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\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}

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