Question

There are 2018 bins along a large circle, and either a white ball or a black ball is alternately put in each bin.

(a) Choose 2 out of the 2018 balls (not bins).
(b) For each chosen ball; If it is white, move it clockwise to the next bin; if black, move it counterclockwise to the next.

By repeating (a) and (b), can we gather all of the balls in one bin?

Answer 1

No, we can't as the number of bins is even number and two balls moved at a time.

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Let's assume we have white balls in the bins with odd numbers and black balls in the bins with even numbers.
1, 3, 5,.....2017- white balls
2, 4, 6, ... 2018- black balls
Now let's divide all bins into 2 groups:1 to 1008 and 1011 to 2018 and start moving balls bu choosing one from each group at a time as per the rule. The white balls will end up in the bin 1009 as the move clockwise and the black balls will end up in the bin 1010. With the next steps we will have white balls moved to bins with even numbers and black balls moved to bins with odd numbers. Eventually all the balls will be moved into opposite bins and not all the balls can be moved to a single bin.
It would be possible to move all balls into one bin if the number of bins were odd not even. So the answer is no, it is not possible.