How many Pages

Number Theory Level pending

While numbering every page of a book, 3077 individual digits are printed. If the first page is numbered 1, how many pages does the book have?


The answer is 1046.

Alex Sammear by Alex Sammear , 23, United Kingdom

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2 solutions

Giorgos K.
Mar 25, 2018

Mathematica

s=n=0;While[s<3077,s+=IntegerLength[n++]];n-1

returns 1046

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Alex Sammear
Mar 24, 2018

For a book, the amount of digits used for each page will increase at the indices of 10, e.g page 27 has 2 digits but page 1937 has 4 digits. The question can be solved by modelling the amount of individual digits printed as:

(9x1)+(90x2)+(900x3)+(9000x4)+.... = Total number of individual digits printed.

(9x1)+(90x2)+(900x3)+(9000x4)+.... = 3077

(9x1)+(90x2)+(900x3)+(9000x4) = 38889, 3077 < 38889 therefore there must be less that 10000 pages, but:

(9x1)+(90x2)+(900x3) = 2889, 3077 > 2889 therefore there must be greater than 999 pages. So:

No. of pages - 999 = n

(9x1)+(90x2)+(900x3)+(nx4) = 3077

nx4 = 3077 - 2889

nx4 = 188

n = 47

No. of pages = n + 999

No. of pages = 1046

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