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    1hG~cu3C#ax(6(T_ng(*mUrPlCLLk6YO}x+G1hzd7QPx={E5fJ>9j* zy+M*m$8aPN^*E301F8r#K0`XvXl7jgTX_jYe!PAj+!kGyB851zy>tGE3gZvf6$%|A zCT*0Bdz!!kwgCk|9P-978!|#&NYF?2YAgkigpK;Uh{oVEFanr%0I0SGK@D^o)buT$ zu&qp}IM#I`GjcU39%2hk6o-Yh%QN=Bl_jD#r7-CVWfo;{XsNCWCm(+$s1Hw=gS46` zHkcK`PYVTtavILPZ|Uf?m#n&53A$N;J~XzO2}*@l(U<`+>6)xmJNUE+gh=X(NotTw zO$-N`J0g(kNeB15NGg**X_y;F*!}nGPCRBCkQEZ#d|){bhZ*F_BE9cPYpV~A%F_n$ z@ajJGyKK5?pRA|h!GRnAM{budMD9uIiOrxNtE=*W8u>N<%(r6U%Lr8ptucnf3vHIe zfD>C^BBTVEiLJ|K^-2Xhuu#ZD7$NTa>>|tN5tzZbGB*YzE21bg*oH-Q61iyn#tC_F zK&G__VzZ2DKBK&vW%d%Jw@lNU!}SvPe$4gzlef%EcuMM90t~uO$bBg;-^64u`syBTWk1+5?mfqbRjp-sHzGDU`bdVx_N5o%%&Y4dizQ8`1bK-(K zo*3iMAZN)9brCb~ehJ$rit(&RGDLwP4B&BbMzRc9BfGXzQM>dCc)E z+~Kz<-atMyyv2&zT>Ace2wSf9+_u(#d|HRDv%T7dAA!Q`x8Z>9KUJ^R4azkbs;=dV z(|`pufmmmVBQ*)ls|?EgRrz!C=*y$Da5BEL}#dS98E$rk+US^ zz89kzDT|#Ys;w7-EsgRg1eWo{u^^Hz#>JlUMMm-YR=!BuEovV= zvjR>y&Qs*@E>o0-j-};xl~gj%tg=KhQ8p3dPqlCa%b5&xU2;g4t({emLJI&i90tR> zHw1EI-A~#cpC(zZFH;azsOE3DmFch zkA~aP7mn|*_(m&|07y_bIpk`K5yh!+gNhlqhmVXtoK~F}`6k9niMs(8quaO=-;@ZO z#fa|73W*R3L6qnNL)jCNL|;DbQZ~|CCB;RGc^bMfLv}y-vr^I4 zv7nU(2Q7JM19^>rFGzLyn)oz>V^FlTGaOL3G}+vZa?P(Z5R=dUYag zMj~dS|GueYZ|_3H$j(8;K%@jiFKK6H|1WX=ZxSKW=3(FzW#ZrvVijXyXA%_UU}Y5( zVG^=ZgN=I7}M@x!Knd~_Hu?;!*?HC1KOh6!WFDd z=N1HO2F3wAFZ(>n;Qec%23-q zb)YW_gy&v23#&j@9<)PgzCz4*$0s?fKuc0r)8>P`H9LP(=4q(iM|R?Fmt4&jT2y$k z*q6R|GUxEVs;xqNF#rYCLcXJI(SKectU`AfK)C@1pzstPXIgg{Qo63f_MoEs6*(YA z_Q0b1B(cJj9dM?5H8~(p3_<96MO(pCTjA8vf@*k!Dk3B75&Y%DES<+n09CSER`)&tHCGEzHL;mU@p9bWs#5 zFuusN`OWcvJr*Xb=W>~jhm~eK9mVx;#agix8}tB^lf33^4GaMo;1!$knchht)5WMhH(^-EM<4CcQ8JP+~l diff --git a/main.tex b/main.tex index 487f2ff..b3dd195 100644 --- a/main.tex +++ b/main.tex @@ -28,6 +28,7 @@ \newtheorem{beob}{Beobachtung} \newcommand{\beobautorefname}{Beobachtung} \newtheorem*{beis}{Beispiel} +\newcommand{\beisautorefname}{Beispiel} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -198,6 +199,7 @@ formalisiert exakt diese Fragestellung. Betrachten wir das Beispiel \(n=2\). \begin{beis}[2-Kaffee-Problem] +\label{beis:2-kaffee-problem} Sei \(K\) ein 2-Kaffeekränzchen und \(k\) eine K-Kaffeekasse. Nehmen wir an, dass bisher \(p_1\) zweimal @@ -343,21 +345,52 @@ analog zu \autoref{def:kaffeekassentransition}. %% Visualisierung des Kaffeeproblems \section{Visualisierung des Kaffeeproblems} \label{sec:visualisierung} +Im folgenden stellen wir Möglichkeiten vor, +das Kaffeeproblem für Gruppen aus 2 oder 3 Personen +grafisch darzustellen bzw. zu lösen. -% \begin{figure} -% \centering -% \includegraphics[scale=1]{2KaffeeProblem} -% \caption{% -% Visualisierung des \(2\)-Kaffee-Problems. -% In diesem Fall schuldet \(p_0\) \(p_1\) -% genau \(2\) Kaffees. -% } -% \label{fig:2_kaffee_problem} -% \end{figure} +\subsection{Darstellung des 2-Kaffee-Problems} +\label{ssec:2-kaffee-problem} +Wie bereits in \autoref{beis:2-kaffee-problem} angesprochen, +wird nur eine einzelne Komponente der Kaffeekasse benötigt. +Somit stellen wir die Kaffeekasse wie in \autoref{fig:2-kaffee-problem} dar. + +\begin{figure}[htbp] + \centering + \includegraphics[scale=1]{2KaffeeProblem} + \caption{% + Visualisierung des \(2\)-Kaffee-Problems. + In diesem Fall schuldet \(p_1\) \(p_2\) + noch \(2\) Kaffees. + } + \label{fig:2-kaffee-problem} +\end{figure} + +In der Abbildung lässt sich durch Verschieben +des Punktes der Zustand aktualisieren. +Hierzu wird der Punkt immer auf diejenige +Person zubewegt, die den Kaffee ausgibt. +\subsection{Darstellungen des 3-Kaffee-Problems} +Analog zu \autoref{ssec:2-kaffee-problem} kann +für die Darstellung des 3-Kaffee-Problems ein +kartesisches Koordinatensystem verwendet werden, +in dem die Werte von \(\Delta_1\) und \(\Delta_2\) +als Koordinaten eingetragen werden. +\begin{figure}[htbp] + \centering + \includegraphics[scale=1]{3KaffeeKartesisch} + \caption{% + Visualisierung des \(3\)-Kaffee-Problems mit kartesischen Koordianten. + } + \label{fig:3-kaffee-kartesisch} +\end{figure} +In \autoref{fig:3-kaffee-kartesisch} ist eine +kartesische Darstellung für die Kaffeekasse +\(k = (2,-1,-1) \) gegeben. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%