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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 rue du Pont & ⑫ rue des Écoles & S0 Place du Marché & \textbf{a} la poste \\ \hline
② rue du Moulin & ⑬ rue Basse & S1 Place de la Fontaine & \textbf{b} l'école \\ \hline
③ avenue de la Gare & ⑭ chemin du Lavoir & S2 Rond-point des Vaches & \textbf{c} la pharmacie \\ \hline
④ rue Verte & ⑮ rue de l'Église & S3 Rond-point de la Gare & \textbf{d} le marché couvert \\ \hline
⑤ rue du Puits & ⑯ rue de la Poste & S4 Rond-point du Pont &  \\ \hline
⑥ impasse des Roses & ⑰ rue des Jardins & S5 Coin du Lavoir &  \\ \hline
⑦ rue du Four & ⑱ allée des Peupliers & S6 Rond-point du Stade &  \\ \hline
⑧ boulevard du Parc & ⑲ rue du Stade & S7 Place des Tanneurs &  \\ \hline
⑨ rue des Lilas & ⑳ chemin des Vignes & S8 Fontaine Notre-Dame &  \\ \hline
⑩ allée des Tilleuls & 21 quai aux Fleurs & S9 Carrefour du Lavoir &  \\ \hline
⑪ venelle du Chat &  & S10 Rond-point de la Poste &  \\ \hline
 &  & S11 Place aux Herbes &  \\ \hline
 &  & S12 Place de l'Église &  \\ \hline
 &  & S13 Carrefour des Quatre-Chemins &  \\ \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}
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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|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 & 2 & 3 & 6 & 4 & 2 & 4 & 2 & 2 & 4 & 4 & 1 & 2 & 4 \\ \hline
Parité & pair & pair & \textbf{impair} & pair & pair & pair & pair & pair & pair & pair & pair & \textbf{impair} & pair & pair \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 42 = 2 \(\times\) 21, 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{S2}, \textbf{S11} — 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{S11}, 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.} Deux carrefours sont impairs, \textbf{S2} et \textbf{S11}, et la poste est justement sur l'un des deux (\textbf{S11}). Il part donc de la poste et \textbf{termine en S2}, l'autre carrefour impair.}\par\addvspace{0.4em}
\newpage
\noindent Voici une tournée possible :\par\addvspace{0.35em}
\begin{enumerate}
  \item Il part de Place aux Herbes \textbf{(S11)}, prend l'\textbf{impasse des Roses ⑥} vers l'est et atteint Carrefour des Quatre-Chemins \textbf{(S13)}.
  \item Il tourne à gauche dans la \textbf{rue du Puits ⑤}, vers le nord, jusqu'à Rond-point de la Poste \textbf{(S10)}.
  \item Il continue tout droit dans la \textbf{rue des Écoles ⑫}, vers le nord, jusqu'à Rond-point du Pont \textbf{(S4)}.
  \item Il continue tout droit dans la \textbf{rue du Moulin ②}, vers le nord, jusqu'à Place du Marché \textbf{(S0)}.
  \item Il tourne à droite dans la \textbf{rue du Pont ①}, vers l'est, jusqu'à Rond-point des Vaches \textbf{(S2)}.
  \item Il continue tout droit dans l'\textbf{avenue de la Gare ③}, vers l'est, jusqu'à Place de la Fontaine \textbf{(S1)}.
  \item Il tourne à droite dans la \textbf{rue Verte ④}, vers le sud-ouest, jusqu'à Rond-point de la Gare \textbf{(S3)}.
  \item Il continue tout droit dans le \textbf{boulevard du Parc ⑧}, vers l'ouest, jusqu'à Rond-point du Pont \textbf{(S4)}.
  \item Il tourne à gauche dans la \textbf{rue Basse ⑬}, vers le sud-est, jusqu'à Rond-point du Stade \textbf{(S6)}.
  \item Il tourne à droite dans la \textbf{rue de l'Église ⑮}, vers l'ouest, jusqu'à Rond-point de la Poste \textbf{(S10)}.
  \item Il continue tout droit dans la \textbf{rue du Stade ⑲}, vers l'ouest, jusqu'à Fontaine Notre-Dame \textbf{(S8)}.
  \item Il tourne à gauche dans le \textbf{chemin des Vignes ⑳}, vers le sud-est, jusqu'à Carrefour des Quatre-Chemins \textbf{(S13)}.
  \item Il tourne à gauche dans la \textbf{rue de la Poste ⑯}, vers le nord-est, jusqu'à Rond-point du Stade \textbf{(S6)}.
  \item Il continue tout droit dans l'\textbf{allée des Tilleuls ⑩}, vers le nord-est, jusqu'à Rond-point de la Gare \textbf{(S3)}.
  \item Il tourne à droite dans la \textbf{rue des Lilas ⑨}, vers l'est, jusqu'à Coin du Lavoir \textbf{(S5)}.
  \item Il tourne à droite dans le \textbf{chemin du Lavoir ⑭}, vers le sud, jusqu'à Carrefour du Lavoir \textbf{(S9)}.
  \item Il tourne à droite dans l'\textbf{allée des Peupliers ⑱}, vers l'ouest, jusqu'à Place des Tanneurs \textbf{(S7)}.
  \item Il tourne à gauche dans la \textbf{rue des Jardins ⑰}, vers le sud, jusqu'à Place de l'Église \textbf{(S12)}.
  \item Il tourne à gauche dans le \textbf{quai aux Fleurs 21}, vers le nord-est, jusqu'à Carrefour du Lavoir \textbf{(S9)}.
  \item Il tourne à gauche dans la \textbf{venelle du Chat ⑪}, vers le nord-ouest, jusqu'à Rond-point de la Gare \textbf{(S3)}.
  \item Il continue tout droit dans la \textbf{rue du Four ⑦}, vers le nord, jusqu'à Rond-point des Vaches \textbf{(S2)}.
\end{enumerate}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries C'était déjà possible. Notons seulement que la poste devait être en \textbf{S2} ou en \textbf{S11} : ailleurs, la tournée n'aurait pas pu commencer.}\par\addvspace{0.4em}
\noindent{\color{green!45!black}\textbf{À remarquer.} Ce quartier a une impasse. On y entre et on en ressort par la même rue : elle serait forcément empruntée deux fois. Une impasse interdit donc à elle seule la tournée qui revient à son point de départ — sans qu'il soit besoin de compter quoi que ce soit.}\par\addvspace{0.4em}
\end{document}
