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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 rue des Lilas & Ⓐ Place de la Fontaine & Ⓚ Place du Marché \\ \hline
① allée des Peupliers & ⑪ venelle du Chat & Ⓑ Place des Tanneurs & Ⓛ Place aux Herbes \\ \hline
② boulevard du Parc & ⑫ rue de l'Église & Ⓒ Carrefour de l'Hôpital & Ⓜ Rond-point de la Poste \\ \hline
③ rue Basse & ⑬ rue des Écoles & Ⓓ Place du Vieux-Puits & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ rue Pasteur & ⑭ chemin Creux & Ⓔ Carrefour de la Mairie & \textbf{a} l'école \\ \hline
⑤ rue des Jardins & ⑮ rue du Four & Ⓕ Place Sainte-Croix & \textbf{b} la pharmacie \\ \hline
⑥ rue du Puits & ⑯ chemin des Vignes & Ⓖ Carrefour Saint-Roch & \textbf{c} le cimetière \\ \hline
⑦ rue Haute & ⑰ chemin du Lavoir & Ⓗ Carrefour du Moulin & \textbf{d} la salle des fêtes \\ \hline
⑧ allée des Tilleuls & ⑱ passage du Marché & Ⓘ Carrefour des Tilleuls &  \\ \hline
⑨ rue Verte & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Rond-point de la Gare &  \\ \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] (8.204,5.07) -- (8.196,4.873) -- (8.183,4.676) -- (8.166,4.48) -- (8.145,4.284) -- (8.118,4.088) -- (8.088,3.893) -- (8.052,3.699) -- (8.012,3.505) -- (7.968,3.313) -- (7.918,3.122) -- (7.865,2.932) -- (7.807,2.743) -- (7.745,2.556) -- (7.678,2.371);
\draw[line width=0.5pt] (11.1,4.995) -- (11.047,4.804) -- (10.998,4.612) -- (10.954,4.419) -- (10.914,4.225) -- (10.879,4.03) -- (10.849,3.834) -- (10.823,3.638) -- (10.801,3.441) -- (10.785,3.243) -- (10.773,3.045) -- (10.765,2.847) -- (10.763,2.649) -- (10.765,2.451) -- (10.771,2.253);
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\draw[line width=0.5pt] (4.801,4.839) -- (4.783,4.599) -- (4.77,4.36) -- (4.762,4.12) -- (4.761,3.88) -- (4.765,3.64) -- (4.774,3.4) -- (4.79,3.16) -- (4.81,2.921) -- (4.837,2.682) -- (4.869,2.444) -- (4.907,2.207) -- (4.95,1.971) -- (4.999,1.736) -- (5.053,1.503);
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\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.676,11.143) -- (11.647,11.286) -- (11.567,11.407) -- (11.445,11.488) -- (11.303,11.517) -- (11.16,11.488) -- (11.039,11.407) -- (10.958,11.286) -- (10.929,11.143) -- (10.958,11.0) -- (11.039,10.879) -- (11.16,10.798) -- (11.303,10.77) -- (11.445,10.798) -- (11.567,10.879) -- (11.647,11.0) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.646,10.889) -- (8.617,11.032) -- (8.537,11.153) -- (8.415,11.234) -- (8.273,11.262) -- (8.13,11.234) -- (8.009,11.153) -- (7.928,11.032) -- (7.899,10.889) -- (7.928,10.746) -- (8.009,10.625) -- (8.13,10.544) -- (8.273,10.516) -- (8.415,10.544) -- (8.537,10.625) -- (8.617,10.746) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.973,8.06) -- (10.944,8.203) -- (10.863,8.324) -- (10.742,8.405) -- (10.599,8.433) -- (10.457,8.405) -- (10.335,8.324) -- (10.255,8.203) -- (10.226,8.06) -- (10.255,7.917) -- (10.335,7.796) -- (10.457,7.715) -- (10.599,7.687) -- (10.742,7.715) -- (10.863,7.796) -- (10.944,7.917) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.359,7.849) -- (5.331,7.992) -- (5.25,8.113) -- (5.129,8.194) -- (4.986,8.222) -- (4.843,8.194) -- (4.722,8.113) -- (4.641,7.992) -- (4.613,7.849) -- (4.641,7.706) -- (4.722,7.585) -- (4.843,7.504) -- (4.986,7.476) -- (5.129,7.504) -- (5.25,7.585) -- (5.331,7.706) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.26,7.718) -- (8.232,7.861) -- (8.151,7.982) -- (8.03,8.063) -- (7.887,8.091) -- (7.744,8.063) -- (7.623,7.982) -- (7.542,7.861) -- (7.514,7.718) -- (7.542,7.575) -- (7.623,7.454) -- (7.744,7.373) -- (7.887,7.345) -- (8.03,7.373) -- (8.151,7.454) -- (8.232,7.575) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.099,5.524) -- (2.071,5.667) -- (1.99,5.788) -- (1.869,5.869) -- (1.726,5.897) -- (1.583,5.869) -- (1.462,5.788) -- (1.381,5.667) -- (1.353,5.524) -- (1.381,5.381) -- (1.462,5.26) -- (1.583,5.179) -- (1.726,5.151) -- (1.869,5.179) -- (1.99,5.26) -- (2.071,5.381) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.577,5.07) -- (8.548,5.213) -- (8.467,5.334) -- (8.346,5.415) -- (8.204,5.443) -- (8.061,5.415) -- (7.94,5.334) -- (7.859,5.213) -- (7.83,5.07) -- (7.859,4.927) -- (7.94,4.806) -- (8.061,4.725) -- (8.204,4.697) -- (8.346,4.725) -- (8.467,4.806) -- (8.548,4.927) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.473,4.995) -- (11.445,5.138) -- (11.364,5.259) -- (11.243,5.34) -- (11.1,5.368) -- (10.957,5.34) -- (10.836,5.259) -- (10.755,5.138) -- (10.727,4.995) -- (10.755,4.852) -- (10.836,4.731) -- (10.957,4.65) -- (11.1,4.622) -- (11.243,4.65) -- (11.364,4.731) -- (11.445,4.852) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.174,4.839) -- (5.146,4.981) -- (5.065,5.102) -- (4.944,5.183) -- (4.801,5.212) -- (4.658,5.183) -- (4.537,5.102) -- (4.456,4.981) -- (4.428,4.839) -- (4.456,4.696) -- (4.537,4.575) -- (4.658,4.494) -- (4.801,4.465) -- (4.944,4.494) -- (5.065,4.575) -- (5.146,4.696) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.051,2.371) -- (8.023,2.513) -- (7.942,2.634) -- (7.821,2.715) -- (7.678,2.744) -- (7.535,2.715) -- (7.414,2.634) -- (7.333,2.513) -- (7.305,2.371) -- (7.333,2.228) -- (7.414,2.107) -- (7.535,2.026) -- (7.678,1.997) -- (7.821,2.026) -- (7.942,2.107) -- (8.023,2.228) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.676,2.344) -- (1.647,2.487) -- (1.567,2.608) -- (1.445,2.689) -- (1.303,2.717) -- (1.16,2.689) -- (1.039,2.608) -- (0.958,2.487) -- (0.929,2.344) -- (0.958,2.201) -- (1.039,2.08) -- (1.16,1.999) -- (1.303,1.971) -- (1.445,1.999) -- (1.567,2.08) -- (1.647,2.201) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.144,2.253) -- (11.116,2.396) -- (11.035,2.517) -- (10.914,2.598) -- (10.771,2.626) -- (10.628,2.598) -- (10.507,2.517) -- (10.426,2.396) -- (10.398,2.253) -- (10.426,2.11) -- (10.507,1.989) -- (10.628,1.908) -- (10.771,1.88) -- (10.914,1.908) -- (11.035,1.989) -- (11.116,2.11) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.426,1.503) -- (5.398,1.645) -- (5.317,1.767) -- (5.196,1.847) -- (5.053,1.876) -- (4.91,1.847) -- (4.789,1.767) -- (4.708,1.645) -- (4.68,1.503) -- (4.708,1.36) -- (4.789,1.239) -- (4.91,1.158) -- (5.053,1.129) -- (5.196,1.158) -- (5.317,1.239) -- (5.398,1.36) -- cycle;
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\filldraw[fill=white, draw=black, line width=0.5pt] (9.745,11.14) -- (9.864,11.159) -- (9.971,11.213) -- (10.056,11.298) -- (10.111,11.406) -- (10.13,11.524) -- (10.111,11.643) -- (10.056,11.75) -- (9.971,11.835) -- (9.864,11.89) -- (9.745,11.909) -- (9.745,11.909) -- (9.626,11.89) -- (9.519,11.835) -- (9.434,11.75) -- (9.379,11.643) -- (9.36,11.524) -- (9.379,11.406) -- (9.434,11.298) -- (9.519,11.213) -- (9.626,11.159) -- (9.745,11.14) -- cycle;
\node at (9.745,11.524) {\scalebox{1.608}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.331,8.889) -- (8.45,8.907) -- (8.557,8.962) -- (8.642,9.047) -- (8.696,9.154) -- (8.715,9.273) -- (8.696,9.392) -- (8.642,9.499) -- (8.557,9.584) -- (8.45,9.639) -- (8.331,9.658) -- (8.331,9.658) -- (8.212,9.639) -- (8.105,9.584) -- (8.02,9.499) -- (7.965,9.392) -- (7.946,9.273) -- (7.965,9.154) -- (8.02,9.047) -- (8.105,8.962) -- (8.212,8.907) -- (8.331,8.889) -- cycle;
\node at (8.331,9.273) {\scalebox{1.608}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.356,6.226) -- (11.475,6.244) -- (11.582,6.299) -- (11.667,6.384) -- (11.721,6.491) -- (11.74,6.61) -- (11.721,6.729) -- (11.667,6.836) -- (11.582,6.921) -- (11.475,6.976) -- (11.356,6.995) -- (11.356,6.995) -- (11.237,6.976) -- (11.13,6.921) -- (11.045,6.836) -- (10.99,6.729) -- (10.971,6.61) -- (10.99,6.491) -- (11.045,6.384) -- (11.13,6.299) -- (11.237,6.244) -- (11.356,6.226) -- cycle;
\node at (11.356,6.61) {\scalebox{1.608}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.448,7.664) -- (6.567,7.682) -- (6.675,7.737) -- (6.76,7.822) -- (6.814,7.929) -- (6.833,8.048) -- (6.814,8.167) -- (6.76,8.274) -- (6.675,8.359) -- (6.567,8.414) -- (6.448,8.433) -- (6.448,8.433) -- (6.33,8.414) -- (6.222,8.359) -- (6.137,8.274) -- (6.083,8.167) -- (6.064,8.048) -- (6.083,7.929) -- (6.137,7.822) -- (6.222,7.737) -- (6.33,7.682) -- (6.448,7.664) -- cycle;
\node at (6.448,8.048) {\scalebox{1.608}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.402,5.928) -- (5.521,5.947) -- (5.628,6.001) -- (5.713,6.087) -- (5.767,6.194) -- (5.786,6.313) -- (5.767,6.431) -- (5.713,6.539) -- (5.628,6.624) -- (5.521,6.678) -- (5.402,6.697) -- (5.402,6.697) -- (5.283,6.678) -- (5.176,6.624) -- (5.091,6.539) -- (5.036,6.431) -- (5.017,6.313) -- (5.036,6.194) -- (5.091,6.087) -- (5.176,6.001) -- (5.283,5.947) -- (5.402,5.928) -- cycle;
\node at (5.402,6.313) {\scalebox{1.608}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.209,7.774) -- (9.328,7.793) -- (9.435,7.848) -- (9.52,7.933) -- (9.575,8.04) -- (9.594,8.159) -- (9.575,8.278) -- (9.52,8.385) -- (9.435,8.47) -- (9.328,8.524) -- (9.209,8.543) -- (9.209,8.543) -- (9.09,8.524) -- (8.983,8.47) -- (8.898,8.385) -- (8.843,8.278) -- (8.825,8.159) -- (8.843,8.04) -- (8.898,7.933) -- (8.983,7.848) -- (9.09,7.793) -- (9.209,7.774) -- cycle;
\node at (9.209,8.159) {\scalebox{1.608}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.536,6.068) -- (8.655,6.087) -- (8.762,6.142) -- (8.847,6.227) -- (8.902,6.334) -- (8.921,6.453) -- (8.902,6.572) -- (8.847,6.679) -- (8.762,6.764) -- (8.655,6.819) -- (8.536,6.837) -- (8.536,6.837) -- (8.417,6.819) -- (8.31,6.764) -- (8.225,6.679) -- (8.171,6.572) -- (8.152,6.453) -- (8.171,6.334) -- (8.225,6.227) -- (8.31,6.142) -- (8.417,6.087) -- (8.536,6.068) -- cycle;
\node at (8.536,6.453) {\scalebox{1.608}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.319,5.045) -- (3.438,5.064) -- (3.545,5.119) -- (3.63,5.204) -- (3.685,5.311) -- (3.704,5.43) -- (3.685,5.549) -- (3.63,5.656) -- (3.545,5.741) -- (3.438,5.796) -- (3.319,5.814) -- (3.319,5.814) -- (3.2,5.796) -- (3.093,5.741) -- (3.008,5.656) -- (2.953,5.549) -- (2.935,5.43) -- (2.953,5.311) -- (3.008,5.204) -- (3.093,5.119) -- (3.2,5.064) -- (3.319,5.045) -- cycle;
\node at (3.319,5.43) {\scalebox{1.608}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.208,3.481) -- (2.327,3.5) -- (2.434,3.555) -- (2.519,3.64) -- (2.573,3.747) -- (2.592,3.866) -- (2.573,3.985) -- (2.519,4.092) -- (2.434,4.177) -- (2.327,4.232) -- (2.208,4.25) -- (1.846,4.25) -- (1.727,4.232) -- (1.62,4.177) -- (1.535,4.092) -- (1.48,3.985) -- (1.461,3.866) -- (1.48,3.747) -- (1.535,3.64) -- (1.62,3.555) -- (1.727,3.5) -- (1.846,3.481) -- cycle;
\node at (2.027,3.866) {\scalebox{1.608}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.84,4.913) -- (9.959,4.932) -- (10.066,4.986) -- (10.151,5.072) -- (10.205,5.179) -- (10.224,5.298) -- (10.205,5.416) -- (10.151,5.524) -- (10.066,5.609) -- (9.959,5.663) -- (9.84,5.682) -- (9.478,5.682) -- (9.359,5.663) -- (9.252,5.609) -- (9.167,5.524) -- (9.112,5.416) -- (9.093,5.298) -- (9.112,5.179) -- (9.167,5.072) -- (9.252,4.986) -- (9.359,4.932) -- (9.478,4.913) -- cycle;
\node at (9.659,5.298) {\scalebox{1.608}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.61,3.241) -- (8.729,3.26) -- (8.836,3.314) -- (8.922,3.399) -- (8.976,3.506) -- (8.995,3.625) -- (8.976,3.744) -- (8.922,3.851) -- (8.836,3.936) -- (8.729,3.991) -- (8.61,4.01) -- (8.249,4.01) -- (8.13,3.991) -- (8.023,3.936) -- (7.937,3.851) -- (7.883,3.744) -- (7.864,3.625) -- (7.883,3.506) -- (7.937,3.399) -- (8.023,3.314) -- (8.13,3.26) -- (8.249,3.241) -- cycle;
\node at (8.429,3.625) {\scalebox{1.608}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.385,3.207) -- (11.504,3.226) -- (11.611,3.281) -- (11.696,3.366) -- (11.751,3.473) -- (11.77,3.592) -- (11.751,3.711) -- (11.696,3.818) -- (11.611,3.903) -- (11.504,3.958) -- (11.385,3.976) -- (11.023,3.976) -- (10.905,3.958) -- (10.797,3.903) -- (10.712,3.818) -- (10.658,3.711) -- (10.639,3.592) -- (10.658,3.473) -- (10.712,3.366) -- (10.797,3.281) -- (10.905,3.226) -- (11.023,3.207) -- cycle;
\node at (11.204,3.592) {\scalebox{1.608}{\small 13}};
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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 & A, E, H, I, J, K & 6 \\ \hline
clairs & B, C, D, F, G, L, 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 l'\textbf{allée des Peupliers ①} vers le nord jusqu'à Place de la Fontaine \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{boulevard du Parc ②}, vers l'ouest, jusqu'à Place des Tanneurs \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Basse ③}, vers le sud, jusqu'à Carrefour de la Mairie \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Jardins ⑤}, vers l'ouest, jusqu'à Place du Vieux-Puits \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Puits ⑥}, vers le sud, jusqu'à Carrefour des Tilleuls \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Verte ⑨}, vers l'ouest, jusqu'à Place Sainte-Croix \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Lilas ⑩}, vers le sud, jusqu'à Place du Marché \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin du Lavoir ⑰}, vers l'est, jusqu'à Rond-point de la Poste \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{passage du Marché ⑱}, vers l'est, jusqu'à Rond-point de la Gare \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de l'Église ⑫}, vers le nord, jusqu'à Carrefour Saint-Roch \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{venelle du Chat ⑪}, vers l'est, jusqu'à Carrefour du Moulin \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Écoles ⑬}, vers le sud, jusqu'à Place aux Herbes \textbf{(L)}.\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}
