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people are standing in a circle. Each is assigned a number between 1 and 12. \par + Participants numbered 1, 2, 3, and 4 hold red, green, yellow, and black balls, respectively. \par + Everyone else is empty-handed. + + \vspace{2mm} + + Each participant can pass their ball to any student that is exactly 5 positions away. \par + Balls cannot be passed to someone who has one in hand. + + \vspace{2mm} + + After a number of passes, the first four participants again hold all the balls. \par + Participant 1 has a black ball. Which balls are held by participants 2, 3, and 4? + + \begin{solution} + \begin{itemize} + \item the graph of possible moves is isomorphic to a circle (since 5 and 12 are coprime), + \item but the balls get passed around, so swapping the place of any two balls is not allowed. \\ + Therefore, the balls will stay in their initial (cyclic) order: + \end{itemize} + + \begin{center} + \includegraphics[width=\textwidth]{pass-sol.png} + \end{center} + \end{solution} + + +\end{document} \ No newline at end of file