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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 C, 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 & ⑩ venelle du Chat & Ⓑ Rond-point de la Poste & Ⓛ Place du Marché \\ \hline
① impasse Verte & ⑪ chemin de Ronde & Ⓒ Place de l'Église & Ⓜ Coin du Lavoir \\ \hline
② chemin des Vignes & ⑫ rue du Stade & Ⓓ Place aux Herbes & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ passage du Marché & ⑬ avenue de la Gare & Ⓔ Rond-point de la Gare & \textbf{a} l'école \\ \hline
④ rue Neuve & ⑭ rue du Moulin & Ⓕ Rond-point des Vaches & \textbf{b} le stade \\ \hline
⑤ rue Haute & ⑮ boulevard du Parc & Ⓖ Carrefour de l'Hôpital & \textbf{c} la boulangerie \\ \hline
⑥ rue du Four & ⑯ rue des Jardins & Ⓗ Place du Vieux-Puits & \textbf{d} la gendarmerie \\ \hline
⑦ rue de l'Église & ⑰ rue de la Poste & Ⓘ Carrefour des Tilleuls &  \\ \hline
⑧ rue des Écoles & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Carrefour du Moulin &  \\ \hline
⑨ allée des Tilleuls & Ⓐ Fontaine Notre-Dame & Ⓚ Rond-point du Stade &  \\ \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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\draw[line width=0.5pt] (6.779,4.188) -- (6.792,4.042) -- (6.802,3.895) -- (6.809,3.749) -- (6.813,3.602) -- (6.812,3.455) -- (6.809,3.309) -- (6.802,3.162) -- (6.791,3.016) -- (6.777,2.87) -- (6.76,2.724) -- (6.739,2.579) -- (6.715,2.434) -- (6.687,2.29) -- (6.657,2.147);
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\filldraw[fill=white, draw=black, line width=0.5pt] (11.977,6.861) -- (11.949,7.004) -- (11.868,7.125) -- (11.747,7.206) -- (11.604,7.234) -- (11.461,7.206) -- (11.34,7.125) -- (11.259,7.004) -- (11.231,6.861) -- (11.259,6.718) -- (11.34,6.597) -- (11.461,6.516) -- (11.604,6.488) -- (11.747,6.516) -- (11.868,6.597) -- (11.949,6.718) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.719,6.838) -- (4.69,6.981) -- (4.609,7.102) -- (4.488,7.183) -- (4.345,7.212) -- (4.203,7.183) -- (4.081,7.102) -- (4.001,6.981) -- (3.972,6.838) -- (4.001,6.696) -- (4.081,6.574) -- (4.203,6.494) -- (4.345,6.465) -- (4.488,6.494) -- (4.609,6.574) -- (4.69,6.696) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.172,6.814) -- (2.144,6.957) -- (2.063,7.078) -- (1.942,7.159) -- (1.799,7.187) -- (1.656,7.159) -- (1.535,7.078) -- (1.454,6.957) -- (1.426,6.814) -- (1.454,6.671) -- (1.535,6.55) -- (1.656,6.469) -- (1.799,6.441) -- (1.942,6.469) -- (2.063,6.55) -- (2.144,6.671) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.79,6.633) -- (6.761,6.775) -- (6.68,6.896) -- (6.559,6.977) -- (6.416,7.006) -- (6.274,6.977) -- (6.153,6.896) -- (6.072,6.775) -- (6.043,6.633) -- (6.072,6.49) -- (6.153,6.369) -- (6.274,6.288) -- (6.416,6.259) -- (6.559,6.288) -- (6.68,6.369) -- (6.761,6.49) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.651,4.819) -- (9.622,4.962) -- (9.541,5.083) -- (9.42,5.164) -- (9.277,5.192) -- (9.135,5.164) -- (9.014,5.083) -- (8.933,4.962) -- (8.904,4.819) -- (8.933,4.676) -- (9.014,4.555) -- (9.135,4.474) -- (9.277,4.446) -- (9.42,4.474) -- (9.541,4.555) -- (9.622,4.676) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.977,4.805) -- (1.949,4.948) -- (1.868,5.069) -- (1.747,5.15) -- (1.604,5.179) -- (1.461,5.15) -- (1.34,5.069) -- (1.259,4.948) -- (1.231,4.805) -- (1.259,4.663) -- (1.34,4.542) -- (1.461,4.461) -- (1.604,4.432) -- (1.747,4.461) -- (1.868,4.542) -- (1.949,4.663) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.422,4.432) -- (4.394,4.574) -- (4.313,4.695) -- (4.192,4.776) -- (4.049,4.805) -- (3.906,4.776) -- (3.785,4.695) -- (3.704,4.574) -- (3.676,4.432) -- (3.704,4.289) -- (3.785,4.168) -- (3.906,4.087) -- (4.049,4.058) -- (4.192,4.087) -- (4.313,4.168) -- (4.394,4.289) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.895,4.321) -- (11.866,4.464) -- (11.785,4.585) -- (11.664,4.666) -- (11.521,4.695) -- (11.379,4.666) -- (11.258,4.585) -- (11.177,4.464) -- (11.148,4.321) -- (11.177,4.179) -- (11.258,4.058) -- (11.379,3.977) -- (11.521,3.948) -- (11.664,3.977) -- (11.785,4.058) -- (11.866,4.179) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.152,4.188) -- (7.123,4.33) -- (7.042,4.451) -- (6.921,4.532) -- (6.779,4.561) -- (6.636,4.532) -- (6.515,4.451) -- (6.434,4.33) -- (6.405,4.188) -- (6.434,4.045) -- (6.515,3.924) -- (6.636,3.843) -- (6.779,3.814) -- (6.921,3.843) -- (7.042,3.924) -- (7.123,4.045) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.03,2.147) -- (7.001,2.29) -- (6.92,2.411) -- (6.799,2.492) -- (6.657,2.52) -- (6.514,2.492) -- (6.393,2.411) -- (6.312,2.29) -- (6.283,2.147) -- (6.312,2.004) -- (6.393,1.883) -- (6.514,1.802) -- (6.657,1.773) -- (6.799,1.802) -- (6.92,1.883) -- (7.001,2.004) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.329,2.073) -- (4.301,2.216) -- (4.22,2.337) -- (4.099,2.418) -- (3.956,2.447) -- (3.813,2.418) -- (3.692,2.337) -- (3.611,2.216) -- (3.583,2.073) -- (3.611,1.931) -- (3.692,1.81) -- (3.813,1.729) -- (3.956,1.7) -- (4.099,1.729) -- (4.22,1.81) -- (4.301,1.931) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.578,1.955) -- (9.549,2.098) -- (9.468,2.219) -- (9.347,2.3) -- (9.204,2.329) -- (9.062,2.3) -- (8.941,2.219) -- (8.86,2.098) -- (8.831,1.955) -- (8.86,1.813) -- (8.941,1.692) -- (9.062,1.611) -- (9.204,1.582) -- (9.347,1.611) -- (9.468,1.692) -- (9.549,1.813) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.378,1.939) -- (2.35,2.082) -- (2.269,2.203) -- (2.148,2.284) -- (2.005,2.313) -- (1.862,2.284) -- (1.741,2.203) -- (1.66,2.082) -- (1.632,1.939) -- (1.66,1.797) -- (1.741,1.676) -- (1.862,1.595) -- (2.005,1.566) -- (2.148,1.595) -- (2.269,1.676) -- (2.35,1.797) -- cycle;
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\filldraw[fill=white, draw=black, line width=0.5pt] (5.427,6.819) -- (5.546,6.838) -- (5.653,6.892) -- (5.738,6.978) -- (5.793,7.085) -- (5.812,7.204) -- (5.793,7.322) -- (5.738,7.43) -- (5.653,7.515) -- (5.546,7.569) -- (5.427,7.588) -- (5.427,7.588) -- (5.309,7.569) -- (5.201,7.515) -- (5.116,7.43) -- (5.062,7.322) -- (5.043,7.204) -- (5.062,7.085) -- (5.116,6.978) -- (5.201,6.892) -- (5.309,6.838) -- (5.427,6.819) -- cycle;
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\filldraw[fill=white, draw=black, line width=0.5pt] (4.479,5.216) -- (4.598,5.234) -- (4.705,5.289) -- (4.79,5.374) -- (4.845,5.481) -- (4.864,5.6) -- (4.845,5.719) -- (4.79,5.826) -- (4.705,5.911) -- (4.598,5.966) -- (4.479,5.985) -- (4.479,5.985) -- (4.361,5.966) -- (4.253,5.911) -- (4.168,5.826) -- (4.114,5.719) -- (4.095,5.6) -- (4.114,5.481) -- (4.168,5.374) -- (4.253,5.289) -- (4.361,5.234) -- (4.479,5.216) -- cycle;
\node at (4.479,5.6) {\scalebox{1.644}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.067,6.931) -- (3.186,6.95) -- (3.293,7.005) -- (3.379,7.09) -- (3.433,7.197) -- (3.452,7.316) -- (3.433,7.435) -- (3.379,7.542) -- (3.293,7.627) -- (3.186,7.682) -- (3.067,7.7) -- (3.067,7.7) -- (2.949,7.682) -- (2.841,7.627) -- (2.756,7.542) -- (2.702,7.435) -- (2.683,7.316) -- (2.702,7.197) -- (2.756,7.09) -- (2.841,7.005) -- (2.949,6.95) -- (3.067,6.931) -- cycle;
\node at (3.067,7.316) {\scalebox{1.644}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.001,5.396) -- (2.12,5.415) -- (2.227,5.47) -- (2.312,5.555) -- (2.367,5.662) -- (2.386,5.781) -- (2.367,5.9) -- (2.312,6.007) -- (2.227,6.092) -- (2.12,6.146) -- (2.001,6.165) -- (2.001,6.165) -- (1.883,6.146) -- (1.775,6.092) -- (1.69,6.007) -- (1.636,5.9) -- (1.617,5.781) -- (1.636,5.662) -- (1.69,5.555) -- (1.775,5.47) -- (1.883,5.415) -- (2.001,5.396) -- cycle;
\node at (2.001,5.781) {\scalebox{1.644}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.079,5.097) -- (7.198,5.116) -- (7.305,5.17) -- (7.39,5.255) -- (7.444,5.363) -- (7.463,5.481) -- (7.444,5.6) -- (7.39,5.707) -- (7.305,5.792) -- (7.198,5.847) -- (7.079,5.866) -- (7.079,5.866) -- (6.96,5.847) -- (6.853,5.792) -- (6.768,5.707) -- (6.713,5.6) -- (6.694,5.481) -- (6.713,5.363) -- (6.768,5.255) -- (6.853,5.17) -- (6.96,5.116) -- (7.079,5.097) -- cycle;
\node at (7.079,5.481) {\scalebox{1.644}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.296,3.718) -- (10.415,3.736) -- (10.522,3.791) -- (10.607,3.876) -- (10.661,3.983) -- (10.68,4.102) -- (10.661,4.221) -- (10.607,4.328) -- (10.522,4.413) -- (10.415,4.468) -- (10.296,4.487) -- (10.296,4.487) -- (10.177,4.468) -- (10.07,4.413) -- (9.985,4.328) -- (9.93,4.221) -- (9.911,4.102) -- (9.93,3.983) -- (9.985,3.876) -- (10.07,3.791) -- (10.177,3.736) -- (10.296,3.718) -- cycle;
\node at (10.296,4.102) {\scalebox{1.644}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.743,2.99) -- (9.862,3.009) -- (9.97,3.063) -- (10.055,3.148) -- (10.109,3.255) -- (10.128,3.374) -- (10.109,3.493) -- (10.055,3.6) -- (9.97,3.685) -- (9.862,3.74) -- (9.743,3.759) -- (9.743,3.759) -- (9.625,3.74) -- (9.517,3.685) -- (9.432,3.6) -- (9.378,3.493) -- (9.359,3.374) -- (9.378,3.255) -- (9.432,3.148) -- (9.517,3.063) -- (9.625,3.009) -- (9.743,2.99) -- cycle;
\node at (9.743,3.374) {\scalebox{1.644}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.703,4.541) -- (2.822,4.559) -- (2.929,4.614) -- (3.014,4.699) -- (3.069,4.806) -- (3.087,4.925) -- (3.069,5.044) -- (3.014,5.151) -- (2.929,5.236) -- (2.822,5.291) -- (2.703,5.31) -- (2.703,5.31) -- (2.584,5.291) -- (2.477,5.236) -- (2.392,5.151) -- (2.337,5.044) -- (2.318,4.925) -- (2.337,4.806) -- (2.392,4.699) -- (2.477,4.614) -- (2.584,4.559) -- (2.703,4.541) -- cycle;
\node at (2.703,4.925) {\scalebox{1.644}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.484,3.058) -- (2.603,3.076) -- (2.711,3.131) -- (2.796,3.216) -- (2.85,3.323) -- (2.869,3.442) -- (2.85,3.561) -- (2.796,3.668) -- (2.711,3.753) -- (2.603,3.808) -- (2.484,3.827) -- (2.123,3.827) -- (2.004,3.808) -- (1.897,3.753) -- (1.812,3.668) -- (1.757,3.561) -- (1.738,3.442) -- (1.757,3.323) -- (1.812,3.216) -- (1.897,3.131) -- (2.004,3.076) -- (2.123,3.058) -- cycle;
\node at (2.304,3.442) {\scalebox{1.644}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.619,4.195) -- (5.738,4.214) -- (5.845,4.269) -- (5.93,4.354) -- (5.984,4.461) -- (6.003,4.58) -- (5.984,4.699) -- (5.93,4.806) -- (5.845,4.891) -- (5.738,4.946) -- (5.619,4.965) -- (5.257,4.965) -- (5.138,4.946) -- (5.031,4.891) -- (4.946,4.806) -- (4.891,4.699) -- (4.872,4.58) -- (4.891,4.461) -- (4.946,4.354) -- (5.031,4.269) -- (5.138,4.214) -- (5.257,4.195) -- cycle;
\node at (5.438,4.58) {\scalebox{1.644}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.738,2.885) -- (3.857,2.904) -- (3.964,2.959) -- (4.049,3.044) -- (4.104,3.151) -- (4.123,3.27) -- (4.104,3.389) -- (4.049,3.496) -- (3.964,3.581) -- (3.857,3.636) -- (3.738,3.655) -- (3.376,3.655) -- (3.257,3.636) -- (3.15,3.581) -- (3.065,3.496) -- (3.011,3.389) -- (2.992,3.27) -- (3.011,3.151) -- (3.065,3.044) -- (3.15,2.959) -- (3.257,2.904) -- (3.376,2.885) -- cycle;
\node at (3.557,3.27) {\scalebox{1.644}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.434,4.634) -- (8.553,4.653) -- (8.66,4.708) -- (8.745,4.793) -- (8.8,4.9) -- (8.819,5.019) -- (8.8,5.137) -- (8.745,5.245) -- (8.66,5.33) -- (8.553,5.384) -- (8.434,5.403) -- (8.072,5.403) -- (7.953,5.384) -- (7.846,5.33) -- (7.761,5.245) -- (7.706,5.137) -- (7.688,5.019) -- (7.706,4.9) -- (7.761,4.793) -- (7.846,4.708) -- (7.953,4.653) -- (8.072,4.634) -- cycle;
\node at (8.253,5.019) {\scalebox{1.644}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.599,2.801) -- (6.718,2.819) -- (6.825,2.874) -- (6.91,2.959) -- (6.965,3.066) -- (6.983,3.185) -- (6.965,3.304) -- (6.91,3.411) -- (6.825,3.496) -- (6.718,3.551) -- (6.599,3.57) -- (6.237,3.57) -- (6.118,3.551) -- (6.011,3.496) -- (5.926,3.411) -- (5.871,3.304) -- (5.852,3.185) -- (5.871,3.066) -- (5.926,2.959) -- (6.011,2.874) -- (6.118,2.819) -- (6.237,2.801) -- cycle;
\node at (6.418,3.185) {\scalebox{1.644}{\small 14}};
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\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 & B, F, H, I, K, L & 6 \\ \hline
clairs & A, C, D, E, G, J, M & 7 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{7} clairs. Il y en a donc \textbf{1 de plus} du côté des clairs.}\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 6 carrefours sombres et 7 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{C} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend la \textbf{rue Neuve ④} vers l'est jusqu'à Rond-point de la Poste \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin des Vignes ②}, vers l'est, jusqu'à Place aux Herbes \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Four ⑥}, vers le sud, jusqu'à Carrefour des Tilleuls \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin de Ronde ⑪}, vers l'ouest, jusqu'à Carrefour de l'Hôpital \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{allée des Tilleuls ⑨}, vers l'ouest, jusqu'à Rond-point des Vaches \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{venelle du Chat ⑩}, vers le sud, jusqu'à Coin du Lavoir \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de la Poste ⑰}, vers l'est, jusqu'à Rond-point du Stade \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Jardins ⑯}, vers l'est, jusqu'à Carrefour du Moulin \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{boulevard du Parc ⑮}, vers l'est, jusqu'à Place du Marché \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Écoles ⑧}, vers le nord, jusqu'à Rond-point de la Gare \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de l'Église ⑦}, vers l'est, jusqu'à Place du Vieux-Puits \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{impasse Verte ①}, vers le nord, jusqu'à Fontaine Notre-Dame \textbf{(A)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 13 carrefours ont été desservis une fois chacun. Le car termine loin de l'école : c'est permis ici, mais pas à la question 3.}\par\addvspace{0.4em}
\end{document}
