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\title{The game of NIM}
%author{The Author}
%\date{10/16/2009}                                           % Activate to display a given date or no date

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NIM is a game played with piles of stones (or pennies). You will need a partner to play this game. Here are the rules:


\begin{enumerate}


\item On your turn, you can take as many stones you want from one pile.

\item You must take at least one stone on each turn.

\item Players alternate taking turns.

\item The player who takes the last stone from the last pile wins.
\end{enumerate}

Try playing NIM with your partner. Keep track of who goes first in each game. There is always one player who is guaranteed to win if he/she plays according to a certain strategy. It is your job to figure out what this strategy is. Decide if Player 1 wins or Player 2 wins in each game.

\bigskip

Game 1: Pile 1 has four stones. \\
Winner is:  Player 1 or Player 2 \\
Strategy:
\medskip

Game 2: Pile 1 has one stone. Pile 2 has two stones. \\
Winner is:  Player 1 or Player 2 \\
Strategy:
\medskip

Game 3: Pile 1 has three stones. Pile 2 has three stones. \\
Winner is:  Player 1 or Player 2 \\
Strategy:
\medskip

Game 4: Pile 1 has eight stones. Pile 2 has five stones. \\
Winner is:  Player 1 or Player 2 \\
Strategy:
\medskip

Game 5: Pile 1 has seven stones. Pile 2 has seven stones. \\
Winner is:  Player 1 or Player 2 \\
Strategy:
\bigskip


\emph{Question 1: If there is only one pile of stones, which player wins? Why?}

\bigskip
\vspace{30pt}



\emph{Question 2: If there are two piles of stones, when does Player 1 win? When does Player 2 win? Why?}

\bigskip

\vspace{20pt}


Game 6: Pile 1 has one stone. Pile 2 has two stones. Pile 3 has three stones. \\
Winner is:  Player 1 or Player 2 \\
Strategy:
\medskip

Game 7: Pile 1 has one stone. Pile 2 has two stones. Pile 3 has two stones. \\
Winner is:  Player 1 or Player 2 \\
Strategy:
\medskip

Game 8: Pile 1 has one stone. Pile 2 has four stones. Pile 3 has six stones. \\
Winner is:  Player 1 or Player 2 \\
Strategy:
\medskip

Game 9: Pile 1 has two stones. Pile 2 has four stones. Pile 3 has six stones. \\
Winner is:  Player 1 or Player 2 \\
Strategy:
\medskip

Game 10: Pile 1 has three stones. Pile 2 has four stones. Pile 3 has eight stones. \\
Winner is:  Player 1 or Player 2 \\
Strategy:
\bigskip

Think about situation when the largest pile has either six, seven or eight stones. \\

\vspace{4pt}

\emph{Question 3: If there are three piles of stones, when does Player 1 win? When does Player 2 win? (This is not an easy question, but any good ideas may be valuable.)}

\bigskip
\vspace{150pt}

\section*{Additional Questions to think about}

\bigskip

\emph{Question 4: If there are four  piles of stones, what then?}

\bigskip

\emph{Question 5: If the rules were changed so that you lose if you take the last stone, how do our strategies change?}


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