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How Many Threes?


Date: 06/12/99 at 18:35:17
From: ken logan
Subject: Problem Solving (Graduate Level)

Dear Dr. Math,

If all the numbers from 1 to 333,333 were written out, how many times 
would the digit 3 be used?


Date: 06/13/99 at 08:07:50
From: Doctor Anthony
Subject: Re: Problem Solving (Graduate Level)

Consider the righthand column alone starting at 000000.

          000000
          000001
          000002
          ......
          333329
                 <-  there are 333330 rows to here
          333330
          333331
          333332
          333333

In the first column up to row 333330 there will be equal numbers of 
all digits 0 to 9, so the number of 3's is 1/10th of this = 33333. 
We then add the final 3 in this column, giving 33334 altogether.

Considering just the second column we have

         000000
         000001
         ......
         333299
                <-  333300 rows to here
         333300
         ......
         333310
         ......
         333320
         ......
         333330
         333331
         333332
         333333
          
Considering the second column alone there are 1/10 * 333300 + 4 = 
33334 threes, which is the same result as for the first column.  
Similarly the other columns each give 33334 lots of 3's.

The total number of 3's is  6 x 33334 = 200004 

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Discrete Mathematics

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