A book has N pages, numbered the usual way, from 1 to N. The total number of digits in the page numbers is 1,095. How many pages does the book have?
ANSWER: Every page number has a digit in the units column. With N pages, that’s N digits right there. All but the first 9 pages have a digit in the tens column. That’s N – 9 more digits.
All but the first 99 pages have a digit in the hundreds column (accounting for N – 99 more digits).
I could go on, but not many books have more than 999 pages. A book with 1,095 digits in its page numbers won’t, anyway.
This means that 1,095 must equal:
N + (N – 9) + (N – 99)
This can be simplified to:
1,095 = 3N – 108.
That means that 3N = 1,203, or N = 401. That’s the answer, 401 pages.