Question

The pages of a book are numbered consecutively: 1, 2, 3, 4 and so on. No pages are missing. If in the page numbers the digit 3 occurs exactly 99 times, what is the number of the last page?

Answer 1

page numbers with the digit 3 occurring in them are :
        03, 13, 23, 30, 31,32,33, 34, 35, 36, 37, 38, 39, ,43,53, 63, 73,83,93,
  1 + 03,13, 23, 30, 31, 32, 33, 34,35, 36, 37, 38, 39, 43, 53, 63, 73, 83, 93,
  2 +
  3 +
  4 + ...

so from 00 to 99: we have  20 of digit 3.
          100 to 199             20      total      up to 199 : 40
          200 to  299 :          20            total up to 299: 60
          300 to  399 :          20 + 100    total up to 399 :  180

From 300: 309 :  we have 10    so total 70  of digit 3.
        310 to 319 :            10 :              80
         320 to 329 :            10:             90

So now,   330, 331, 332, 333, : there are 9 of the digit 3.
So the answer is 333.