\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 378}

\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=378}\\[2pt]
    {\small\ttfamily\bfseries\color{red} EULC0378}
\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=0.752, line join=round, line cap=round]
\draw[line width=0.5pt] (4.737,11.412) -- (4.991,11.313) -- (5.248,11.22) -- (5.506,11.133) -- (5.767,11.052) -- (6.03,10.978) -- (6.294,10.909) -- (6.56,10.847) -- (6.827,10.791) -- (7.096,10.741) -- (7.365,10.697) -- (7.636,10.66) -- (7.907,10.63) -- (8.179,10.605) -- (8.451,10.588);
\draw[line width=0.5pt] (4.737,11.412) -- (4.796,11.157) -- (4.85,10.9) -- (4.897,10.642) -- (4.938,10.383) -- (4.973,10.123) -- (5.002,9.862) -- (5.025,9.6) -- (5.041,9.338) -- (5.052,9.076) -- (5.056,8.814) -- (5.054,8.551) -- (5.046,8.289) -- (5.032,8.027) -- (5.011,7.766);
\draw[line width=0.5pt] (8.451,10.588) -- (8.564,10.393) -- (8.682,10.201) -- (8.805,10.012) -- (8.932,9.825) -- (9.064,9.642) -- (9.2,9.462) -- (9.34,9.285) -- (9.484,9.111) -- (9.632,8.941) -- (9.784,8.774) -- (9.94,8.611) -- (10.1,8.452) -- (10.263,8.297) -- (10.43,8.145);
\draw[line width=0.5pt] (8.451,10.588) -- (8.238,10.35) -- (8.019,10.117) -- (7.795,9.89) -- (7.566,9.668) -- (7.332,9.451) -- (7.092,9.24) -- (6.847,9.035) -- (6.598,8.835) -- (6.344,8.641) -- (6.086,8.454) -- (5.823,8.272) -- (5.557,8.097) -- (5.286,7.928) -- (5.011,7.766);
\draw[line width=0.5pt] (4.46,1.708) -- (4.248,1.719) -- (4.035,1.725) -- (3.822,1.727) -- (3.609,1.723) -- (3.397,1.714) -- (3.184,1.7) -- (2.972,1.682) -- (2.76,1.658) -- (2.549,1.629) -- (2.339,1.596) -- (2.13,1.557) -- (1.921,1.514) -- (1.714,1.465) -- (1.508,1.412);
\draw[line width=0.5pt] (7.439,8.348) -- (7.274,8.28) -- (7.106,8.216) -- (6.938,8.156) -- (6.767,8.1) -- (6.596,8.048) -- (6.423,8.0) -- (6.249,7.956) -- (6.075,7.917) -- (5.899,7.881) -- (5.722,7.85) -- (5.545,7.822) -- (5.368,7.799) -- (5.19,7.78) -- (5.011,7.766);
\draw[line width=0.5pt] (7.439,8.348) -- (7.406,8.151) -- (7.377,7.954) -- (7.352,7.756) -- (7.332,7.557) -- (7.317,7.358) -- (7.307,7.159) -- (7.301,6.959) -- (7.3,6.759) -- (7.304,6.56) -- (7.312,6.36) -- (7.326,6.161) -- (7.343,5.962) -- (7.366,5.764) -- (7.393,5.567);
\draw[line width=0.5pt] (10.43,8.145) -- (10.243,7.929) -- (10.051,7.718) -- (9.854,7.511) -- (9.652,7.309) -- (9.445,7.111) -- (9.234,6.918) -- (9.018,6.731) -- (8.798,6.548) -- (8.573,6.371) -- (8.345,6.199) -- (8.112,6.032) -- (7.876,5.871) -- (7.636,5.716) -- (7.393,5.567);
\draw[line width=0.5pt] (10.43,8.145) -- (10.399,7.955) -- (10.373,7.763) -- (10.351,7.572) -- (10.334,7.379) -- (10.321,7.186) -- (10.313,6.993) -- (10.31,6.8) -- (10.31,6.606) -- (10.316,6.413) -- (10.326,6.22) -- (10.34,6.027) -- (10.359,5.835) -- (10.383,5.644) -- (10.411,5.453);
\draw[line width=0.5pt] (2.322,7.885) -- (2.514,7.906) -- (2.706,7.923) -- (2.899,7.934) -- (3.092,7.942) -- (3.285,7.945) -- (3.478,7.943) -- (3.671,7.936) -- (3.864,7.926) -- (4.057,7.91) -- (4.249,7.89) -- (4.441,7.866) -- (4.632,7.837) -- (4.822,7.804) -- (5.011,7.766);
\draw[line width=0.5pt] (2.322,7.885) -- (2.265,7.735) -- (2.211,7.584) -- (2.162,7.431) -- (2.116,7.278) -- (2.073,7.123) -- (2.034,6.967) -- (1.999,6.81) -- (1.968,6.653) -- (1.94,6.495) -- (1.916,6.336) -- (1.895,6.177) -- (1.879,6.017) -- (1.866,5.857) -- (1.857,5.697);
\draw[line width=0.5pt] (5.011,7.766) -- (5.031,7.617) -- (5.046,7.469) -- (5.058,7.32) -- (5.067,7.17) -- (5.072,7.021) -- (5.074,6.871) -- (5.072,6.721) -- (5.067,6.572) -- (5.058,6.422) -- (5.046,6.273) -- (5.03,6.124) -- (5.01,5.976) -- (4.987,5.828) -- (4.961,5.681) -- (4.882,5.405) -- (4.81,5.127) -- (4.744,4.847) -- (4.685,4.566) -- (4.632,4.284) -- (4.586,4.0) -- (4.547,3.715) -- (4.514,3.429) -- (4.488,3.143) -- (4.469,2.856) -- (4.457,2.569) -- (4.451,2.282) -- (4.452,1.995) -- (4.46,1.708);
\draw[line width=0.5pt] (5.011,7.766) -- (4.765,7.654) -- (4.522,7.536) -- (4.281,7.412) -- (4.043,7.283) -- (3.808,7.148) -- (3.577,7.007) -- (3.349,6.861) -- (3.124,6.71) -- (2.903,6.554) -- (2.686,6.392) -- (2.472,6.226) -- (2.263,6.054) -- (2.058,5.878) -- (1.857,5.697);
\draw[line width=0.5pt] (5.011,7.766) -- (5.204,7.636) -- (5.394,7.502) -- (5.58,7.363) -- (5.764,7.22) -- (5.944,7.072) -- (6.12,6.92) -- (6.293,6.764) -- (6.462,6.605) -- (6.627,6.441) -- (6.789,6.273) -- (6.946,6.102) -- (7.099,5.927) -- (7.248,5.749) -- (7.393,5.567);
\draw[line width=0.5pt] (1.857,5.697) -- (1.998,5.386) -- (2.146,5.078) -- (2.302,4.774) -- (2.464,4.473) -- (2.634,4.176) -- (2.811,3.883) -- (2.994,3.595) -- (3.184,3.311) -- (3.381,3.031) -- (3.584,2.756) -- (3.794,2.486) -- (4.01,2.221) -- (4.232,1.962) -- (4.46,1.708);
\draw[line width=0.5pt] (1.857,5.697) -- (1.879,5.389) -- (1.894,5.081) -- (1.902,4.773) -- (1.902,4.465) -- (1.895,4.156) -- (1.881,3.848) -- (1.859,3.54) -- (1.831,3.233) -- (1.795,2.926) -- (1.752,2.621) -- (1.702,2.317) -- (1.644,2.013) -- (1.58,1.712) -- (1.508,1.412);
\draw[line width=0.5pt] (7.393,5.567) -- (7.143,5.325) -- (6.899,5.078) -- (6.66,4.825) -- (6.428,4.566) -- (6.201,4.302) -- (5.981,4.033) -- (5.767,3.758) -- (5.56,3.479) -- (5.36,3.194) -- (5.166,2.906) -- (4.979,2.612) -- (4.799,2.315) -- (4.626,2.013) -- (4.46,1.708);
\draw[line width=0.5pt] (7.393,5.567) -- (7.608,5.592) -- (7.824,5.611) -- (8.04,5.626) -- (8.256,5.636) -- (8.473,5.64) -- (8.69,5.64) -- (8.907,5.634) -- (9.123,5.623) -- (9.34,5.608) -- (9.555,5.587) -- (9.771,5.561) -- (9.985,5.53) -- (10.199,5.494) -- (10.411,5.453);
\draw[line width=0.5pt] (7.393,5.567) -- (7.35,5.347) -- (7.312,5.126) -- (7.279,4.905) -- (7.252,4.682) -- (7.23,4.459) -- (7.213,4.236) -- (7.201,4.012) -- (7.194,3.788) -- (7.193,3.564) -- (7.197,3.34) -- (7.207,3.116) -- (7.221,2.892) -- (7.241,2.669) -- (7.266,2.447);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.897,11.412) -- (4.875,11.492) -- (4.817,11.551) -- (4.737,11.572) -- (4.657,11.551) -- (4.598,11.492) -- (4.577,11.412) -- (4.598,11.332) -- (4.657,11.274) -- (4.737,11.252) -- (4.817,11.274) -- (4.875,11.332) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.611,10.588) -- (8.589,10.668) -- (8.531,10.726) -- (8.451,10.748) -- (8.371,10.726) -- (8.312,10.668) -- (8.291,10.588) -- (8.312,10.508) -- (8.371,10.449) -- (8.451,10.428) -- (8.531,10.449) -- (8.589,10.508) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.599,8.348) -- (7.578,8.428) -- (7.519,8.486) -- (7.439,8.508) -- (7.359,8.486) -- (7.301,8.428) -- (7.279,8.348) -- (7.301,8.268) -- (7.359,8.209) -- (7.439,8.188) -- (7.519,8.209) -- (7.578,8.268) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.59,8.145) -- (10.569,8.225) -- (10.51,8.284) -- (10.43,8.305) -- (10.35,8.284) -- (10.291,8.225) -- (10.27,8.145) -- (10.291,8.065) -- (10.35,8.007) -- (10.43,7.985) -- (10.51,8.007) -- (10.569,8.065) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.482,7.885) -- (2.46,7.965) -- (2.402,8.024) -- (2.322,8.045) -- (2.242,8.024) -- (2.183,7.965) -- (2.162,7.885) -- (2.183,7.805) -- (2.242,7.747) -- (2.322,7.725) -- (2.402,7.747) -- (2.46,7.805) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.171,7.766) -- (5.15,7.846) -- (5.091,7.904) -- (5.011,7.926) -- (4.931,7.904) -- (4.873,7.846) -- (4.851,7.766) -- (4.873,7.686) -- (4.931,7.627) -- (5.011,7.606) -- (5.091,7.627) -- (5.15,7.686) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.017,5.697) -- (1.996,5.777) -- (1.937,5.835) -- (1.857,5.857) -- (1.777,5.835) -- (1.718,5.777) -- (1.697,5.697) -- (1.718,5.617) -- (1.777,5.558) -- (1.857,5.537) -- (1.937,5.558) -- (1.996,5.617) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.553,5.567) -- (7.531,5.647) -- (7.473,5.705) -- (7.393,5.727) -- (7.313,5.705) -- (7.254,5.647) -- (7.233,5.567) -- (7.254,5.487) -- (7.313,5.428) -- (7.393,5.407) -- (7.473,5.428) -- (7.531,5.487) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.571,5.453) -- (10.55,5.533) -- (10.491,5.591) -- (10.411,5.613) -- (10.331,5.591) -- (10.273,5.533) -- (10.251,5.453) -- (10.273,5.373) -- (10.331,5.314) -- (10.411,5.293) -- (10.491,5.314) -- (10.55,5.373) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.426,2.447) -- (7.405,2.527) -- (7.346,2.585) -- (7.266,2.607) -- (7.186,2.585) -- (7.128,2.527) -- (7.106,2.447) -- (7.128,2.367) -- (7.186,2.308) -- (7.266,2.287) -- (7.346,2.308) -- (7.405,2.367) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.62,1.708) -- (4.599,1.788) -- (4.54,1.846) -- (4.46,1.868) -- (4.38,1.846) -- (4.322,1.788) -- (4.3,1.708) -- (4.322,1.628) -- (4.38,1.569) -- (4.46,1.548) -- (4.54,1.569) -- (4.599,1.628) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.668,1.412) -- (1.647,1.492) -- (1.588,1.551) -- (1.508,1.572) -- (1.428,1.551) -- (1.37,1.492) -- (1.348,1.412) -- (1.37,1.332) -- (1.428,1.274) -- (1.508,1.252) -- (1.588,1.274) -- (1.647,1.332) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.969,11.159) -- (6.924,11.329) -- (6.799,11.454) -- (6.629,11.499) -- (6.459,11.454) -- (6.335,11.329) -- (6.289,11.159) -- (6.335,10.989) -- (6.459,10.865) -- (6.629,10.819) -- (6.799,10.865) -- (6.924,10.989) -- cycle;
\node at (6.629,11.159) {\scalebox{1.33}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.684,9.624) -- (5.638,9.794) -- (5.514,9.919) -- (5.344,9.964) -- (5.174,9.919) -- (5.049,9.794) -- (5.004,9.624) -- (5.049,9.454) -- (5.174,9.33) -- (5.344,9.284) -- (5.514,9.33) -- (5.638,9.454) -- cycle;
\node at (5.344,9.624) {\scalebox{1.33}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.928,9.486) -- (9.883,9.656) -- (9.758,9.781) -- (9.588,9.826) -- (9.418,9.781) -- (9.294,9.656) -- (9.248,9.486) -- (9.294,9.316) -- (9.418,9.192) -- (9.588,9.146) -- (9.758,9.192) -- (9.883,9.316) -- cycle;
\node at (9.588,9.486) {\scalebox{1.33}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.984,9.282) -- (6.939,9.452) -- (6.814,9.577) -- (6.644,9.622) -- (6.474,9.577) -- (6.35,9.452) -- (6.304,9.282) -- (6.35,9.112) -- (6.474,8.988) -- (6.644,8.942) -- (6.814,8.988) -- (6.939,9.112) -- cycle;
\node at (6.644,9.282) {\scalebox{1.33}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.344,1.363) -- (3.298,1.533) -- (3.174,1.658) -- (3.004,1.703) -- (2.834,1.658) -- (2.71,1.533) -- (2.664,1.363) -- (2.71,1.193) -- (2.834,1.069) -- (3.004,1.023) -- (3.174,1.069) -- (3.298,1.193) -- cycle;
\node at (3.004,1.363) {\scalebox{1.33}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.664,7.645) -- (6.618,7.815) -- (6.494,7.94) -- (6.324,7.985) -- (6.154,7.94) -- (6.029,7.815) -- (5.984,7.645) -- (6.029,7.475) -- (6.154,7.351) -- (6.324,7.305) -- (6.494,7.351) -- (6.618,7.475) -- cycle;
\node at (6.324,7.645) {\scalebox{1.33}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.961,6.954) -- (7.916,7.124) -- (7.791,7.248) -- (7.621,7.294) -- (7.451,7.248) -- (7.327,7.124) -- (7.281,6.954) -- (7.327,6.784) -- (7.451,6.659) -- (7.621,6.614) -- (7.791,6.659) -- (7.916,6.784) -- cycle;
\node at (7.621,6.954) {\scalebox{1.33}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.565,6.487) -- (9.519,6.657) -- (9.395,6.781) -- (9.225,6.827) -- (9.055,6.781) -- (8.931,6.657) -- (8.885,6.487) -- (8.931,6.317) -- (9.055,6.192) -- (9.225,6.147) -- (9.395,6.192) -- (9.519,6.317) -- cycle;
\node at (9.225,6.487) {\scalebox{1.33}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.97,6.798) -- (10.924,6.968) -- (10.8,7.092) -- (10.63,7.138) -- (10.46,7.092) -- (10.335,6.968) -- (10.29,6.798) -- (10.335,6.628) -- (10.46,6.503) -- (10.63,6.458) -- (10.8,6.503) -- (10.924,6.628) -- cycle;
\node at (10.63,6.798) {\scalebox{1.33}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.026,8.256) -- (3.98,8.426) -- (3.856,8.551) -- (3.686,8.596) -- (3.516,8.551) -- (3.391,8.426) -- (3.346,8.256) -- (3.391,8.086) -- (3.516,7.962) -- (3.686,7.916) -- (3.856,7.962) -- (3.98,8.086) -- cycle;
\node at (3.686,8.256) {\scalebox{1.33}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.652,6.744) -- (2.607,6.914) -- (2.482,7.038) -- (2.312,7.084) -- (2.142,7.038) -- (2.018,6.914) -- (1.972,6.744) -- (2.018,6.574) -- (2.142,6.449) -- (2.312,6.404) -- (2.482,6.449) -- (2.607,6.574) -- cycle;
\node at (2.312,6.744) {\scalebox{1.33}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.612,5.604) -- (5.566,5.774) -- (5.442,5.898) -- (5.272,5.944) -- (5.102,5.898) -- (4.977,5.774) -- (4.932,5.604) -- (4.977,5.434) -- (5.102,5.309) -- (5.272,5.264) -- (5.442,5.309) -- (5.566,5.434) -- cycle;
\node at (5.272,5.604) {\scalebox{1.33}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.864,6.594) -- (3.819,6.764) -- (3.694,6.888) -- (3.524,6.934) -- (3.354,6.888) -- (3.23,6.764) -- (3.184,6.594) -- (3.23,6.424) -- (3.354,6.299) -- (3.524,6.254) -- (3.694,6.299) -- (3.819,6.424) -- cycle;
\node at (3.524,6.594) {\scalebox{1.33}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.416,6.529) -- (6.37,6.699) -- (6.246,6.824) -- (6.076,6.869) -- (5.906,6.824) -- (5.781,6.699) -- (5.736,6.529) -- (5.781,6.359) -- (5.906,6.235) -- (6.076,6.189) -- (6.246,6.235) -- (6.37,6.359) -- cycle;
\node at (6.076,6.529) {\scalebox{1.33}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.602,3.77) -- (3.557,3.94) -- (3.432,4.064) -- (3.262,4.11) -- (3.092,4.064) -- (2.968,3.94) -- (2.922,3.77) -- (2.968,3.6) -- (3.092,3.475) -- (3.262,3.43) -- (3.432,3.475) -- (3.557,3.6) -- cycle;
\node at (3.262,3.77) {\scalebox{1.33}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.518,3.514) -- (2.473,3.684) -- (2.348,3.809) -- (2.178,3.854) -- (2.008,3.809) -- (1.884,3.684) -- (1.838,3.514) -- (1.884,3.344) -- (2.008,3.22) -- (2.178,3.174) -- (2.348,3.22) -- (2.473,3.344) -- cycle;
\node at (2.178,3.514) {\scalebox{1.33}{\small 16}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.362,3.565) -- (6.317,3.735) -- (6.192,3.859) -- (6.022,3.905) -- (5.852,3.859) -- (5.728,3.735) -- (5.682,3.565) -- (5.728,3.395) -- (5.852,3.27) -- (6.022,3.225) -- (6.192,3.27) -- (6.317,3.395) -- cycle;
\node at (6.022,3.565) {\scalebox{1.33}{\small 17}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.235,5.314) -- (9.189,5.484) -- (9.065,5.609) -- (8.895,5.654) -- (8.725,5.609) -- (8.6,5.484) -- (8.555,5.314) -- (8.6,5.144) -- (8.725,5.02) -- (8.895,4.974) -- (9.065,5.02) -- (9.189,5.144) -- cycle;
\node at (8.895,5.314) {\scalebox{1.33}{\small 18}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.861,3.999) -- (7.815,4.169) -- (7.691,4.293) -- (7.521,4.339) -- (7.351,4.293) -- (7.226,4.169) -- (7.181,3.999) -- (7.226,3.829) -- (7.351,3.704) -- (7.521,3.659) -- (7.691,3.704) -- (7.815,3.829) -- cycle;
\node at (7.521,3.999) {\scalebox{1.33}{\small 19}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (6.966,2.147) -- (7.566,2.147) -- (7.566,2.747) -- (6.966,2.747) -- cycle;
\node at (7.266,2.447) {\scalebox{1.33}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.074,6.012) -- (9.674,6.012) -- (9.674,6.612) -- (9.074,6.612) -- cycle;
\node at (9.374,6.312) {\scalebox{1.33}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (6.078,7.122) -- (6.678,7.122) -- (6.678,7.722) -- (6.078,7.722) -- cycle;
\node at (6.378,7.422) {\scalebox{1.33}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (6.896,8.31) -- (7.496,8.31) -- (7.496,8.91) -- (6.896,8.91) -- cycle;
\node at (7.196,8.61) {\scalebox{1.33}{\small d}};
\node at (4.593,12.015) {\scalebox{1.33}{\small S0}};
\node at (8.768,11.12) {\scalebox{1.33}{\small S1}};
\node at (7.814,8.842) {\scalebox{1.33}{\small S2}};
\node at (11.006,8.374) {\scalebox{1.33}{\small S3}};
\node at (1.75,8.125) {\scalebox{1.33}{\small S4}};
\node at (4.668,8.282) {\scalebox{1.33}{\small S5}};
\node at (1.245,5.596) {\scalebox{1.33}{\small S6}};
\node at (7.936,5.267) {\scalebox{1.33}{\small S7}};
\node at (11.019,5.328) {\scalebox{1.33}{\small S8}};
\node at (7.465,1.859) {\scalebox{1.33}{\small S9}};
\node at (4.273,1.117) {\scalebox{1.33}{\small S10}};
\node at (1.095,0.95) {\scalebox{1.33}{\small S11}};
\draw[line width=0.5pt] (11.73,10.462) -- (11.73,11.562);
\draw[line width=0.5pt] (11.499,11.177) -- (11.73,11.562) -- (11.961,11.177);
\draw[line width=0.5pt] (11.427,11.012) -- (12.032,11.012);
\node at (11.73,11.947) {\scalebox{1.33}{\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 rue Pasteur & ⑪ allée des Tilleuls & S0 Place aux Herbes & \textbf{a} la poste \\ \hline
② rue Neuve & ⑫ rue du Four & S1 Rond-point des Vaches & \textbf{b} la pharmacie \\ \hline
③ avenue de la Gare & ⑬ chemin Creux & S2 Place Sainte-Croix & \textbf{c} le gymnase \\ \hline
④ boulevard du Parc & ⑭ rue du Pont & S3 Rond-point du Pont & \textbf{d} la bibliothèque \\ \hline
⑤ rue du Moulin & ⑮ passage du Marché & S4 Place des Tanneurs &  \\ \hline
⑥ rue Haute & ⑯ venelle du Chat & S5 Carrefour du Lavoir &  \\ \hline
⑦ quai aux Fleurs & ⑰ sentier des Prés & S6 Rond-point du Stade &  \\ \hline
⑧ chemin de Ronde & ⑱ rue des Lilas & S7 Carrefour du Moulin &  \\ \hline
⑨ chemin du Lavoir & ⑲ impasse du Puits & S8 Placette des Ormes &  \\ \hline
⑩ chemin des Vignes &  & S9 Place de l'Église &  \\ \hline
 &  & S10 Carrefour des Tilleuls &  \\ \hline
 &  & S11 Coin du Lavoir &  \\ \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=0.6729, line join=round, line cap=round]
\draw[line width=0.5pt] (4.737,11.412) -- (4.991,11.313) -- (5.248,11.22) -- (5.506,11.133) -- (5.767,11.052) -- (6.03,10.978) -- (6.294,10.909) -- (6.56,10.847) -- (6.827,10.791) -- (7.096,10.741) -- (7.365,10.697) -- (7.636,10.66) -- (7.907,10.63) -- (8.179,10.605) -- (8.451,10.588);
\draw[line width=0.5pt] (4.737,11.412) -- (4.796,11.157) -- (4.85,10.9) -- (4.897,10.642) -- (4.938,10.383) -- (4.973,10.123) -- (5.002,9.862) -- (5.025,9.6) -- (5.041,9.338) -- (5.052,9.076) -- (5.056,8.814) -- (5.054,8.551) -- (5.046,8.289) -- (5.032,8.027) -- (5.011,7.766);
\draw[line width=0.5pt] (8.451,10.588) -- (8.564,10.393) -- (8.682,10.201) -- (8.805,10.012) -- (8.932,9.825) -- (9.064,9.642) -- (9.2,9.462) -- (9.34,9.285) -- (9.484,9.111) -- (9.632,8.941) -- (9.784,8.774) -- (9.94,8.611) -- (10.1,8.452) -- (10.263,8.297) -- (10.43,8.145);
\draw[line width=0.5pt] (8.451,10.588) -- (8.238,10.35) -- (8.019,10.117) -- (7.795,9.89) -- (7.566,9.668) -- (7.332,9.451) -- (7.092,9.24) -- (6.847,9.035) -- (6.598,8.835) -- (6.344,8.641) -- (6.086,8.454) -- (5.823,8.272) -- (5.557,8.097) -- (5.286,7.928) -- (5.011,7.766);
\draw[line width=0.5pt] (4.46,1.708) -- (4.248,1.719) -- (4.035,1.725) -- (3.822,1.727) -- (3.609,1.723) -- (3.397,1.714) -- (3.184,1.7) -- (2.972,1.682) -- (2.76,1.658) -- (2.549,1.629) -- (2.339,1.596) -- (2.13,1.557) -- (1.921,1.514) -- (1.714,1.465) -- (1.508,1.412);
\draw[line width=0.5pt] (7.439,8.348) -- (7.274,8.28) -- (7.106,8.216) -- (6.938,8.156) -- (6.767,8.1) -- (6.596,8.048) -- (6.423,8.0) -- (6.249,7.956) -- (6.075,7.917) -- (5.899,7.881) -- (5.722,7.85) -- (5.545,7.822) -- (5.368,7.799) -- (5.19,7.78) -- (5.011,7.766);
\draw[line width=0.5pt] (7.439,8.348) -- (7.406,8.151) -- (7.377,7.954) -- (7.352,7.756) -- (7.332,7.557) -- (7.317,7.358) -- (7.307,7.159) -- (7.301,6.959) -- (7.3,6.759) -- (7.304,6.56) -- (7.312,6.36) -- (7.326,6.161) -- (7.343,5.962) -- (7.366,5.764) -- (7.393,5.567);
\draw[line width=0.5pt] (10.43,8.145) -- (10.243,7.929) -- (10.051,7.718) -- (9.854,7.511) -- (9.652,7.309) -- (9.445,7.111) -- (9.234,6.918) -- (9.018,6.731) -- (8.798,6.548) -- (8.573,6.371) -- (8.345,6.199) -- (8.112,6.032) -- (7.876,5.871) -- (7.636,5.716) -- (7.393,5.567);
\draw[line width=0.5pt] (10.43,8.145) -- (10.399,7.955) -- (10.373,7.763) -- (10.351,7.572) -- (10.334,7.379) -- (10.321,7.186) -- (10.313,6.993) -- (10.31,6.8) -- (10.31,6.606) -- (10.316,6.413) -- (10.326,6.22) -- (10.34,6.027) -- (10.359,5.835) -- (10.383,5.644) -- (10.411,5.453);
\draw[line width=0.5pt] (2.322,7.885) -- (2.514,7.906) -- (2.706,7.923) -- (2.899,7.934) -- (3.092,7.942) -- (3.285,7.945) -- (3.478,7.943) -- (3.671,7.936) -- (3.864,7.926) -- (4.057,7.91) -- (4.249,7.89) -- (4.441,7.866) -- (4.632,7.837) -- (4.822,7.804) -- (5.011,7.766);
\draw[line width=0.5pt] (2.322,7.885) -- (2.265,7.735) -- (2.211,7.584) -- (2.162,7.431) -- (2.116,7.278) -- (2.073,7.123) -- (2.034,6.967) -- (1.999,6.81) -- (1.968,6.653) -- (1.94,6.495) -- (1.916,6.336) -- (1.895,6.177) -- (1.879,6.017) -- (1.866,5.857) -- (1.857,5.697);
\draw[line width=0.5pt] (5.011,7.766) -- (5.031,7.617) -- (5.046,7.469) -- (5.058,7.32) -- (5.067,7.17) -- (5.072,7.021) -- (5.074,6.871) -- (5.072,6.721) -- (5.067,6.572) -- (5.058,6.422) -- (5.046,6.273) -- (5.03,6.124) -- (5.01,5.976) -- (4.987,5.828) -- (4.961,5.681) -- (4.882,5.405) -- (4.81,5.127) -- (4.744,4.847) -- (4.685,4.566) -- (4.632,4.284) -- (4.586,4.0) -- (4.547,3.715) -- (4.514,3.429) -- (4.488,3.143) -- (4.469,2.856) -- (4.457,2.569) -- (4.451,2.282) -- (4.452,1.995) -- (4.46,1.708);
\draw[line width=0.5pt] (5.011,7.766) -- (4.765,7.654) -- (4.522,7.536) -- (4.281,7.412) -- (4.043,7.283) -- (3.808,7.148) -- (3.577,7.007) -- (3.349,6.861) -- (3.124,6.71) -- (2.903,6.554) -- (2.686,6.392) -- (2.472,6.226) -- (2.263,6.054) -- (2.058,5.878) -- (1.857,5.697);
\draw[line width=0.5pt] (5.011,7.766) -- (5.204,7.636) -- (5.394,7.502) -- (5.58,7.363) -- (5.764,7.22) -- (5.944,7.072) -- (6.12,6.92) -- (6.293,6.764) -- (6.462,6.605) -- (6.627,6.441) -- (6.789,6.273) -- (6.946,6.102) -- (7.099,5.927) -- (7.248,5.749) -- (7.393,5.567);
\draw[line width=0.5pt] (1.857,5.697) -- (1.998,5.386) -- (2.146,5.078) -- (2.302,4.774) -- (2.464,4.473) -- (2.634,4.176) -- (2.811,3.883) -- (2.994,3.595) -- (3.184,3.311) -- (3.381,3.031) -- (3.584,2.756) -- (3.794,2.486) -- (4.01,2.221) -- (4.232,1.962) -- (4.46,1.708);
\draw[line width=0.5pt] (1.857,5.697) -- (1.879,5.389) -- (1.894,5.081) -- (1.902,4.773) -- (1.902,4.465) -- (1.895,4.156) -- (1.881,3.848) -- (1.859,3.54) -- (1.831,3.233) -- (1.795,2.926) -- (1.752,2.621) -- (1.702,2.317) -- (1.644,2.013) -- (1.58,1.712) -- (1.508,1.412);
\draw[line width=0.5pt] (7.393,5.567) -- (7.143,5.325) -- (6.899,5.078) -- (6.66,4.825) -- (6.428,4.566) -- (6.201,4.302) -- (5.981,4.033) -- (5.767,3.758) -- (5.56,3.479) -- (5.36,3.194) -- (5.166,2.906) -- (4.979,2.612) -- (4.799,2.315) -- (4.626,2.013) -- (4.46,1.708);
\draw[line width=0.5pt] (7.393,5.567) -- (7.608,5.592) -- (7.824,5.611) -- (8.04,5.626) -- (8.256,5.636) -- (8.473,5.64) -- (8.69,5.64) -- (8.907,5.634) -- (9.123,5.623) -- (9.34,5.608) -- (9.555,5.587) -- (9.771,5.561) -- (9.985,5.53) -- (10.199,5.494) -- (10.411,5.453);
\draw[line width=0.5pt] (7.393,5.567) -- (7.35,5.347) -- (7.312,5.126) -- (7.279,4.905) -- (7.252,4.682) -- (7.23,4.459) -- (7.213,4.236) -- (7.201,4.012) -- (7.194,3.788) -- (7.193,3.564) -- (7.197,3.34) -- (7.207,3.116) -- (7.221,2.892) -- (7.241,2.669) -- (7.266,2.447);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.897,11.412) -- (4.875,11.492) -- (4.817,11.551) -- (4.737,11.572) -- (4.657,11.551) -- (4.598,11.492) -- (4.577,11.412) -- (4.598,11.332) -- (4.657,11.274) -- (4.737,11.252) -- (4.817,11.274) -- (4.875,11.332) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.611,10.588) -- (8.589,10.668) -- (8.531,10.726) -- (8.451,10.748) -- (8.371,10.726) -- (8.312,10.668) -- (8.291,10.588) -- (8.312,10.508) -- (8.371,10.449) -- (8.451,10.428) -- (8.531,10.449) -- (8.589,10.508) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.599,8.348) -- (7.578,8.428) -- (7.519,8.486) -- (7.439,8.508) -- (7.359,8.486) -- (7.301,8.428) -- (7.279,8.348) -- (7.301,8.268) -- (7.359,8.209) -- (7.439,8.188) -- (7.519,8.209) -- (7.578,8.268) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.59,8.145) -- (10.569,8.225) -- (10.51,8.284) -- (10.43,8.305) -- (10.35,8.284) -- (10.291,8.225) -- (10.27,8.145) -- (10.291,8.065) -- (10.35,8.007) -- (10.43,7.985) -- (10.51,8.007) -- (10.569,8.065) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.482,7.885) -- (2.46,7.965) -- (2.402,8.024) -- (2.322,8.045) -- (2.242,8.024) -- (2.183,7.965) -- (2.162,7.885) -- (2.183,7.805) -- (2.242,7.747) -- (2.322,7.725) -- (2.402,7.747) -- (2.46,7.805) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.171,7.766) -- (5.15,7.846) -- (5.091,7.904) -- (5.011,7.926) -- (4.931,7.904) -- (4.873,7.846) -- (4.851,7.766) -- (4.873,7.686) -- (4.931,7.627) -- (5.011,7.606) -- (5.091,7.627) -- (5.15,7.686) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.017,5.697) -- (1.996,5.777) -- (1.937,5.835) -- (1.857,5.857) -- (1.777,5.835) -- (1.718,5.777) -- (1.697,5.697) -- (1.718,5.617) -- (1.777,5.558) -- (1.857,5.537) -- (1.937,5.558) -- (1.996,5.617) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.553,5.567) -- (7.531,5.647) -- (7.473,5.705) -- (7.393,5.727) -- (7.313,5.705) -- (7.254,5.647) -- (7.233,5.567) -- (7.254,5.487) -- (7.313,5.428) -- (7.393,5.407) -- (7.473,5.428) -- (7.531,5.487) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.571,5.453) -- (10.55,5.533) -- (10.491,5.591) -- (10.411,5.613) -- (10.331,5.591) -- (10.273,5.533) -- (10.251,5.453) -- (10.273,5.373) -- (10.331,5.314) -- (10.411,5.293) -- (10.491,5.314) -- (10.55,5.373) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.426,2.447) -- (7.405,2.527) -- (7.346,2.585) -- (7.266,2.607) -- (7.186,2.585) -- (7.128,2.527) -- (7.106,2.447) -- (7.128,2.367) -- (7.186,2.308) -- (7.266,2.287) -- (7.346,2.308) -- (7.405,2.367) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.62,1.708) -- (4.599,1.788) -- (4.54,1.846) -- (4.46,1.868) -- (4.38,1.846) -- (4.322,1.788) -- (4.3,1.708) -- (4.322,1.628) -- (4.38,1.569) -- (4.46,1.548) -- (4.54,1.569) -- (4.599,1.628) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.668,1.412) -- (1.647,1.492) -- (1.588,1.551) -- (1.508,1.572) -- (1.428,1.551) -- (1.37,1.492) -- (1.348,1.412) -- (1.37,1.332) -- (1.428,1.274) -- (1.508,1.252) -- (1.588,1.274) -- (1.647,1.332) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.969,11.159) -- (6.924,11.329) -- (6.799,11.454) -- (6.629,11.499) -- (6.459,11.454) -- (6.335,11.329) -- (6.289,11.159) -- (6.335,10.989) -- (6.459,10.865) -- (6.629,10.819) -- (6.799,10.865) -- (6.924,10.989) -- cycle;
\node at (6.629,11.159) {\scalebox{1.486}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.684,9.624) -- (5.638,9.794) -- (5.514,9.919) -- (5.344,9.964) -- (5.174,9.919) -- (5.049,9.794) -- (5.004,9.624) -- (5.049,9.454) -- (5.174,9.33) -- (5.344,9.284) -- (5.514,9.33) -- (5.638,9.454) -- cycle;
\node at (5.344,9.624) {\scalebox{1.486}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.928,9.486) -- (9.883,9.656) -- (9.758,9.781) -- (9.588,9.826) -- (9.418,9.781) -- (9.294,9.656) -- (9.248,9.486) -- (9.294,9.316) -- (9.418,9.192) -- (9.588,9.146) -- (9.758,9.192) -- (9.883,9.316) -- cycle;
\node at (9.588,9.486) {\scalebox{1.486}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.984,9.282) -- (6.939,9.452) -- (6.814,9.577) -- (6.644,9.622) -- (6.474,9.577) -- (6.35,9.452) -- (6.304,9.282) -- (6.35,9.112) -- (6.474,8.988) -- (6.644,8.942) -- (6.814,8.988) -- (6.939,9.112) -- cycle;
\node at (6.644,9.282) {\scalebox{1.486}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.344,1.363) -- (3.298,1.533) -- (3.174,1.658) -- (3.004,1.703) -- (2.834,1.658) -- (2.71,1.533) -- (2.664,1.363) -- (2.71,1.193) -- (2.834,1.069) -- (3.004,1.023) -- (3.174,1.069) -- (3.298,1.193) -- cycle;
\node at (3.004,1.363) {\scalebox{1.486}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.664,7.645) -- (6.618,7.815) -- (6.494,7.94) -- (6.324,7.985) -- (6.154,7.94) -- (6.029,7.815) -- (5.984,7.645) -- (6.029,7.475) -- (6.154,7.351) -- (6.324,7.305) -- (6.494,7.351) -- (6.618,7.475) -- cycle;
\node at (6.324,7.645) {\scalebox{1.486}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.961,6.954) -- (7.916,7.124) -- (7.791,7.248) -- (7.621,7.294) -- (7.451,7.248) -- (7.327,7.124) -- (7.281,6.954) -- (7.327,6.784) -- (7.451,6.659) -- (7.621,6.614) -- (7.791,6.659) -- (7.916,6.784) -- cycle;
\node at (7.621,6.954) {\scalebox{1.486}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.565,6.487) -- (9.519,6.657) -- (9.395,6.781) -- (9.225,6.827) -- (9.055,6.781) -- (8.931,6.657) -- (8.885,6.487) -- (8.931,6.317) -- (9.055,6.192) -- (9.225,6.147) -- (9.395,6.192) -- (9.519,6.317) -- cycle;
\node at (9.225,6.487) {\scalebox{1.486}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.97,6.798) -- (10.924,6.968) -- (10.8,7.092) -- (10.63,7.138) -- (10.46,7.092) -- (10.335,6.968) -- (10.29,6.798) -- (10.335,6.628) -- (10.46,6.503) -- (10.63,6.458) -- (10.8,6.503) -- (10.924,6.628) -- cycle;
\node at (10.63,6.798) {\scalebox{1.486}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.026,8.256) -- (3.98,8.426) -- (3.856,8.551) -- (3.686,8.596) -- (3.516,8.551) -- (3.391,8.426) -- (3.346,8.256) -- (3.391,8.086) -- (3.516,7.962) -- (3.686,7.916) -- (3.856,7.962) -- (3.98,8.086) -- cycle;
\node at (3.686,8.256) {\scalebox{1.486}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.652,6.744) -- (2.607,6.914) -- (2.482,7.038) -- (2.312,7.084) -- (2.142,7.038) -- (2.018,6.914) -- (1.972,6.744) -- (2.018,6.574) -- (2.142,6.449) -- (2.312,6.404) -- (2.482,6.449) -- (2.607,6.574) -- cycle;
\node at (2.312,6.744) {\scalebox{1.486}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.612,5.604) -- (5.566,5.774) -- (5.442,5.898) -- (5.272,5.944) -- (5.102,5.898) -- (4.977,5.774) -- (4.932,5.604) -- (4.977,5.434) -- (5.102,5.309) -- (5.272,5.264) -- (5.442,5.309) -- (5.566,5.434) -- cycle;
\node at (5.272,5.604) {\scalebox{1.486}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.864,6.594) -- (3.819,6.764) -- (3.694,6.888) -- (3.524,6.934) -- (3.354,6.888) -- (3.23,6.764) -- (3.184,6.594) -- (3.23,6.424) -- (3.354,6.299) -- (3.524,6.254) -- (3.694,6.299) -- (3.819,6.424) -- cycle;
\node at (3.524,6.594) {\scalebox{1.486}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.416,6.529) -- (6.37,6.699) -- (6.246,6.824) -- (6.076,6.869) -- (5.906,6.824) -- (5.781,6.699) -- (5.736,6.529) -- (5.781,6.359) -- (5.906,6.235) -- (6.076,6.189) -- (6.246,6.235) -- (6.37,6.359) -- cycle;
\node at (6.076,6.529) {\scalebox{1.486}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.602,3.77) -- (3.557,3.94) -- (3.432,4.064) -- (3.262,4.11) -- (3.092,4.064) -- (2.968,3.94) -- (2.922,3.77) -- (2.968,3.6) -- (3.092,3.475) -- (3.262,3.43) -- (3.432,3.475) -- (3.557,3.6) -- cycle;
\node at (3.262,3.77) {\scalebox{1.486}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.518,3.514) -- (2.473,3.684) -- (2.348,3.809) -- (2.178,3.854) -- (2.008,3.809) -- (1.884,3.684) -- (1.838,3.514) -- (1.884,3.344) -- (2.008,3.22) -- (2.178,3.174) -- (2.348,3.22) -- (2.473,3.344) -- cycle;
\node at (2.178,3.514) {\scalebox{1.486}{\small 16}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.362,3.565) -- (6.317,3.735) -- (6.192,3.859) -- (6.022,3.905) -- (5.852,3.859) -- (5.728,3.735) -- (5.682,3.565) -- (5.728,3.395) -- (5.852,3.27) -- (6.022,3.225) -- (6.192,3.27) -- (6.317,3.395) -- cycle;
\node at (6.022,3.565) {\scalebox{1.486}{\small 17}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.235,5.314) -- (9.189,5.484) -- (9.065,5.609) -- (8.895,5.654) -- (8.725,5.609) -- (8.6,5.484) -- (8.555,5.314) -- (8.6,5.144) -- (8.725,5.02) -- (8.895,4.974) -- (9.065,5.02) -- (9.189,5.144) -- cycle;
\node at (8.895,5.314) {\scalebox{1.486}{\small 18}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.861,3.999) -- (7.815,4.169) -- (7.691,4.293) -- (7.521,4.339) -- (7.351,4.293) -- (7.226,4.169) -- (7.181,3.999) -- (7.226,3.829) -- (7.351,3.704) -- (7.521,3.659) -- (7.691,3.704) -- (7.815,3.829) -- cycle;
\node at (7.521,3.999) {\scalebox{1.486}{\small 19}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (6.966,2.147) -- (7.566,2.147) -- (7.566,2.747) -- (6.966,2.747) -- cycle;
\node at (7.266,2.447) {\scalebox{1.486}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.074,6.012) -- (9.674,6.012) -- (9.674,6.612) -- (9.074,6.612) -- cycle;
\node at (9.374,6.312) {\scalebox{1.486}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (6.078,7.122) -- (6.678,7.122) -- (6.678,7.722) -- (6.078,7.722) -- cycle;
\node at (6.378,7.422) {\scalebox{1.486}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (6.896,8.31) -- (7.496,8.31) -- (7.496,8.91) -- (6.896,8.91) -- cycle;
\node at (7.196,8.61) {\scalebox{1.486}{\small d}};
\node at (4.593,12.015) {\scalebox{1.486}{\small S0}};
\node at (8.768,11.12) {\scalebox{1.486}{\small S1}};
\node at (7.814,8.842) {\scalebox{1.486}{\small S2}};
\node at (11.006,8.374) {\scalebox{1.486}{\small S3}};
\node at (1.75,8.125) {\scalebox{1.486}{\small S4}};
\node at (4.668,8.282) {\scalebox{1.486}{\small S5}};
\node at (1.245,5.596) {\scalebox{1.486}{\small S6}};
\node at (7.936,5.267) {\scalebox{1.486}{\small S7}};
\node at (11.019,5.328) {\scalebox{1.486}{\small S8}};
\node at (7.465,1.859) {\scalebox{1.486}{\small S9}};
\node at (4.273,1.117) {\scalebox{1.486}{\small S10}};
\node at (1.095,0.95) {\scalebox{1.486}{\small S11}};
\draw[line width=0.5pt] (11.73,10.462) -- (11.73,11.562);
\draw[line width=0.5pt] (11.499,11.177) -- (11.73,11.562) -- (11.961,11.177);
\draw[line width=0.5pt] (11.427,11.012) -- (12.032,11.012);
\node at (11.73,11.947) {\scalebox{1.486}{\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|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 \\ \hline
Nombre de rues & 2 & 3 & 2 & 3 & 2 & 7 & 4 & 6 & 2 & 1 & 4 & 2 \\ \hline
Parité & pair & \textbf{impair} & pair & \textbf{impair} & pair & \textbf{impair} & pair & pair & pair & \textbf{impair} & pair & pair \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 38 = 2 \(\times\) 19, 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{S1}, \textbf{S3}, \textbf{S5}, \textbf{S9} — 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{S9}, 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} (S1, S3, S5, S9).}\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}
