47 lines
1.3 KiB
TeX
Executable File
47 lines
1.3 KiB
TeX
Executable File
\documentclass[
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solutions,
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singlenumbering,
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nopagenumber
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]{../../resources/ormc_handout}
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\usepackage{../../resources/macros}
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\usepackage{graphicx}
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\title{Warm-Up: Passing balls}
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\uptitler{\smallurl{}}
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\subtitle{Prepared by Mark on \today}
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\begin{document}
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\maketitle
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\problem{}
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Twelve people are standing in a circle. Each is assigned a number between 1 and 12. \par
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Participants numbered 1, 2, 3, and 4 hold red, green, yellow, and black balls, respectively. \par
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Everyone else is empty-handed.
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\vspace{2mm}
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Each participant can pass their ball to any student that is exactly 5 positions away. \par
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Balls cannot be passed to someone who has one in hand.
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\vspace{2mm}
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After a number of passes, the first four participants again hold all the balls. \par
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Participant 1 has a black ball. Which balls are held by participants 2, 3, and 4?
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\begin{solution}
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\begin{itemize}
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\item the graph of possible moves is isomorphic to a circle (since 5 and 12 are coprime),
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\item but the balls get passed around, so swapping the place of any two balls is not allowed. \\
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Therefore, the balls will stay in their initial (cyclic) order:
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\end{itemize}
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\begin{center}
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\includegraphics[width=\textwidth]{pass-sol.png}
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\end{center}
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\end{solution}
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\end{document} |