\documentclass[a4paper,11pt]{article}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[french]{babel}
\usepackage[a4paper,top=3.2cm,bottom=2cm,left=2cm,right=2cm]{geometry}
\usepackage{graphicx}
\usepackage[export]{adjustbox}
\usepackage{fancyhdr}
\usepackage{qrcode}
\usepackage[table]{xcolor}
\usepackage{amsmath,amssymb}
\usepackage{enumitem}
% Puces nettes (babel-french remplace sinon la puce par un tiret « – » peu pro).
% Vaut pour TOUTES les fiches qui importent ce module. AtBeginDocument pour
% passer après la personnalisation des listes faite par babel-french.
\AtBeginDocument{%
  \setlist[itemize]{leftmargin=1.8em,itemsep=0.5ex,topsep=0.4ex,parsep=0pt}%
  \setlist[itemize,1]{label=\textcolor{red!75!black}{\small\textbullet}}%
  \setlist[itemize,2]{label=\textcolor{red!75!black}{\scriptsize\textbullet}}%
}
% Titres compacts (gain de place vertical, fiches plus denses) — vaut pour tous.
\usepackage{titlesec}
\titlespacing*{\section}{0pt}{1.4ex}{0.6ex}
\titlespacing*{\subsection}{0pt}{1.0ex}{0.3ex}
% multicol : listes denses (calcul mental) sur 2 colonnes — neutre tant qu'il
% n'est pas utilisé, dispo pour tous les générateurs.
\usepackage{multicol}
\setlength{\columnsep}{1.5em}
% needspace : empêche un titre d'exercice de rester orphelin en bas de page
% (réserve quelques lignes ; sinon le titre passe à la page suivante avec sa
% liste). Neutre tant qu'il n'est pas appelé — dispo pour tous les générateurs.
\usepackage{needspace}
\usepackage{tikz}
\usetikzlibrary{decorations.text}

\usepackage{tgheros}
\pagestyle{fancy}
\fancyhf{}
\setlength{\headheight}{75pt}
\setlength{\headsep}{10pt}

\newcommand{\logowouf}{%
\begin{tikzpicture}[baseline={(current bounding box.center)},scale=0.55,
                    line cap=round,line join=round]
    \path[use as bounding box] (-2.20,-1.45) rectangle (2.20,2.20);
    \path[
        postaction={decorate},
        decoration={
            text along path,
            text={|\fontfamily{qhv}\bfseries\fontsize{9}{9}\selectfont|site2wouf.fr},
            text align={center},
            raise=0.03ex
        }
    ] (-1.29,0.48) arc[start angle=160,end angle=20,radius=1.29];
    \begin{scope}[xshift=-0.06cm]
        \fill[black] (-0.66,0.14) ellipse (0.21 and 0.31);
        \fill[black] ( 0.66,0.14) ellipse (0.21 and 0.31);
        \fill[black] (-0.23,0.49) ellipse (0.22 and 0.35);
        \fill[black] ( 0.23,0.49) ellipse (0.22 and 0.35);
        \fill[black]
            (0,-0.10)
            .. controls (-0.36,-0.02) and (-0.63,-0.24) .. (-0.58,-0.58)
            .. controls (-0.52,-0.93) and (-0.16,-1.12) .. (0,-0.90)
            .. controls (0.16,-1.12) and (0.52,-0.93) .. (0.58,-0.58)
            .. controls (0.63,-0.24) and (0.36,-0.02) .. (0,-0.10)
            -- cycle;
    \end{scope}
\end{tikzpicture}%
}
\DeclareUnicodeCharacter{2460}{\textcircled{\scriptsize 1}}
\DeclareUnicodeCharacter{2461}{\textcircled{\scriptsize 2}}
\DeclareUnicodeCharacter{2462}{\textcircled{\scriptsize 3}}
\DeclareUnicodeCharacter{2463}{\textcircled{\scriptsize 4}}
\DeclareUnicodeCharacter{2464}{\textcircled{\scriptsize 5}}
\DeclareUnicodeCharacter{2465}{\textcircled{\scriptsize 6}}
\DeclareUnicodeCharacter{2466}{\textcircled{\scriptsize 7}}
\DeclareUnicodeCharacter{2467}{\textcircled{\scriptsize 8}}
\DeclareUnicodeCharacter{2468}{\textcircled{\scriptsize 9}}
\DeclareUnicodeCharacter{2469}{\textcircled{\scriptsize 10}}
\DeclareUnicodeCharacter{246A}{\textcircled{\scriptsize 11}}
\DeclareUnicodeCharacter{246B}{\textcircled{\scriptsize 12}}
\DeclareUnicodeCharacter{246C}{\textcircled{\scriptsize 13}}
\DeclareUnicodeCharacter{246D}{\textcircled{\scriptsize 14}}
\DeclareUnicodeCharacter{246E}{\textcircled{\scriptsize 15}}
\DeclareUnicodeCharacter{246F}{\textcircled{\scriptsize 16}}
\DeclareUnicodeCharacter{2470}{\textcircled{\scriptsize 17}}
\DeclareUnicodeCharacter{2471}{\textcircled{\scriptsize 18}}
\DeclareUnicodeCharacter{2472}{\textcircled{\scriptsize 19}}
\DeclareUnicodeCharacter{2473}{\textcircled{\scriptsize 20}}
\DeclareUnicodeCharacter{24B6}{\textcircled{\scriptsize A}}
\DeclareUnicodeCharacter{24B7}{\textcircled{\scriptsize B}}
\DeclareUnicodeCharacter{24B8}{\textcircled{\scriptsize C}}
\DeclareUnicodeCharacter{24B9}{\textcircled{\scriptsize D}}
\DeclareUnicodeCharacter{24BA}{\textcircled{\scriptsize E}}
\DeclareUnicodeCharacter{24BB}{\textcircled{\scriptsize F}}
\DeclareUnicodeCharacter{24BC}{\textcircled{\scriptsize G}}
\DeclareUnicodeCharacter{24BD}{\textcircled{\scriptsize H}}
\DeclareUnicodeCharacter{24BE}{\textcircled{\scriptsize I}}
\DeclareUnicodeCharacter{24BF}{\textcircled{\scriptsize J}}
\DeclareUnicodeCharacter{24C0}{\textcircled{\scriptsize K}}
\DeclareUnicodeCharacter{24C1}{\textcircled{\scriptsize L}}
\DeclareUnicodeCharacter{24C2}{\textcircled{\scriptsize M}}
\DeclareUnicodeCharacter{24C3}{\textcircled{\scriptsize N}}
\DeclareUnicodeCharacter{24C4}{\textcircled{\scriptsize O}}
\DeclareUnicodeCharacter{24C5}{\textcircled{\scriptsize P}}
\DeclareUnicodeCharacter{24C6}{\textcircled{\scriptsize Q}}
\DeclareUnicodeCharacter{24C7}{\textcircled{\scriptsize R}}
\DeclareUnicodeCharacter{24C8}{\textcircled{\scriptsize S}}
\DeclareUnicodeCharacter{24C9}{\textcircled{\scriptsize T}}
\DeclareUnicodeCharacter{24CA}{\textcircled{\scriptsize U}}
\DeclareUnicodeCharacter{24CB}{\textcircled{\scriptsize V}}
\DeclareUnicodeCharacter{24CC}{\textcircled{\scriptsize W}}
\DeclareUnicodeCharacter{24CD}{\textcircled{\scriptsize X}}
\DeclareUnicodeCharacter{24CE}{\textcircled{\scriptsize Y}}
\DeclareUnicodeCharacter{24CF}{\textcircled{\scriptsize Z}}
\DeclareUnicodeCharacter{24D0}{\textcircled{\scriptsize a}}
\DeclareUnicodeCharacter{24D1}{\textcircled{\scriptsize b}}
\DeclareUnicodeCharacter{24D2}{\textcircled{\scriptsize c}}
\DeclareUnicodeCharacter{24D3}{\textcircled{\scriptsize d}}
\DeclareUnicodeCharacter{24D4}{\textcircled{\scriptsize e}}
\DeclareUnicodeCharacter{24D5}{\textcircled{\scriptsize f}}
\DeclareUnicodeCharacter{24D6}{\textcircled{\scriptsize g}}
\DeclareUnicodeCharacter{24D7}{\textcircled{\scriptsize h}}
\DeclareUnicodeCharacter{24D8}{\textcircled{\scriptsize i}}
\DeclareUnicodeCharacter{24D9}{\textcircled{\scriptsize j}}

\newcommand{\titre}{La tournée du facteur}
\newcommand{\soustitre}{Probl\`eme --- Fiche n\textdegree 93}

\fancyhead[C]{%
\raisebox{4pt}{%
\begin{minipage}[c]{0.15\textwidth}
    \centering
    \logowouf
\end{minipage}
\begin{minipage}[c]{0.62\textwidth}
    \centering
    {\Large\bfseries \titre\par}
    {\small\color{gray} \soustitre}
\end{minipage}
\begin{minipage}[c]{0.21\textwidth}
    \centering
    \qrcode[height=1.5cm]{https://site2wouf.fr/facteur_sans_retour.php?x=93}\\[2pt]
    {\small\ttfamily\bfseries\color{red} EULC0093}
\end{minipage}%
}
}

\begin{document}

\begin{center}
    \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.
\end{center}
\vspace{0.5em}
\sloppy
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{La tournée du facteur}}
\noindent Voici le plan d'un quartier.\par\addvspace{0.35em}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.6968, line join=round, line cap=round]
\draw[line width=0.5pt] (2.125,8.357) -- (2.27,8.321) -- (2.415,8.288) -- (2.562,8.258) -- (2.709,8.232) -- (2.856,8.21) -- (3.005,8.191) -- (3.153,8.175) -- (3.302,8.163) -- (3.451,8.154) -- (3.601,8.149) -- (3.75,8.147) -- (3.899,8.149) -- (4.048,8.155) -- (4.197,8.164) -- (4.435,8.171) -- (4.673,8.172) -- (4.911,8.168) -- (5.149,8.158) -- (5.386,8.142) -- (5.623,8.121) -- (5.86,8.094) -- (6.096,8.062) -- (6.331,8.024) -- (6.565,7.981) -- (6.798,7.932) -- (7.03,7.878) -- (7.26,7.818) -- (7.489,7.753);
\draw[line width=0.5pt] (2.125,8.357) -- (2.318,8.124) -- (2.505,7.886) -- (2.687,7.643) -- (2.862,7.397) -- (3.033,7.146) -- (3.197,6.891) -- (3.355,6.633) -- (3.507,6.37) -- (3.653,6.105) -- (3.792,5.836) -- (3.925,5.564) -- (4.052,5.289) -- (4.172,5.011) -- (4.286,4.73);
\draw[line width=0.5pt] (2.125,8.357) -- (2.028,8.08) -- (1.937,7.8) -- (1.853,7.519) -- (1.776,7.235) -- (1.705,6.949) -- (1.641,6.662) -- (1.584,6.374) -- (1.533,6.084) -- (1.49,5.793) -- (1.453,5.501) -- (1.423,5.209) -- (1.4,4.916) -- (1.384,4.622) -- (1.375,4.329);
\draw[line width=0.5pt] (7.489,7.753) -- (7.743,7.789) -- (7.997,7.819) -- (8.252,7.842) -- (8.508,7.86) -- (8.765,7.872) -- (9.021,7.877) -- (9.278,7.877) -- (9.534,7.871) -- (9.79,7.858) -- (10.046,7.84) -- (10.302,7.816) -- (10.556,7.785) -- (10.81,7.749) -- (11.063,7.706);
\draw[line width=0.5pt] (7.489,7.753) -- (7.505,7.589) -- (7.525,7.425) -- (7.55,7.262) -- (7.578,7.099) -- (7.609,6.937) -- (7.645,6.776) -- (7.684,6.616) -- (7.727,6.457) -- (7.774,6.298) -- (7.825,6.141) -- (7.879,5.985) -- (7.937,5.831) -- (7.998,5.678) -- (8.064,5.526);
\draw[line width=0.5pt] (7.489,7.753) -- (7.295,7.504) -- (7.095,7.259) -- (6.889,7.019) -- (6.678,6.784) -- (6.461,6.554) -- (6.239,6.329) -- (6.012,6.109) -- (5.779,5.895) -- (5.542,5.686) -- (5.3,5.483) -- (5.053,5.286) -- (4.801,5.094) -- (4.546,4.909) -- (4.286,4.73);
\draw[line width=0.5pt] (11.063,7.706) -- (11.061,7.545) -- (11.062,7.384) -- (11.067,7.223) -- (11.077,7.062) -- (11.09,6.901) -- (11.106,6.74) -- (11.127,6.58) -- (11.151,6.421) -- (11.179,6.262) -- (11.211,6.103) -- (11.246,5.946) -- (11.285,5.789) -- (11.328,5.634) -- (11.375,5.479);
\draw[line width=0.5pt] (8.064,5.526) -- (8.299,5.559) -- (8.535,5.587) -- (8.771,5.608) -- (9.008,5.624) -- (9.246,5.635) -- (9.483,5.64) -- (9.721,5.64) -- (9.959,5.633) -- (10.196,5.622) -- (10.433,5.604) -- (10.67,5.581) -- (10.906,5.553) -- (11.141,5.519) -- (11.375,5.479);
\draw[line width=0.5pt] (8.064,5.526) -- (8.047,5.288) -- (8.037,5.05) -- (8.032,4.811) -- (8.033,4.572) -- (8.039,4.334) -- (8.051,4.095) -- (8.068,3.857) -- (8.092,3.619) -- (8.12,3.382) -- (8.155,3.146) -- (8.194,2.91) -- (8.24,2.676) -- (8.291,2.442) -- (8.347,2.211);
\draw[line width=0.5pt] (11.375,5.479) -- (11.196,5.215) -- (11.011,4.954) -- (10.82,4.698) -- (10.622,4.446) -- (10.419,4.199) -- (10.21,3.957) -- (9.996,3.72) -- (9.776,3.488) -- (9.55,3.261) -- (9.319,3.04) -- (9.084,2.824) -- (8.843,2.614) -- (8.597,2.409) -- (8.347,2.211);
\draw[line width=0.5pt] (4.286,4.73) -- (4.075,4.733) -- (3.864,4.732) -- (3.653,4.725) -- (3.443,4.714) -- (3.233,4.697) -- (3.023,4.676) -- (2.814,4.649) -- (2.605,4.618) -- (2.397,4.582) -- (2.19,4.541) -- (1.984,4.495) -- (1.78,4.444) -- (1.577,4.389) -- (1.375,4.329);
\draw[line width=0.5pt] (4.286,4.73) -- (4.383,4.513) -- (4.475,4.293) -- (4.562,4.072) -- (4.643,3.848) -- (4.72,3.622) -- (4.791,3.395) -- (4.856,3.166) -- (4.917,2.936) -- (4.972,2.704) -- (5.021,2.471) -- (5.065,2.237) -- (5.103,2.002) -- (5.136,1.766) -- (5.163,1.53);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.285,8.357) -- (2.263,8.437) -- (2.205,8.496) -- (2.125,8.517) -- (2.045,8.496) -- (1.986,8.437) -- (1.965,8.357) -- (1.986,8.277) -- (2.045,8.219) -- (2.125,8.197) -- (2.205,8.219) -- (2.263,8.277) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.649,7.753) -- (7.627,7.833) -- (7.569,7.892) -- (7.489,7.913) -- (7.409,7.892) -- (7.35,7.833) -- (7.329,7.753) -- (7.35,7.673) -- (7.409,7.614) -- (7.489,7.593) -- (7.569,7.614) -- (7.627,7.673) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.223,7.706) -- (11.201,7.786) -- (11.143,7.845) -- (11.063,7.866) -- (10.983,7.845) -- (10.924,7.786) -- (10.903,7.706) -- (10.924,7.626) -- (10.983,7.568) -- (11.063,7.546) -- (11.143,7.568) -- (11.201,7.626) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.224,5.526) -- (8.202,5.606) -- (8.144,5.665) -- (8.064,5.686) -- (7.984,5.665) -- (7.925,5.606) -- (7.904,5.526) -- (7.925,5.446) -- (7.984,5.388) -- (8.064,5.366) -- (8.144,5.388) -- (8.202,5.446) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.535,5.479) -- (11.513,5.559) -- (11.455,5.618) -- (11.375,5.639) -- (11.295,5.618) -- (11.236,5.559) -- (11.215,5.479) -- (11.236,5.399) -- (11.295,5.341) -- (11.375,5.319) -- (11.455,5.341) -- (11.513,5.399) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.446,4.73) -- (4.424,4.81) -- (4.366,4.868) -- (4.286,4.89) -- (4.206,4.868) -- (4.147,4.81) -- (4.126,4.73) -- (4.147,4.65) -- (4.206,4.591) -- (4.286,4.57) -- (4.366,4.591) -- (4.424,4.65) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.535,4.329) -- (1.513,4.409) -- (1.455,4.467) -- (1.375,4.489) -- (1.295,4.467) -- (1.236,4.409) -- (1.215,4.329) -- (1.236,4.249) -- (1.295,4.19) -- (1.375,4.169) -- (1.455,4.19) -- (1.513,4.249) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.507,2.211) -- (8.486,2.291) -- (8.427,2.349) -- (8.347,2.371) -- (8.267,2.349) -- (8.208,2.291) -- (8.187,2.211) -- (8.208,2.131) -- (8.267,2.072) -- (8.347,2.051) -- (8.427,2.072) -- (8.486,2.131) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.323,1.53) -- (5.302,1.61) -- (5.243,1.668) -- (5.163,1.69) -- (5.083,1.668) -- (5.025,1.61) -- (5.003,1.53) -- (5.025,1.45) -- (5.083,1.391) -- (5.163,1.37) -- (5.243,1.391) -- (5.302,1.45) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.524,8.484) -- (4.479,8.654) -- (4.354,8.778) -- (4.184,8.824) -- (4.014,8.778) -- (3.89,8.654) -- (3.844,8.484) -- (3.89,8.314) -- (4.014,8.189) -- (4.184,8.144) -- (4.354,8.189) -- (4.479,8.314) -- cycle;
\node at (4.184,8.484) {\scalebox{1.435}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.97,6.796) -- (3.924,6.966) -- (3.8,7.091) -- (3.63,7.136) -- (3.46,7.091) -- (3.335,6.966) -- (3.29,6.796) -- (3.335,6.626) -- (3.46,6.502) -- (3.63,6.456) -- (3.8,6.502) -- (3.924,6.626) -- cycle;
\node at (3.63,6.796) {\scalebox{1.435}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.238,6.315) -- (2.193,6.485) -- (2.068,6.61) -- (1.898,6.655) -- (1.728,6.61) -- (1.604,6.485) -- (1.558,6.315) -- (1.604,6.145) -- (1.728,6.021) -- (1.898,5.975) -- (2.068,6.021) -- (2.193,6.145) -- cycle;
\node at (1.898,6.315) {\scalebox{1.435}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.622,8.197) -- (9.576,8.367) -- (9.452,8.492) -- (9.282,8.537) -- (9.112,8.492) -- (8.987,8.367) -- (8.942,8.197) -- (8.987,8.027) -- (9.112,7.903) -- (9.282,7.857) -- (9.452,7.903) -- (9.576,8.027) -- cycle;
\node at (9.282,8.197) {\scalebox{1.435}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.334,6.696) -- (8.289,6.866) -- (8.164,6.99) -- (7.994,7.036) -- (7.824,6.99) -- (7.7,6.866) -- (7.654,6.696) -- (7.7,6.526) -- (7.824,6.401) -- (7.994,6.356) -- (8.164,6.401) -- (8.289,6.526) -- cycle;
\node at (7.994,6.696) {\scalebox{1.435}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.571,5.877) -- (6.526,6.047) -- (6.401,6.171) -- (6.231,6.217) -- (6.061,6.171) -- (5.937,6.047) -- (5.891,5.877) -- (5.937,5.707) -- (6.061,5.582) -- (6.231,5.537) -- (6.401,5.582) -- (6.526,5.707) -- cycle;
\node at (6.231,5.877) {\scalebox{1.435}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.784,6.624) -- (11.738,6.794) -- (11.614,6.919) -- (11.444,6.964) -- (11.274,6.919) -- (11.149,6.794) -- (11.104,6.624) -- (11.149,6.454) -- (11.274,6.33) -- (11.444,6.284) -- (11.614,6.33) -- (11.738,6.454) -- cycle;
\node at (11.444,6.624) {\scalebox{1.435}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.066,5.959) -- (10.02,6.129) -- (9.896,6.254) -- (9.726,6.299) -- (9.556,6.254) -- (9.431,6.129) -- (9.386,5.959) -- (9.431,5.789) -- (9.556,5.665) -- (9.726,5.619) -- (9.896,5.665) -- (10.02,5.789) -- cycle;
\node at (9.726,5.959) {\scalebox{1.435}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.727,3.884) -- (8.682,4.054) -- (8.557,4.179) -- (8.387,4.224) -- (8.217,4.179) -- (8.093,4.054) -- (8.047,3.884) -- (8.093,3.714) -- (8.217,3.59) -- (8.387,3.544) -- (8.557,3.59) -- (8.682,3.714) -- cycle;
\node at (8.387,3.884) {\scalebox{1.435}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.57,3.503) -- (10.525,3.673) -- (10.4,3.797) -- (10.23,3.843) -- (10.06,3.797) -- (9.936,3.673) -- (9.89,3.503) -- (9.936,3.333) -- (10.06,3.208) -- (10.23,3.163) -- (10.4,3.208) -- (10.525,3.333) -- cycle;
\node at (10.23,3.503) {\scalebox{1.435}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.197,4.332) -- (3.152,4.502) -- (3.027,4.627) -- (2.857,4.672) -- (2.687,4.627) -- (2.563,4.502) -- (2.517,4.332) -- (2.563,4.162) -- (2.687,4.038) -- (2.857,3.992) -- (3.027,4.038) -- (3.152,4.162) -- cycle;
\node at (2.857,4.332) {\scalebox{1.435}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.505,3.251) -- (5.46,3.421) -- (5.335,3.545) -- (5.165,3.591) -- (4.995,3.545) -- (4.871,3.421) -- (4.825,3.251) -- (4.871,3.081) -- (4.995,2.956) -- (5.165,2.911) -- (5.335,2.956) -- (5.46,3.081) -- cycle;
\node at (5.165,3.251) {\scalebox{1.435}{\small 12}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (11.075,5.179) -- (11.675,5.179) -- (11.675,5.779) -- (11.075,5.779) -- cycle;
\node at (11.375,5.479) {\scalebox{1.435}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (12.434,6.642) -- (13.034,6.642) -- (13.034,7.242) -- (12.434,7.242) -- cycle;
\node at (12.734,6.942) {\scalebox{1.435}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (0.75,6.209) -- (1.35,6.209) -- (1.35,6.809) -- (0.75,6.809) -- cycle;
\node at (1.05,6.509) {\scalebox{1.435}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.517,2.071) -- (5.117,2.071) -- (5.117,2.671) -- (4.517,2.671) -- cycle;
\node at (4.817,2.371) {\scalebox{1.435}{\small d}};
\node at (1.614,8.708) {\scalebox{1.435}{\small A}};
\node at (7.702,8.335) {\scalebox{1.435}{\small B}};
\node at (11.608,8.001) {\scalebox{1.435}{\small C}};
\node at (8.676,5.624) {\scalebox{1.435}{\small D}};
\node at (11.994,5.504) {\scalebox{1.435}{\small E}};
\node at (3.683,4.583) {\scalebox{1.435}{\small F}};
\node at (0.765,4.216) {\scalebox{1.435}{\small G}};
\node at (8.654,1.672) {\scalebox{1.435}{\small H}};
\node at (4.944,0.95) {\scalebox{1.435}{\small I}};
\draw[line width=0.5pt] (12.675,7.407) -- (12.675,8.507);
\draw[line width=0.5pt] (12.444,8.122) -- (12.675,8.507) -- (12.906,8.122);
\draw[line width=0.5pt] (12.372,7.957) -- (12.977,7.957);
\node at (12.675,8.892) {\scalebox{1.435}{\small N}};
\end{tikzpicture}
\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 Tilleuls & ⑦ rue de l'Église & Ⓐ Carrefour du Moulin & \textbf{a} la poste \\ \hline
② rue du Stade & ⑧ rue Haute & Ⓑ Carrefour de l'Hôpital & \textbf{b} la bibliothèque \\ \hline
③ sentier des Prés & ⑨ venelle du Chat & Ⓒ Place aux Herbes & \textbf{c} la pharmacie \\ \hline
④ rue des Écoles & ⑩ chemin des Vignes & Ⓓ Rond-point de la Poste & \textbf{d} la boulangerie \\ \hline
⑤ quai aux Fleurs & ⑪ rue du Four & Ⓔ Rond-point du Pont &  \\ \hline
⑥ chemin de Ronde & ⑫ impasse Sainte-Anne & Ⓕ Carrefour de la Mairie &  \\ \hline
 &  & Ⓖ Placette des Ormes &  \\ \hline
 &  & Ⓗ Place Sainte-Croix &  \\ \hline
 &  & Ⓘ Place de la Fontaine &  \\ \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}
\begin{tikzpicture}[scale=0.6234, line join=round, line cap=round]
\draw[line width=0.5pt] (2.125,8.357) -- (2.27,8.321) -- (2.415,8.288) -- (2.562,8.258) -- (2.709,8.232) -- (2.856,8.21) -- (3.005,8.191) -- (3.153,8.175) -- (3.302,8.163) -- (3.451,8.154) -- (3.601,8.149) -- (3.75,8.147) -- (3.899,8.149) -- (4.048,8.155) -- (4.197,8.164) -- (4.435,8.171) -- (4.673,8.172) -- (4.911,8.168) -- (5.149,8.158) -- (5.386,8.142) -- (5.623,8.121) -- (5.86,8.094) -- (6.096,8.062) -- (6.331,8.024) -- (6.565,7.981) -- (6.798,7.932) -- (7.03,7.878) -- (7.26,7.818) -- (7.489,7.753);
\draw[line width=0.5pt] (2.125,8.357) -- (2.318,8.124) -- (2.505,7.886) -- (2.687,7.643) -- (2.862,7.397) -- (3.033,7.146) -- (3.197,6.891) -- (3.355,6.633) -- (3.507,6.37) -- (3.653,6.105) -- (3.792,5.836) -- (3.925,5.564) -- (4.052,5.289) -- (4.172,5.011) -- (4.286,4.73);
\draw[line width=0.5pt] (2.125,8.357) -- (2.028,8.08) -- (1.937,7.8) -- (1.853,7.519) -- (1.776,7.235) -- (1.705,6.949) -- (1.641,6.662) -- (1.584,6.374) -- (1.533,6.084) -- (1.49,5.793) -- (1.453,5.501) -- (1.423,5.209) -- (1.4,4.916) -- (1.384,4.622) -- (1.375,4.329);
\draw[line width=0.5pt] (7.489,7.753) -- (7.743,7.789) -- (7.997,7.819) -- (8.252,7.842) -- (8.508,7.86) -- (8.765,7.872) -- (9.021,7.877) -- (9.278,7.877) -- (9.534,7.871) -- (9.79,7.858) -- (10.046,7.84) -- (10.302,7.816) -- (10.556,7.785) -- (10.81,7.749) -- (11.063,7.706);
\draw[line width=0.5pt] (7.489,7.753) -- (7.505,7.589) -- (7.525,7.425) -- (7.55,7.262) -- (7.578,7.099) -- (7.609,6.937) -- (7.645,6.776) -- (7.684,6.616) -- (7.727,6.457) -- (7.774,6.298) -- (7.825,6.141) -- (7.879,5.985) -- (7.937,5.831) -- (7.998,5.678) -- (8.064,5.526);
\draw[line width=0.5pt] (7.489,7.753) -- (7.295,7.504) -- (7.095,7.259) -- (6.889,7.019) -- (6.678,6.784) -- (6.461,6.554) -- (6.239,6.329) -- (6.012,6.109) -- (5.779,5.895) -- (5.542,5.686) -- (5.3,5.483) -- (5.053,5.286) -- (4.801,5.094) -- (4.546,4.909) -- (4.286,4.73);
\draw[line width=0.5pt] (11.063,7.706) -- (11.061,7.545) -- (11.062,7.384) -- (11.067,7.223) -- (11.077,7.062) -- (11.09,6.901) -- (11.106,6.74) -- (11.127,6.58) -- (11.151,6.421) -- (11.179,6.262) -- (11.211,6.103) -- (11.246,5.946) -- (11.285,5.789) -- (11.328,5.634) -- (11.375,5.479);
\draw[line width=0.5pt] (8.064,5.526) -- (8.299,5.559) -- (8.535,5.587) -- (8.771,5.608) -- (9.008,5.624) -- (9.246,5.635) -- (9.483,5.64) -- (9.721,5.64) -- (9.959,5.633) -- (10.196,5.622) -- (10.433,5.604) -- (10.67,5.581) -- (10.906,5.553) -- (11.141,5.519) -- (11.375,5.479);
\draw[line width=0.5pt] (8.064,5.526) -- (8.047,5.288) -- (8.037,5.05) -- (8.032,4.811) -- (8.033,4.572) -- (8.039,4.334) -- (8.051,4.095) -- (8.068,3.857) -- (8.092,3.619) -- (8.12,3.382) -- (8.155,3.146) -- (8.194,2.91) -- (8.24,2.676) -- (8.291,2.442) -- (8.347,2.211);
\draw[line width=0.5pt] (11.375,5.479) -- (11.196,5.215) -- (11.011,4.954) -- (10.82,4.698) -- (10.622,4.446) -- (10.419,4.199) -- (10.21,3.957) -- (9.996,3.72) -- (9.776,3.488) -- (9.55,3.261) -- (9.319,3.04) -- (9.084,2.824) -- (8.843,2.614) -- (8.597,2.409) -- (8.347,2.211);
\draw[line width=0.5pt] (4.286,4.73) -- (4.075,4.733) -- (3.864,4.732) -- (3.653,4.725) -- (3.443,4.714) -- (3.233,4.697) -- (3.023,4.676) -- (2.814,4.649) -- (2.605,4.618) -- (2.397,4.582) -- (2.19,4.541) -- (1.984,4.495) -- (1.78,4.444) -- (1.577,4.389) -- (1.375,4.329);
\draw[line width=0.5pt] (4.286,4.73) -- (4.383,4.513) -- (4.475,4.293) -- (4.562,4.072) -- (4.643,3.848) -- (4.72,3.622) -- (4.791,3.395) -- (4.856,3.166) -- (4.917,2.936) -- (4.972,2.704) -- (5.021,2.471) -- (5.065,2.237) -- (5.103,2.002) -- (5.136,1.766) -- (5.163,1.53);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.285,8.357) -- (2.263,8.437) -- (2.205,8.496) -- (2.125,8.517) -- (2.045,8.496) -- (1.986,8.437) -- (1.965,8.357) -- (1.986,8.277) -- (2.045,8.219) -- (2.125,8.197) -- (2.205,8.219) -- (2.263,8.277) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.649,7.753) -- (7.627,7.833) -- (7.569,7.892) -- (7.489,7.913) -- (7.409,7.892) -- (7.35,7.833) -- (7.329,7.753) -- (7.35,7.673) -- (7.409,7.614) -- (7.489,7.593) -- (7.569,7.614) -- (7.627,7.673) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.223,7.706) -- (11.201,7.786) -- (11.143,7.845) -- (11.063,7.866) -- (10.983,7.845) -- (10.924,7.786) -- (10.903,7.706) -- (10.924,7.626) -- (10.983,7.568) -- (11.063,7.546) -- (11.143,7.568) -- (11.201,7.626) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.224,5.526) -- (8.202,5.606) -- (8.144,5.665) -- (8.064,5.686) -- (7.984,5.665) -- (7.925,5.606) -- (7.904,5.526) -- (7.925,5.446) -- (7.984,5.388) -- (8.064,5.366) -- (8.144,5.388) -- (8.202,5.446) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.535,5.479) -- (11.513,5.559) -- (11.455,5.618) -- (11.375,5.639) -- (11.295,5.618) -- (11.236,5.559) -- (11.215,5.479) -- (11.236,5.399) -- (11.295,5.341) -- (11.375,5.319) -- (11.455,5.341) -- (11.513,5.399) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.446,4.73) -- (4.424,4.81) -- (4.366,4.868) -- (4.286,4.89) -- (4.206,4.868) -- (4.147,4.81) -- (4.126,4.73) -- (4.147,4.65) -- (4.206,4.591) -- (4.286,4.57) -- (4.366,4.591) -- (4.424,4.65) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.535,4.329) -- (1.513,4.409) -- (1.455,4.467) -- (1.375,4.489) -- (1.295,4.467) -- (1.236,4.409) -- (1.215,4.329) -- (1.236,4.249) -- (1.295,4.19) -- (1.375,4.169) -- (1.455,4.19) -- (1.513,4.249) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.507,2.211) -- (8.486,2.291) -- (8.427,2.349) -- (8.347,2.371) -- (8.267,2.349) -- (8.208,2.291) -- (8.187,2.211) -- (8.208,2.131) -- (8.267,2.072) -- (8.347,2.051) -- (8.427,2.072) -- (8.486,2.131) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.323,1.53) -- (5.302,1.61) -- (5.243,1.668) -- (5.163,1.69) -- (5.083,1.668) -- (5.025,1.61) -- (5.003,1.53) -- (5.025,1.45) -- (5.083,1.391) -- (5.163,1.37) -- (5.243,1.391) -- (5.302,1.45) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.524,8.484) -- (4.479,8.654) -- (4.354,8.778) -- (4.184,8.824) -- (4.014,8.778) -- (3.89,8.654) -- (3.844,8.484) -- (3.89,8.314) -- (4.014,8.189) -- (4.184,8.144) -- (4.354,8.189) -- (4.479,8.314) -- cycle;
\node at (4.184,8.484) {\scalebox{1.604}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.97,6.796) -- (3.924,6.966) -- (3.8,7.091) -- (3.63,7.136) -- (3.46,7.091) -- (3.335,6.966) -- (3.29,6.796) -- (3.335,6.626) -- (3.46,6.502) -- (3.63,6.456) -- (3.8,6.502) -- (3.924,6.626) -- cycle;
\node at (3.63,6.796) {\scalebox{1.604}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.238,6.315) -- (2.193,6.485) -- (2.068,6.61) -- (1.898,6.655) -- (1.728,6.61) -- (1.604,6.485) -- (1.558,6.315) -- (1.604,6.145) -- (1.728,6.021) -- (1.898,5.975) -- (2.068,6.021) -- (2.193,6.145) -- cycle;
\node at (1.898,6.315) {\scalebox{1.604}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.622,8.197) -- (9.576,8.367) -- (9.452,8.492) -- (9.282,8.537) -- (9.112,8.492) -- (8.987,8.367) -- (8.942,8.197) -- (8.987,8.027) -- (9.112,7.903) -- (9.282,7.857) -- (9.452,7.903) -- (9.576,8.027) -- cycle;
\node at (9.282,8.197) {\scalebox{1.604}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.334,6.696) -- (8.289,6.866) -- (8.164,6.99) -- (7.994,7.036) -- (7.824,6.99) -- (7.7,6.866) -- (7.654,6.696) -- (7.7,6.526) -- (7.824,6.401) -- (7.994,6.356) -- (8.164,6.401) -- (8.289,6.526) -- cycle;
\node at (7.994,6.696) {\scalebox{1.604}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.571,5.877) -- (6.526,6.047) -- (6.401,6.171) -- (6.231,6.217) -- (6.061,6.171) -- (5.937,6.047) -- (5.891,5.877) -- (5.937,5.707) -- (6.061,5.582) -- (6.231,5.537) -- (6.401,5.582) -- (6.526,5.707) -- cycle;
\node at (6.231,5.877) {\scalebox{1.604}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.784,6.624) -- (11.738,6.794) -- (11.614,6.919) -- (11.444,6.964) -- (11.274,6.919) -- (11.149,6.794) -- (11.104,6.624) -- (11.149,6.454) -- (11.274,6.33) -- (11.444,6.284) -- (11.614,6.33) -- (11.738,6.454) -- cycle;
\node at (11.444,6.624) {\scalebox{1.604}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.066,5.959) -- (10.02,6.129) -- (9.896,6.254) -- (9.726,6.299) -- (9.556,6.254) -- (9.431,6.129) -- (9.386,5.959) -- (9.431,5.789) -- (9.556,5.665) -- (9.726,5.619) -- (9.896,5.665) -- (10.02,5.789) -- cycle;
\node at (9.726,5.959) {\scalebox{1.604}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.727,3.884) -- (8.682,4.054) -- (8.557,4.179) -- (8.387,4.224) -- (8.217,4.179) -- (8.093,4.054) -- (8.047,3.884) -- (8.093,3.714) -- (8.217,3.59) -- (8.387,3.544) -- (8.557,3.59) -- (8.682,3.714) -- cycle;
\node at (8.387,3.884) {\scalebox{1.604}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.57,3.503) -- (10.525,3.673) -- (10.4,3.797) -- (10.23,3.843) -- (10.06,3.797) -- (9.936,3.673) -- (9.89,3.503) -- (9.936,3.333) -- (10.06,3.208) -- (10.23,3.163) -- (10.4,3.208) -- (10.525,3.333) -- cycle;
\node at (10.23,3.503) {\scalebox{1.604}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.197,4.332) -- (3.152,4.502) -- (3.027,4.627) -- (2.857,4.672) -- (2.687,4.627) -- (2.563,4.502) -- (2.517,4.332) -- (2.563,4.162) -- (2.687,4.038) -- (2.857,3.992) -- (3.027,4.038) -- (3.152,4.162) -- cycle;
\node at (2.857,4.332) {\scalebox{1.604}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.505,3.251) -- (5.46,3.421) -- (5.335,3.545) -- (5.165,3.591) -- (4.995,3.545) -- (4.871,3.421) -- (4.825,3.251) -- (4.871,3.081) -- (4.995,2.956) -- (5.165,2.911) -- (5.335,2.956) -- (5.46,3.081) -- cycle;
\node at (5.165,3.251) {\scalebox{1.604}{\small 12}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (11.075,5.179) -- (11.675,5.179) -- (11.675,5.779) -- (11.075,5.779) -- cycle;
\node at (11.375,5.479) {\scalebox{1.604}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (12.434,6.642) -- (13.034,6.642) -- (13.034,7.242) -- (12.434,7.242) -- cycle;
\node at (12.734,6.942) {\scalebox{1.604}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (0.75,6.209) -- (1.35,6.209) -- (1.35,6.809) -- (0.75,6.809) -- cycle;
\node at (1.05,6.509) {\scalebox{1.604}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.517,2.071) -- (5.117,2.071) -- (5.117,2.671) -- (4.517,2.671) -- cycle;
\node at (4.817,2.371) {\scalebox{1.604}{\small d}};
\node at (1.614,8.708) {\scalebox{1.604}{\small A}};
\node at (7.702,8.335) {\scalebox{1.604}{\small B}};
\node at (11.608,8.001) {\scalebox{1.604}{\small C}};
\node at (8.676,5.624) {\scalebox{1.604}{\small D}};
\node at (11.994,5.504) {\scalebox{1.604}{\small E}};
\node at (3.683,4.583) {\scalebox{1.604}{\small F}};
\node at (0.765,4.216) {\scalebox{1.604}{\small G}};
\node at (8.654,1.672) {\scalebox{1.604}{\small H}};
\node at (4.944,0.95) {\scalebox{1.604}{\small I}};
\draw[line width=0.5pt] (12.675,7.407) -- (12.675,8.507);
\draw[line width=0.5pt] (12.444,8.122) -- (12.675,8.507) -- (12.906,8.122);
\draw[line width=0.5pt] (12.372,7.957) -- (12.977,7.957);
\node at (12.675,8.892) {\scalebox{1.604}{\small N}};
\end{tikzpicture}
\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|}
\hline
\cellcolor{blue!7}\bfseries Carrefour & \cellcolor{blue!7}\bfseries A & \cellcolor{blue!7}\bfseries B & \cellcolor{blue!7}\bfseries C & \cellcolor{blue!7}\bfseries D & \cellcolor{blue!7}\bfseries E & \cellcolor{blue!7}\bfseries F & \cellcolor{blue!7}\bfseries G & \cellcolor{blue!7}\bfseries H & \cellcolor{blue!7}\bfseries I \\ \hline
Nombre de rues & 3 & 4 & 2 & 3 & 3 & 4 & 2 & 2 & 1 \\ \hline
Parité & \textbf{impair} & pair & pair & \textbf{impair} & \textbf{impair} & pair & pair & pair & \textbf{impair} \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 24 = 2 \(\times\) 12, 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{A}, \textbf{D}, \textbf{E}, \textbf{I} — 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{E}, 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} (A, D, E, I).}\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}
\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}
