From 877872325fb1c601e122ca6a9bb0af9a6332ab8f Mon Sep 17 00:00:00 2001 From: Christoph Daniel Schulze Date: Sun, 14 Sep 2014 16:24:05 +0200 Subject: [PATCH] Korrekturen und Definition n-Kaffee-Problem. 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Wichtig ist 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. +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. + Sei \(n\in\naturals_{\geq 2}\) + und \(K\) ein \(n\)-Kaffeekränzchen. + Eine \emph{\(K\)-Kaffeekasse} ist ein Tupel \(k\in\integers^n\), + \(k= (\Delta_1,\ldots,\Delta_n)\), + für das gilt: + \[ \sum_{i=1}^n \Delta_i = 0. \] + Für \(1 \leq i \leq n\) + bezeichnet \(\Delta_i\) die Differenz + der von \(p_i\) getrunkenen und ausgegebenen Kaffees. \end{defi} -Aus der Definition der \(\Delta_i\) ergibt sich +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. +Wir können nun +das allgemeine \(n\)-Kaffee-Problem formulieren. \begin{defi}[\(n\)-Kaffee-Problem] - foo \todo[inline]{allgemeines Problem formulieren} + Sei \(n\in\naturals_{\geq 2}\) + und \(k\) eine Kaffeekasse + über dem \(n\)-Kaffeekränzchen \(K\). + Gegeben zwei Personen \(p_i,p_j \in K\), + die einen Kaffee zusammen trinken wollen. + Das \(n\)-Kaffee-Problem besteht darin, + zu entscheiden, + ob \(p_i\) \(p_j\) einen Kaffee ausgeben muss + oder umgekehrt. \end{defi} \begin{beis}[2-Kaffee-Problem]