\documentclass[11pt]{article}
\usepackage{jeffe,handout,multicol,tikz,kgm}

\newcommand{\kgmcomment}[1]{}
\renewcommand{\gcd}{\mathrm{gcd}}

\begin{document}
\headers{Math378}{Quiz 2}{Wed. Feb 4, 2026}
\setlength{\columnseprule}{0.4pt}

%\makeatletter
%\def\@oddfoot{\hfill \tikz \node[draw] {over} ; }
%\makeatother

%\newcommand{\seq}[1]{\left\langle #1\right\rangle}

\newcommand{\compicon}%
{%
	\begin{tikzpicture}[scale=0.15]%
		\draw[double] (0,0) rectangle (1,1) ;
		\begin{scope}[yshift=-5pt]
			\draw (-0.5, -0.5) rectangle (1.5, 0) ;
		\end{scope}
	\end{tikzpicture}
}

\makeatletter
\def\@oddfoot{}
\makeatother

\noindent
\hfill Name: \underline{\hspace{3.5in}} 

\noindent
\textbf{Directions:} Show all work.  

\begin{enumerate}

\item In a game of Nim, tokens are placed in various piles.  In each turn, players removing one or more tokens from a selected pile.  In this variant, the game begins with $n$ piles, each starting with 10 tokens.  When no tokens remain, the player to move loses.
\begin{enumerate}
	\item\points{3} Prove that if $n$ is even, then Player 2 has a winning strategy.  (Hint: Analyzing the general game of Nim is tricky; give an explicit strategy for Player 2 in this special case.)
	\vfill
	
	\item\points{2} Use part (a) to show that if $n$ is odd, then Player 1 has a winning strategy.
	\vspace{5cm}
\end{enumerate}

\newpage
\item\points{5} Let $a_0 = 0$ and $a_n = \frac{1}{2-a_{n-1}}$ for $n\ge 1$.  Guess a formula for $a_n$ and prove your formula is correct.
\vfill

\end{enumerate}
\end{document}
