Show that for every positive integer n, the total number of digits in the sequence 1,2,3,...,10^n is equal to the number of zero digits in the sequence 1,2,3,...,10^(n+1).

I'm going to do like Cantor, and rather than counting the digits and zeros, I'll match each digit in the first set to each zero in the second set.

Take any number in the second set with at least one zero digit.  Pick any zero digit, drop it from the number, and underline the digit preceding it.  This way, each zero in the second set corresponds to a digit in the first.  Likewise, you may take any digit from a number in the first set and add an underlined zero after it, forming a number from the second set.  This is the one-to-one correspondence we were looking for.

Posted by Tristan on 2005-02-26 03:13:00