SOLUTION: how many digits does it take in total to number a book from page 16 to page 667

page 16 to page 667

Of the 667 pages from page 1 to page 667, we subtract
the 15 pages from 1 through 15 that we aren't counting 
and get 667-15 = 652 pages.  So that's 652 numbers.  
They all have units digits and tens digits.
So there are 652 units digits and 652 tens digits.  

So all that's left to count are the hundreds digits,
and they only occur on pages 100 through 667.

Of the 667 pages from page 1 to page 667, we subtract
the 99 pages that do not have hundreds digits, from 1 
through 99 and get 667-99 = 568 pages that have a 
hundreds digit in their page numbers.

Answer: 

652 units digits + 652 units digits + 568 hundreds digits =

1872 digits.

Edwin