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 " and continue numbering the bins clockwise( 2 , 3 , 4 , . . . , 2 0 1 8 ).
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 … − 2 0 1 7 + 2 0 1 8 = 1 0 0 9 , and regardless of our actions, this sum must stay an odd number because moving a ball changes the sum by an odd number( 1 or 2 0 1 7 ) 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.