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### COME 2018

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• #24708
swastik pramanik
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Alice has two boxes $$A$$ and $$B$$. Initially box $$A$$ contains $$n$$ coins and box $$B$$ is empty. On each turn, she may either move a coin from box $$A$$ to box $$B$$, or remove $$k$$ coins from box $$A$$, where $$k$$ is the current number of coins in box $$B$$ . She wins when box $$A$$ is empty.

$$(a)$$ If initially box $$A$$ contains $$6$$ coins, show that Alice can win in $$4$$ turns.
$$(b)$$ If initially box $$B$$ contains $$31$$ coins, show that Alice cannot win in $$10$$ turns.
$$(c)$$ What is the minimum number of turns needed for Alice to win if box $$A$$ initially contains $$2018$$ coins?

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