Ball Collector

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?

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1 solution

Pick a bin with a black ball in it, label it as " 1 1 " and continue numbering the bins clockwise( 2 , 3 , 4 , . . . , 2018 2, 3, 4, ... , 2018 ).

Next, let each white ball represent a positive(+) sign, and black ball a negative(-) sign.

Then, the "sum" of the balls is initially 1 + 2 3 + 4 2017 + 2018 = 1009 -1+2-3+4 … -2017+2018 = 1009 , and regardless of our actions, this sum must stay an odd number because moving a ball changes the sum by an odd number( 1 1 or 2017 2017 ) and we move two balls at a time.

If all of the balls go into the same bin the sum will be zero, which is never going to happen.

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