\documentclass[12pt,ifthen]{article} \usepackage{url} \usepackage{comment} \newcommand{\und}{\_\_\_\_\_\_\_\_\_} \newcommand{\Z}{\mathbb{Z}} \usepackage{amsmath} \usepackage{amssymb} % for \nmid \begin{document} \centerline{\textbf{Honors HW05. Morally DUE Mon Apr 4}} Nim Mark is the variant where the person who can't move WINS (so when there are 0 stones, player 1 wins). In this problem we consider 1-pile Nim Mark where the set of moves is $\{1,2,3\}$. \begin{enumerate} \item (20 points) Work out by (by hand or by code) who wins this game if the number of stones is $0,1,\ldots,20$. Give us your table. It should be of the form, though the {\it Who Wins} column will be filled in. $\begin{array}{|c|c|} \hline n & \hbox{ Who Wins } \cr \hline 0 & \cr 1 & \cr \vdots & \vdots \cr 20 & \cr \hline \end{array}$ NO proof or reasoning required, but if its not perfect you get 0. \item (10 points) Make a conjecture of the form If BLANK1 then P1 wins. If BLANK2 then P2 wins. Your conjecture should covers all cases. \item (70 points) Prove your conjecture. \end{enumerate} \end{document}