I am compiling the BrainBashers world almanac and it now contains quite a few pages. But just how many?
I know that it takes 121 digits to print all of the page numbers in sequence.
So, how many numbered pages does the book have, and how many times does the number 9 appear?
Puzzle Copyright © Kevin Stone
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Answer
There are 65 pages, and the number 9 appears six times.
Reasoning
The first 9 pages each require one digit per page, which leaves 112 digits remaining. This requires an additional 112 ÷ 2 = 56 more pages. For a total of 9 + 56 = 65 pages.
Each block of ten contains a single 9, for a total of six 9's (as we've only reached page 65, and we don't have to worry about the 9's in the 90's either).
Double-Checking
The 65 pages are (with the six 9's highlighted):
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65.
If we remove the spaces, we get:
1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465. Which is 121 digits.