Counting zero-digits between 1 and 1 million I just remembered a problem I read years ago but never found an answer:

Find how many 0-digits exist in natural numbers between 1 and 1 million.

I am a programmer, so a quick brute-force would easily give me the answer, but I am more interested in a pen-and-paper solution.

Just to show there is more than one way to do it:
How many zero digits are there in all six-digit numbers? The first digit is never zero, but if we pool all of the non-first digits together, no value occurs more often than the others, so exactly one-tenth of them will be zeroes. There are $9\cdot 5\cdot 10^5$ such digits all of all, and a tenth of them is $9\cdot 5 \cdot 10^4$.
Repeating that reasoning for each possible length, the number of zero digits we find between $1$ and $999999$ inclusive is
$$\sum_{n=2}^6 9(n-1)10^{n-2} = 9\cdot 54321 = 488889 $$
To that we may (depending on how we interpret "between" in the problem statement) need to add the 6 zeroes from 1,000,000 itself, giving a total of 488,895.
