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Problem of the Week 1018

Balanced Numbers

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Call a positive integer n "balanced" if the average number of 1s that occur in the digits of 1, 2, 3, ..., n is EXACTLY 1. Then 1 is balanced. But 2 is not balanced since the average of {1, 0} is 1/2; and 11 is unbalanced because the average of {1,0,0,0,0,0,0,0,0,1,2} is 4/11.

Is 1 the only balanced integer?

© Copyright 2004 Stan Wagon. Reproduced with permission.

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3 November 2004