2. In this question we will consider the game Nim 4, played between Alice and Bob. In Nim 4, we start with n stones. A player moves by removing 1, 2, or 4 stones such that there is at least one (≥ 1) stone leftover. If a player cannot make such a move, they lose. For example, suppose Alice starts with 3 stones, she may remove 1 or 2 stones. If she removes 1 stone, Bob then has 2 stones on his turn and his only valid move is to remove 1 stone. Alice then cannot make a valid move and loses. A valid game is any game where both players make valid moves until one player loses. Note that players do not need to play optimally. (a) Let an be the number of valid games if the game starts with n stones and Alice is the first player. Find a recurrence relation with initial conditions for an. (b) Find the closed form solution for the generating function for an. (c) Find all n where Alice has a winning strategy. Explain your reasoning.

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Chapter2: Second-order Linear Odes
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2. In this question we will consider the game Nim 4, played between Alice and Bob. In Nim 4, we start
with n stones. A player moves by removing 1, 2, or 4 stones such that there is at least one (≥ 1) stone
leftover. If a player cannot make such a move, they lose.
For example, suppose Alice starts with 3 stones, she may remove 1 or 2 stones. If she removes 1 stone,
Bob then has 2 stones on his turn and his only valid move is to remove 1 stone. Alice then cannot make
a valid move and loses.
A valid game is any game where both players make valid moves until one player loses. Note that players
do not need to play optimally.
(a) Let an be the number of valid games if the game starts with n stones and Alice is the first player.
Find a recurrence relation with initial conditions for an.
(b) Find the closed form solution for the generating function for an.
(c) Find all n where Alice has a winning strategy. Explain your reasoning.
Transcribed Image Text:2. In this question we will consider the game Nim 4, played between Alice and Bob. In Nim 4, we start with n stones. A player moves by removing 1, 2, or 4 stones such that there is at least one (≥ 1) stone leftover. If a player cannot make such a move, they lose. For example, suppose Alice starts with 3 stones, she may remove 1 or 2 stones. If she removes 1 stone, Bob then has 2 stones on his turn and his only valid move is to remove 1 stone. Alice then cannot make a valid move and loses. A valid game is any game where both players make valid moves until one player loses. Note that players do not need to play optimally. (a) Let an be the number of valid games if the game starts with n stones and Alice is the first player. Find a recurrence relation with initial conditions for an. (b) Find the closed form solution for the generating function for an. (c) Find all n where Alice has a winning strategy. Explain your reasoning.
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