# 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
1 year, 5 months 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.

- 1 year, 5 months ago