The game of Scoring is a two-player game. Start by creating a pile of $n\geq 1$ objects (feel free to choose $n$ however you want). On each player's turn, they must choose 1, 2, or 3 items from the pile. Players alternate taking turns until someone takes the last object. Whoever takes the last object wins.

1. Play this game several times with various values of $n$.
2. State all values of $n$ for which the first player has a winning strategy (meaning they are guaranteed to win if they play correctly). Remember to always fully justify your answers.
3. State all values of $n$ for which the second player has a winning strategy? Why?
4. State all values of $n$ for which the first player has a winning strategy if we change the rules so that now each player must choose 1, 2, 3, or 4 items from the pile.
5. We'll now change the rules and require a player to take anywhere from 1 to $k$ objects each turn. Conjecture the values of $n$ for which the first player has a winning strategy.

• None.

problem.the_game_of_scoring.tex

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\begin{problem}
The game of \textit{Scoring} is a two-player game. Start by creating a pile of $n\geq 1$ objects (feel free to choose $n$ however you want). On each player's turn, they must choose 1, 2, or 3 items from the pile.  Players alternate taking turns until someone takes the last object. Whoever takes the last object wins.
\begin{enumerate}
\item Play this game several times with various values of $n$. 
\item  State all values of $n$ for which the first player has a winning strategy (meaning they are guaranteed to win if they play correctly). Remember to always fully justify your answers. 
\item  State all values of $n$ for which the second player has a winning strategy? Why?
\item  State all values of $n$ for which the first player has a winning strategy if we change the rules so that now each player must choose 1, 2, 3, or 4 items from the pile. 
\item  We'll now change the rules and require a player to take anywhere from 1 to $k$ objects each turn. Conjecture the values of $n$ for which the first player has a winning strategy.
\end{enumerate}
\end{problem}

problem ben