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lediglich, dass die beteiligten Personen sich gegenseitig Kaffee ausgeben. +Zusätzlich zu dem Kaffeekränzchen benötigen wir noch +eine Möglichkeit die Kaffeeschulden innerhalb +der Gruppe zu dokumentieren. +\begin{defi}[Kaffeekasse] + Sei \(n\in\naturals_{\geq 2}\) und + \(K\) ein \(n\)-Kaffeekränzchen. + Als \(K\)-Kaffeekasse definieren wir das Tupel \(k\in\integers^n\) durch + \[ k = (\Delta_1,\ldots,\Delta_n) \text{ mit } \sum_{i=1}^n \Delta_i = 0. \] + Die Komponenten \(\Delta_i\) berechnen sich hierbei + als die Differenz der von \(p_i\) bezahlten und + getrunkenen Kaffees. +\end{defi} +Aus der Definition der \(\Delta_i\) ergibt sich +\begin{align*} + \Delta_i < 0 &: p_i \text{ bekommt noch } |\Delta_i| \text{ Kaffees} \\ + \Delta_i > 0 &: p_i \text{ schuldet noch } \Delta_i \text{ Kaffees} +\end{align*} - +Mit diesen Definitionen können wir nun das +allgemeine \(n\)-Kaffee-Problem formulieren. +\begin{defi}[\(n\)-Kaffee-Problem] + foo \todo[inline]{allgemeines Problem formulieren} +\end{defi} - \begin{beis}[2-Kaffee-Problem] - Sind nur 2 Personen \( p_0 \) und \( p_1 \) an der Kaffeerunde beteiligt, lässt sich das 2-Kaffee-Problem als \( x \in \mathbb{Z} \) beschreiben. Hierbei gilt: - \begin{description} - \item[x = 0] Das Verhältnis ist ausgeglichen, niemand hat Kaffeeschulden. - \item[x $>$ 0] \( p_1 \) schuldet \( p_0 \) noch x Kaffees. - \item[x $<$ 0] \( p_0 \) schuldet \( p_1 \) noch x Kaffees. - \end{description} +\begin{beis}[2-Kaffee-Problem] + Sind nur 2 Personen \( p_0 \) und \( p_1 \) an der Kaffeerunde beteiligt, lässt sich das 2-Kaffee-Problem als \( x \in \mathbb{Z} \) beschreiben. Hierbei gilt: + \begin{description} + \item[x = 0] Das Verhältnis ist ausgeglichen, niemand hat Kaffeeschulden. + \item[x $>$ 0] \( p_1 \) schuldet \( p_0 \) noch x Kaffees. + \item[x $<$ 0] \( p_0 \) schuldet \( p_1 \) noch x Kaffees. + \end{description} - \begin{figure}[hbtp] - \centering - \includegraphics[scale=1]{2KaffeeProblem} - \caption{Beispiel für das 2-Körper-Problem, $p_1$ schuldet $p_0$ noch 2 Kaffee} - \end{figure} - \end{beis} + \begin{figure}[hbtp] + \centering + \includegraphics[scale=1]{2KaffeeProblem} + \caption{Beispiel für das 2-Körper-Problem, $p_1$ schuldet $p_0$ noch 2 Kaffee} + \end{figure} +\end{beis} - \begin{defi}[n-Kaffee-Problem] - \todo{Formel korrigieren} - Das allgemeine \textbf{n-Kaffee-Problem} zu einer Kafferunde K mit n Personen lässt sich definieren als \( x \in \mathbb{Z}^{n-1} \), also \( x = (x_0, \ldots, x_{n-1}) \). Zu diesem Tupel werden die Kaffeefunktion \( k_{i,j} : \mathbb{Z}^{n-1} \rightarrow \mathbb{Z}^{n-1} \) definiert durch - \[ k_{i,j}(x_0,\ldots,x_{n-1}) = - \left\{ \begin{array}{cc} - (x_0,\ldots,x_{i-1},x_i +1, x_{i+1},\ldots,x_{n-1}) & $, falls $ i < n \\ - (x_0-1, \ldots, x_{n-1}-1) & $, falls $ i = n - \end{array} \right. - \] - Diese Kaffeefunktion \( k_i \) gibt die Veränderung der Guthaben-Schulden-Verhältnisse innerhalb der Kaffeerunde an, wenn \( p_i \) ein Kaffee \textbf{ausgegeben wird}. - \end{defi} +\begin{defi}[n-Kaffee-Problem] +\todo{Formel korrigieren} + Das allgemeine \textbf{n-Kaffee-Problem} zu einer Kafferunde K mit n Personen lässt sich definieren als \( x \in \mathbb{Z}^{n-1} \), also \( x = (x_0, \ldots, x_{n-1}) \). Zu diesem Tupel werden die Kaffeefunktion \( k_{i,j} : \mathbb{Z}^{n-1} \rightarrow \mathbb{Z}^{n-1} \) definiert durch + \[ k_{i,j}(x_0,\ldots,x_{n-1}) = + \left\{ \begin{array}{cc} + (x_0,\ldots,x_{i-1},x_i +1, x_{i+1},\ldots,x_{n-1}) & $, falls $ i < n \\ + (x_0-1, \ldots, x_{n-1}-1) & $, falls $ i = n + \end{array} \right. + \] + Diese Kaffeefunktion \( k_i \) gibt die Veränderung der Guthaben-Schulden-Verhältnisse innerhalb der Kaffeerunde an, wenn \( p_i \) ein Kaffee \textbf{ausgegeben wird}. +\end{defi} + +\begin{beis}[3-Kaffee-Problem] +\todo{Text ergänzen} + \begin{figure}[hbtp] + \centering + \includegraphics[scale=1]{3KaffeeProblem} + \caption{} + \end{figure} +\end{beis} - \begin{beis}[3-Kaffee-Problem] - \todo{Text ergänzen} - \begin{figure}[hbtp] - \centering - \includegraphics[scale=1]{3KaffeeProblem} - \caption{} - \end{figure} - \end{beis} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%