The first nine pages are numbered with a single digit, so the first nine pages use 9 digits to number them.
Pages 10 through 99 are numbered with two digits, 99 minus 10 is 89 plus 1 is 90 pages that use two digits to number them.
There are 90 pages that each use two digits, so that is 180 digits used.
So far we have used 9 plus 180 digits, or 189 digits of the 522 total used. 522 minus 189 is 333. Each of the pages subsequent to page 99 uses three digits, so if the number of digits used to number three digit pages is 333, there must be 333 divided by 3 = 111 three-digit pages.
That makes a total of 9 plus 180 plus 111 equals 210 pages.
