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Number Theory/Game Theory

Brian and Zayn play a game with a box, initially with \( N > 3 \) cups numbered from \( 1 \) to \( N \) . Brian starts the game; they play alternatly, and at each move one of the players takes one of the cups still in the box. The game ends when there are only \( 2 \) cups in the box. Brian wins the game if the numbers of the remaining two cups are coprime. If they are not, Zayn wins. Determine all the values of \( N > 3 \) for which Zayn has a strategy that will make him win.

Note by John Smith
9 months, 4 weeks ago

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I tried thinking of it like this: if \( N \)is odd then Brian ends the game which means that the only way he won't win is if the three numbers all have a common divisor ( different than \( 1 \) ). If it is even, then Zayn ends which means the only way he won't win is if the three numbers are coprime. I think that building on this it's possible to get somewhere but this is still very rudimentary.

John Smith - 9 months, 4 weeks ago

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