Find No. of Pages in a Book

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?

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.