Question 1176475
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Given that a total of 149 1's are used for all the page numbers in a book, 
at least how many pages are there in the book?
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            Actually, this problem requires very accurate counting of digits - more accurate than in the solution by other tutor.


            So I place this accurate solution below.


            Find several distinctions from the solution of the other tutor.



<pre>
(1)  Consider one-digit numbers from 1 to 9 inclusive.

     There is only one digit "1" in this set of numbers.



(2)  Consider two-digit numbers from 10 to 99 inclusive.

     In this set, there are 10 (ten)  digits "1" in the "tens" position (from 10 to 19),

                         and 9 (nine) digits "1" in the "ones" position (11, 21, 31, . . . 91).


     Thus we have 10+9 = 19 digits  of  "1"  in two-digit numbers.

     Hence, so far we have 1 + 19 = 20 digits  of  "1"  in one-digit and two-digit numbers.


     We need to get  149-20 = 129  additional digits  of  "1".




(3)  Consider three-digit numbers from 100 to 199 inclusive (100, 101, 102, . . . , 199)

     In this set, there are 100 (one hundred)  digits  "1"  in the "hundreds" position (from 100 to 199),

                             10 (ten)          digits  "1"  in the "tens" position (110, 111, 112, . . . , 119),

                        and  10 (ten)          digits  "1"  in the "ones" position (101, 111, 121, . . . , 191).


     Thus we have additional 100+10+10 = 120 digits "1" in three-digit numbers from 100 to 199 inclusive.

     Hence, so far (arriving to 199) we have accumulated  20 + 120 = 140 digits of "1".


     140 is still smaller comparing with 149, so we will continue moving forward from the number of 200.

            We still should collect 9 more digits of "1".



(4)  So, at the position of 199, where we have 140 digits of "1", we will move farther in slow pace, accurately counting digits "1".


         The numbers from 200 to 209 inclusive add  1  digits of "1" to 140,  giving the total of 140 + 1 = 141.

              We still need accumulate 149-141 = 8 digits of "1"

         The numbers 210, 211, 212, 213, 214, 215, 216  will add  these 8 digits of "1",  giving finally the total of 149 at  N = 216.


<U>ANSWER</U>.  The book has 216 pages.


<U>CHECK</U>.   The numbers from 1 to 9 inclusive give one digit "1".

         The numbers from  10 to 99    add   19  digits "1", making the cumulative sum equal to  1 + 19   = 20.

         The numbers from  100 to 199  add  120  digits "1", making the cumulative sum equal to  20 + 120 = 140.

         The numbers from  200 to 209  add    1  digit  "1", making the cumulative sum equal to  140 + 1 = 141.

         The numbers  210, 211, 212, 213, 214, 215, 216  add  8 digits of "1", making the cumulative sum equal to 141 + 8 = 149.


<U>ANSWER</U>.  There are  216  pages in the book.
</pre>

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The problem is solved and the solution is completed.


As you see, there is no any advanced super-idea in the solution.


It requires accurate counting, only.


So, let this solution serves as a TEMPLATE for you, showing how to solve problems accurately.