Problem #1
One day an extremely bored student began writing down the positive
integers in order: 1,2,3,... When she got to 20 she noticed that she had
used the digit 1 twelve times (once each for 1,10,12,13,...,19 and twice
for 11). She asked herself if there was some number n (greater than
1) such that in writing down the integers from 1 to n she would
have used the digit 1 exactly n times.

Is there such an n?

If so, is there a largest such n?

What if the digit 1 is replaced by another digit?

What if we work in a number base other than base 10?
The solution will be posted shortly.
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