\documentclass[
	solutions,
	singlenumbering,
	nopagenumber
]{../../resources/ormc_handout}
\usepackage{../../resources/macros}

\usepackage{graphicx}

\title{Warm-Up: Passing Balls}
\uptitler{\smallurl{}}
\subtitle{Prepared by Mark on \today}

\begin{document}

	\maketitle

	\problem{}

	Twelve 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}