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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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\noindent \textbf{Le car part de l'école, au carrefour G, 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 & ⑨ rue des Jardins & Ⓐ Place Sainte-Croix & Ⓙ Carrefour Saint-Roch \\ \hline
① rue Neuve & ⑩ chemin des Vignes & Ⓑ Place de la Fontaine & Ⓚ Rond-point des Vaches \\ \hline
② impasse Sainte-Anne & ⑪ chemin du Lavoir & Ⓒ Place du Vieux-Puits & Ⓛ Coin du Lavoir \\ \hline
③ quai aux Fleurs & ⑫ rue du Pont & Ⓓ Rond-point du Stade & Ⓜ Placette des Ormes \\ \hline
④ rue du Puits & ⑬ allée des Peupliers & Ⓔ Place de l'Église & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
⑤ rue Pasteur & ⑭ rue de l'Église & Ⓕ Carrefour des Tilleuls & \textbf{a} l'école \\ \hline
⑥ rue du Moulin & ⑮ rue du Four & Ⓖ Place du Marché & \textbf{b} le lavoir \\ \hline
⑦ chemin Creux & ⑯ avenue de la Gare & Ⓗ Carrefour de l'Hôpital & \textbf{c} la gendarmerie \\ \hline
⑧ allée des Tilleuls & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Place des Tanneurs & \textbf{d} le cimetière \\ \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] (1.765,1.486) -- (1.972,1.543) -- (2.18,1.595) -- (2.389,1.642) -- (2.6,1.684) -- (2.811,1.72) -- (3.023,1.752) -- (3.236,1.779) -- (3.45,1.801) -- (3.664,1.818) -- (3.878,1.83) -- (4.092,1.837) -- (4.307,1.839) -- (4.521,1.836) -- (4.735,1.827);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (10.931,11.486) -- (10.903,11.629) -- (10.822,11.75) -- (10.701,11.831) -- (10.558,11.86) -- (10.415,11.831) -- (10.294,11.75) -- (10.213,11.629) -- (10.185,11.486) -- (10.213,11.344) -- (10.294,11.223) -- (10.415,11.142) -- (10.558,11.113) -- (10.701,11.142) -- (10.822,11.223) -- (10.903,11.344) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.898,11.188) -- (1.87,11.331) -- (1.789,11.452) -- (1.668,11.533) -- (1.525,11.561) -- (1.382,11.533) -- (1.261,11.452) -- (1.18,11.331) -- (1.152,11.188) -- (1.18,11.045) -- (1.261,10.924) -- (1.382,10.843) -- (1.525,10.815) -- (1.668,10.843) -- (1.789,10.924) -- (1.87,11.045) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.074,11.121) -- (8.046,11.264) -- (7.965,11.385) -- (7.844,11.466) -- (7.701,11.494) -- (7.558,11.466) -- (7.437,11.385) -- (7.356,11.264) -- (7.328,11.121) -- (7.356,10.978) -- (7.437,10.857) -- (7.558,10.776) -- (7.701,10.748) -- (7.844,10.776) -- (7.965,10.857) -- (8.046,10.978) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.985,8.455) -- (7.956,8.598) -- (7.875,8.719) -- (7.754,8.8) -- (7.611,8.828) -- (7.469,8.8) -- (7.347,8.719) -- (7.267,8.598) -- (7.238,8.455) -- (7.267,8.312) -- (7.347,8.191) -- (7.469,8.11) -- (7.611,8.081) -- (7.754,8.11) -- (7.875,8.191) -- (7.956,8.312) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.711,8.374) -- (1.683,8.517) -- (1.602,8.638) -- (1.481,8.719) -- (1.338,8.747) -- (1.195,8.719) -- (1.074,8.638) -- (0.993,8.517) -- (0.965,8.374) -- (0.993,8.231) -- (1.074,8.11) -- (1.195,8.029) -- (1.338,8.001) -- (1.481,8.029) -- (1.602,8.11) -- (1.683,8.231) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.0,8.321) -- (4.971,8.464) -- (4.89,8.585) -- (4.769,8.666) -- (4.626,8.694) -- (4.484,8.666) -- (4.362,8.585) -- (4.282,8.464) -- (4.253,8.321) -- (4.282,8.178) -- (4.362,8.057) -- (4.484,7.976) -- (4.626,7.948) -- (4.769,7.976) -- (4.89,8.057) -- (4.971,8.178) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.9,8.011) -- (10.872,8.154) -- (10.791,8.275) -- (10.67,8.356) -- (10.527,8.384) -- (10.384,8.356) -- (10.263,8.275) -- (10.182,8.154) -- (10.154,8.011) -- (10.182,7.868) -- (10.263,7.747) -- (10.384,7.666) -- (10.527,7.638) -- (10.67,7.666) -- (10.791,7.747) -- (10.872,7.868) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.652,5.085) -- (7.624,5.228) -- (7.543,5.349) -- (7.422,5.43) -- (7.279,5.459) -- (7.136,5.43) -- (7.015,5.349) -- (6.934,5.228) -- (6.906,5.085) -- (6.934,4.942) -- (7.015,4.821) -- (7.136,4.741) -- (7.279,4.712) -- (7.422,4.741) -- (7.543,4.821) -- (7.624,4.942) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.347,4.73) -- (2.319,4.873) -- (2.238,4.994) -- (2.117,5.075) -- (1.974,5.103) -- (1.831,5.075) -- (1.71,4.994) -- (1.629,4.873) -- (1.6,4.73) -- (1.629,4.587) -- (1.71,4.466) -- (1.831,4.385) -- (1.974,4.357) -- (2.117,4.385) -- (2.238,4.466) -- (2.319,4.587) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.146,4.607) -- (5.118,4.749) -- (5.037,4.87) -- (4.916,4.951) -- (4.773,4.98) -- (4.63,4.951) -- (4.509,4.87) -- (4.428,4.749) -- (4.4,4.607) -- (4.428,4.464) -- (4.509,4.343) -- (4.63,4.262) -- (4.773,4.233) -- (4.916,4.262) -- (5.037,4.343) -- (5.118,4.464) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.108,1.827) -- (5.08,1.97) -- (4.999,2.091) -- (4.878,2.172) -- (4.735,2.201) -- (4.592,2.172) -- (4.471,2.091) -- (4.39,1.97) -- (4.362,1.827) -- (4.39,1.685) -- (4.471,1.563) -- (4.592,1.483) -- (4.735,1.454) -- (4.878,1.483) -- (4.999,1.563) -- (5.08,1.685) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.352,1.511) -- (8.323,1.654) -- (8.242,1.775) -- (8.121,1.856) -- (7.978,1.884) -- (7.836,1.856) -- (7.715,1.775) -- (7.634,1.654) -- (7.605,1.511) -- (7.634,1.368) -- (7.715,1.247) -- (7.836,1.166) -- (7.978,1.138) -- (8.121,1.166) -- (8.242,1.247) -- (8.323,1.368) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.138,1.486) -- (2.11,1.629) -- (2.029,1.75) -- (1.908,1.831) -- (1.765,1.86) -- (1.622,1.831) -- (1.501,1.75) -- (1.42,1.629) -- (1.392,1.486) -- (1.42,1.344) -- (1.501,1.223) -- (1.622,1.142) -- (1.765,1.113) -- (1.908,1.142) -- (2.029,1.223) -- (2.11,1.344) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.784,9.362) -- (10.902,9.381) -- (11.01,9.435) -- (11.095,9.521) -- (11.149,9.628) -- (11.168,9.747) -- (11.149,9.865) -- (11.095,9.973) -- (11.01,10.058) -- (10.902,10.112) -- (10.784,10.131) -- (10.784,10.131) -- (10.665,10.112) -- (10.558,10.058) -- (10.473,9.973) -- (10.418,9.865) -- (10.399,9.747) -- (10.418,9.628) -- (10.473,9.521) -- (10.558,9.435) -- (10.665,9.381) -- (10.784,9.362) -- cycle;
\node at (10.784,9.747) {\scalebox{1.519}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.931,9.363) -- (2.05,9.382) -- (2.157,9.437) -- (2.242,9.522) -- (2.297,9.629) -- (2.316,9.748) -- (2.297,9.867) -- (2.242,9.974) -- (2.157,10.059) -- (2.05,10.113) -- (1.931,10.132) -- (1.931,10.132) -- (1.812,10.113) -- (1.705,10.059) -- (1.62,9.974) -- (1.566,9.867) -- (1.547,9.748) -- (1.566,9.629) -- (1.62,9.522) -- (1.705,9.437) -- (1.812,9.382) -- (1.931,9.363) -- cycle;
\node at (1.931,9.748) {\scalebox{1.519}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.096,11.183) -- (9.215,11.202) -- (9.322,11.256) -- (9.407,11.341) -- (9.461,11.448) -- (9.48,11.567) -- (9.461,11.686) -- (9.407,11.793) -- (9.322,11.878) -- (9.215,11.933) -- (9.096,11.952) -- (9.096,11.952) -- (8.977,11.933) -- (8.87,11.878) -- (8.785,11.793) -- (8.73,11.686) -- (8.711,11.567) -- (8.73,11.448) -- (8.785,11.341) -- (8.87,11.256) -- (8.977,11.202) -- (9.096,11.183) -- cycle;
\node at (9.096,11.567) {\scalebox{1.519}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.15,9.387) -- (8.269,9.406) -- (8.376,9.46) -- (8.461,9.545) -- (8.516,9.652) -- (8.535,9.771) -- (8.516,9.89) -- (8.461,9.997) -- (8.376,10.082) -- (8.269,10.137) -- (8.15,10.156) -- (8.15,10.156) -- (8.032,10.137) -- (7.924,10.082) -- (7.839,9.997) -- (7.785,9.89) -- (7.766,9.771) -- (7.785,9.652) -- (7.839,9.545) -- (7.924,9.46) -- (8.032,9.406) -- (8.15,9.387) -- cycle;
\node at (8.15,9.771) {\scalebox{1.519}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.109,8.108) -- (9.227,8.127) -- (9.335,8.182) -- (9.42,8.267) -- (9.474,8.374) -- (9.493,8.493) -- (9.474,8.612) -- (9.42,8.719) -- (9.335,8.804) -- (9.227,8.858) -- (9.109,8.877) -- (9.109,8.877) -- (8.99,8.858) -- (8.883,8.804) -- (8.798,8.719) -- (8.743,8.612) -- (8.724,8.493) -- (8.743,8.374) -- (8.798,8.267) -- (8.883,8.182) -- (8.99,8.127) -- (9.109,8.108) -- cycle;
\node at (9.109,8.493) {\scalebox{1.519}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.967,6.334) -- (8.086,6.353) -- (8.193,6.407) -- (8.278,6.493) -- (8.333,6.6) -- (8.351,6.719) -- (8.333,6.837) -- (8.278,6.945) -- (8.193,7.03) -- (8.086,7.084) -- (7.967,7.103) -- (7.967,7.103) -- (7.848,7.084) -- (7.741,7.03) -- (7.656,6.945) -- (7.601,6.837) -- (7.582,6.719) -- (7.601,6.6) -- (7.656,6.493) -- (7.741,6.407) -- (7.848,6.353) -- (7.967,6.334) -- cycle;
\node at (7.967,6.719) {\scalebox{1.519}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.986,8.212) -- (3.105,8.23) -- (3.212,8.285) -- (3.297,8.37) -- (3.352,8.477) -- (3.371,8.596) -- (3.352,8.715) -- (3.297,8.822) -- (3.212,8.907) -- (3.105,8.962) -- (2.986,8.981) -- (2.986,8.981) -- (2.867,8.962) -- (2.76,8.907) -- (2.675,8.822) -- (2.621,8.715) -- (2.602,8.596) -- (2.621,8.477) -- (2.675,8.37) -- (2.76,8.285) -- (2.867,8.23) -- (2.986,8.212) -- cycle;
\node at (2.986,8.596) {\scalebox{1.519}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.096,8.51) -- (6.215,8.529) -- (6.322,8.584) -- (6.407,8.669) -- (6.462,8.776) -- (6.481,8.895) -- (6.462,9.014) -- (6.407,9.121) -- (6.322,9.206) -- (6.215,9.261) -- (6.096,9.28) -- (6.096,9.28) -- (5.977,9.261) -- (5.87,9.206) -- (5.785,9.121) -- (5.73,9.014) -- (5.712,8.895) -- (5.73,8.776) -- (5.785,8.669) -- (5.87,8.584) -- (5.977,8.529) -- (6.096,8.51) -- cycle;
\node at (6.096,8.895) {\scalebox{1.519}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.931,6.088) -- (5.05,6.107) -- (5.157,6.162) -- (5.242,6.247) -- (5.296,6.354) -- (5.315,6.473) -- (5.296,6.592) -- (5.242,6.699) -- (5.157,6.784) -- (5.05,6.839) -- (4.931,6.857) -- (4.931,6.857) -- (4.812,6.839) -- (4.705,6.784) -- (4.62,6.699) -- (4.565,6.592) -- (4.546,6.473) -- (4.565,6.354) -- (4.62,6.247) -- (4.705,6.162) -- (4.812,6.107) -- (4.931,6.088) -- cycle;
\node at (4.931,6.473) {\scalebox{1.519}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.335,3.016) -- (8.453,3.035) -- (8.561,3.09) -- (8.646,3.175) -- (8.7,3.282) -- (8.719,3.401) -- (8.7,3.52) -- (8.646,3.627) -- (8.561,3.712) -- (8.453,3.767) -- (8.335,3.785) -- (7.973,3.785) -- (7.854,3.767) -- (7.747,3.712) -- (7.662,3.627) -- (7.607,3.52) -- (7.588,3.401) -- (7.607,3.282) -- (7.662,3.175) -- (7.747,3.09) -- (7.854,3.035) -- (7.973,3.016) -- cycle;
\node at (8.154,3.401) {\scalebox{1.519}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.566,4.552) -- (3.685,4.571) -- (3.792,4.626) -- (3.877,4.711) -- (3.932,4.818) -- (3.951,4.937) -- (3.932,5.056) -- (3.877,5.163) -- (3.792,5.248) -- (3.685,5.303) -- (3.566,5.321) -- (3.204,5.321) -- (3.085,5.303) -- (2.978,5.248) -- (2.893,5.163) -- (2.839,5.056) -- (2.82,4.937) -- (2.839,4.818) -- (2.893,4.711) -- (2.978,4.626) -- (3.085,4.571) -- (3.204,4.552) -- cycle;
\node at (3.385,4.937) {\scalebox{1.519}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.568,2.69) -- (2.687,2.709) -- (2.794,2.764) -- (2.879,2.849) -- (2.934,2.956) -- (2.952,3.075) -- (2.934,3.194) -- (2.879,3.301) -- (2.794,3.386) -- (2.687,3.441) -- (2.568,3.459) -- (2.206,3.459) -- (2.087,3.441) -- (1.98,3.386) -- (1.895,3.301) -- (1.84,3.194) -- (1.821,3.075) -- (1.84,2.956) -- (1.895,2.849) -- (1.98,2.764) -- (2.087,2.709) -- (2.206,2.69) -- cycle;
\node at (2.387,3.075) {\scalebox{1.519}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.155,4.736) -- (6.273,4.755) -- (6.381,4.809) -- (6.466,4.894) -- (6.52,5.001) -- (6.539,5.12) -- (6.52,5.239) -- (6.466,5.346) -- (6.381,5.431) -- (6.273,5.486) -- (6.155,5.505) -- (5.793,5.505) -- (5.674,5.486) -- (5.567,5.431) -- (5.482,5.346) -- (5.427,5.239) -- (5.408,5.12) -- (5.427,5.001) -- (5.482,4.894) -- (5.567,4.809) -- (5.674,4.755) -- (5.793,4.736) -- cycle;
\node at (5.974,5.12) {\scalebox{1.519}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.434,2.826) -- (5.553,2.844) -- (5.66,2.899) -- (5.745,2.984) -- (5.8,3.091) -- (5.819,3.21) -- (5.8,3.329) -- (5.745,3.436) -- (5.66,3.521) -- (5.553,3.576) -- (5.434,3.595) -- (5.072,3.595) -- (4.954,3.576) -- (4.846,3.521) -- (4.761,3.436) -- (4.707,3.329) -- (4.688,3.21) -- (4.707,3.091) -- (4.761,2.984) -- (4.846,2.899) -- (4.954,2.844) -- (5.072,2.826) -- cycle;
\node at (5.253,3.21) {\scalebox{1.519}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.011,1.5) -- (7.129,1.519) -- (7.237,1.574) -- (7.322,1.659) -- (7.376,1.766) -- (7.395,1.885) -- (7.376,2.004) -- (7.322,2.111) -- (7.237,2.196) -- (7.129,2.251) -- (7.011,2.269) -- (6.649,2.269) -- (6.53,2.251) -- (6.423,2.196) -- (6.338,2.111) -- (6.283,2.004) -- (6.264,1.885) -- (6.283,1.766) -- (6.338,1.659) -- (6.423,1.574) -- (6.53,1.519) -- (6.649,1.5) -- cycle;
\node at (6.83,1.885) {\scalebox{1.519}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.461,1.013) -- (3.58,1.032) -- (3.687,1.086) -- (3.772,1.171) -- (3.827,1.279) -- (3.845,1.397) -- (3.827,1.516) -- (3.772,1.623) -- (3.687,1.708) -- (3.58,1.763) -- (3.461,1.782) -- (3.099,1.782) -- (2.98,1.763) -- (2.873,1.708) -- (2.788,1.623) -- (2.733,1.516) -- (2.714,1.397) -- (2.733,1.279) -- (2.788,1.171) -- (2.873,1.086) -- (2.98,1.032) -- (3.099,1.013) -- cycle;
\node at (3.28,1.397) {\scalebox{1.519}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (10.227,7.711) -- (10.827,7.711) -- (10.827,8.311) -- (10.227,8.311) -- cycle;
\node at (10.527,8.011) {\scalebox{1.519}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.754,0.74) -- (6.354,0.74) -- (6.354,1.34) -- (5.754,1.34) -- cycle;
\node at (6.054,1.04) {\scalebox{1.519}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.346,7.014) -- (4.946,7.014) -- (4.946,7.614) -- (4.346,7.614) -- cycle;
\node at (4.646,7.314) {\scalebox{1.519}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (1.086,2.039) -- (1.686,2.039) -- (1.686,2.639) -- (1.086,2.639) -- cycle;
\node at (1.386,2.339) {\scalebox{1.519}{\small d}};
\node at (11.003,11.919) {\scalebox{1.519}{\small A}};
\node at (1.113,11.652) {\scalebox{1.519}{\small B}};
\node at (7.967,11.681) {\scalebox{1.519}{\small C}};
\node at (8.074,8.868) {\scalebox{1.519}{\small D}};
\node at (0.765,8.61) {\scalebox{1.519}{\small E}};
\node at (4.324,8.862) {\scalebox{1.519}{\small F}};
\node at (11.124,8.177) {\scalebox{1.519}{\small G}};
\node at (7.739,4.67) {\scalebox{1.519}{\small H}};
\node at (1.426,4.44) {\scalebox{1.519}{\small I}};
\node at (4.546,4.03) {\scalebox{1.519}{\small J}};
\node at (4.629,1.216) {\scalebox{1.519}{\small K}};
\node at (8.243,0.95) {\scalebox{1.519}{\small L}};
\node at (1.396,0.988) {\scalebox{1.519}{\small M}};
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\end{center}
\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, D, E, J, L, M & 6 \\ \hline
clairs & B, C, F, G, H, I, K & 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{G} :}\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 ①} vers le nord jusqu'à Place Sainte-Croix \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{quai aux Fleurs ③}, vers l'ouest, jusqu'à Place du Vieux-Puits \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Puits ④}, vers le sud, jusqu'à Rond-point du Stade \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Moulin ⑥}, vers le sud, jusqu'à Carrefour de l'Hôpital \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin des Vignes ⑩}, vers le sud, jusqu'à Coin du Lavoir \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Four ⑮}, vers l'ouest, jusqu'à Rond-point des Vaches \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{avenue de la Gare ⑯}, vers l'ouest, jusqu'à Placette des Ormes \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Pont ⑫}, vers le nord, jusqu'à Place des Tanneurs \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin du Lavoir ⑪}, vers l'est, jusqu'à Carrefour Saint-Roch \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Jardins ⑨}, vers le nord, jusqu'à Carrefour des Tilleuls \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin Creux ⑦}, vers l'ouest, jusqu'à Place de l'Église \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{impasse Sainte-Anne ②}, vers le nord, jusqu'à Place de la Fontaine \textbf{(B)}.\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}
