Champernowne Puzzle

[BMAD]

Which of these is larger?

 

the total number of digits in the sequence 1, 2, 3, ..., 10¹¹⁸⁰

or

the number of zero digits in the sequence 1, 2, 3, ... , 10¹¹⁸¹

[k-man]

It's the same.

 

The number of digits in the first range is SUM_n=[1,1180](9*10^n-1*n) + 1181. ¹¹⁸⁰] into sub ranges based on the number of digits and call the number of digits n.

Let's break the range [1..10

For single digit numbers (n=1) the sub range is 1..9. There are 9 total digits.

For n=2, the sub range is 10-99. It contains 90*2 or 90*n digits.

For n=3, the sub range is 100-999. It contains 900*3 or 900*n digits

It's easy to see that for each sub range n the total number of digits is 9*n*10^n-1

Adding up these sub ranges accounts for all except the last number, which has 1181 digits, so we need to add it to the sum.

Similarly, the number of zeroes in the second range is SUM_n=[2,1181](9*10^n-2*(n-1)) + 1181, so they are the same.