue Neuve & S10 Place du Marché &  \\ \hline
⑫ rue du Stade &  & S11 Place du Vieux-Puits &  \\ \hline
 &  & S12 Place de la Fontaine &  \\ \hline
 &  & S13 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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\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|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 & \cellcolor{blue!7}\bfseries S11 & \cellcolor{blue!7}\bfseries S12 & \cellcolor{blue!7}\bfseries S13 \\ \hline
Nombre de rues & 2 & 3 & 3 & 4 & 6 & 3 & 3 & 2 & 4 & 8 & 2 & 2 & 2 & 2 \\ \hline
Parité & pair & \textbf{impair} & \textbf{impair} & pair & pair & \textbf{impair} & \textbf{impair} & pair & pair & pair & pair & pair & pair & pair \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 46 = 2 \(\times\) 23, 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{S1}, \textbf{S2}, \textbf{S5}, \textbf{S6} — il y en a 4.}\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{Non.} Le quartier compte \textbf{4 carrefours impairs} (S1, S2, S5, S6).}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Chacun d'eux exigerait d'être un départ ou une arrivée. Une tournée n'a qu'un départ et qu'une arrivée : au moins 2 d'entre eux resteraient sans solution.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non plus.} Où que l'on installe la poste, il reste 4 carrefours impairs, et une tournée n'en tolère que deux : un pour partir, un pour arriver.}\par\addvspace{0.4em}
\noindent{\color{green!45!black}C'est la différence avec un quartier à deux carrefours impairs, où déplacer la poste suffisait. Ici, le problème ne vient pas du point de départ mais du plan lui-même.}\par\addvspace{0.4em}
\noindent{\color{green!45!black}\textbf{Combien de rues en double, au minimum ?} Il faut apparier les carrefours impairs deux à deux, ce qui demande 1 passage(s) supplémentaire(s) au mieux pour une tournée ouverte, un de plus pour revenir à la poste. C'est le « problème du postier chinois », étudié bien après Euler.}\par\addvspace{0.4em}
\end{document}
