White and black balls
You are playing a game with 2016 bins arranged in a circle. At the start, each bin has a white or black ball inside, with the two colors alternating between adjacent bins all throughout the circle.
The rules are as follows:
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Each turn, you select one white ball and one black ball. The white ball moves clockwise to the next bin and the black ball you moves counterclockwise to the next bin.
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You may repeat this process as many times as you like.
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The two balls selected can come from the same bin at later stages when a single bin can have multiple balls.
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You win the game if all 2016 balls are in the same bin after a turn.
Is it possible to win this game?
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1 solution
Let's label the bins 0, 1, 2 ,3, ..., 2017 (0 and 2017 are adjacent since the bins are arranged in a circle). At any point of the game, the sum of all the labels of the bins of the 2016 balls stays the same modulo 2016, since at every turn one white ball get +1 and one black ball get -1. At the start of the game, this sum equal ∑ i = 1 2 0 1 6 i = 2 2 0 1 6 ∗ 2 0 1 7 ≡ 1 0 0 8 ( m o d 2 0 1 6 ) . In the position where all the balls are in in the bin labeled n , the sum equal 2 0 1 6 n ≡ 0 ( m o d 2 0 1 6 )
Thus, it is impossible to go from the starting position to a position where all the balls are in the same bin.