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The digit 3 appears 87 times in the page numbers of a dictionary. How many pages does the dictionary have?

 May 13, 2020
 #1
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We can do this by the long, slow process of counting ...

 

From 0 through 99 it appears 19 times: 

     3, 13, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 43, 53, 63, 73, 83, 93

 

From 100 through 199, it appears another 19 times, for a total of 38 times.

 

From 200 through 299, it appears another 19 times, for a total of 57 times.

 

From 300 through 309, it appears another 11 times, for a total of 68 times:

     300, 301, 302, 303, 304, 305, 306, 307, 308, 309

 

From 310 through 319, it appears another 11 times, for a total of 79 times.

 

Then there are 320, 321, 322, 323, 324, 325, 326 for another 8 times, for a total of 87 times.

 May 13, 2020
 #2
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Here is a computer count of the all 3's from 1 and up:

 

Between 1 and 9 = 1
Between 10 and 99 =19
Between 100 and 199 =40 
Between 300 and 323 =27
So: 1 + 19 + 40 + 27 =87 "3's"
So, the dictionary is 323 pages.

 May 13, 2020

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