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Problem 1 (The Game Of Scoring)
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.
- Play this game several times with various values of $n$.
- For which values of $n$ does the first player have a winning strategy (meaning they are guaranteed to win if they play correctly). Remember to always fully justify your answers.
- For which values of $n$ does the second player have a winning strategy? Why?
- For which values of $n$ does the first player have a winning strategy if we change the rules so that now each player must choose 1, 2, 3, or 4 items from the pile.
- 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.
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