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Missing Pages (Posted on 2009-11-13) Difficulty: 2 of 5
A chunk of pages is missing from a book. The sum of the missing page numbers is 9808. What are those page numbers?

See The Solution Submitted by Brian Smith    
Rating: 3.5000 (2 votes)

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Solution Spreadsheet Solution | Comment 6 of 7 |
For an even sum the pages  must be multiples of 4 and in the order of odd-even.

The sum of pages is given by the triangular sequence N*(N+1)/2 while the sequence of sums of multiples of 4 is given by 8N^2+2N.

At any point "y" the sum may be expressed by a triangular number plus a constant [K] and the sum at the next term "y+1" will be the next triangular number plus K.

Using those principles in a spreadsheet and by reiteration I subtracted successive 4N multiples from 9808 and then divided by 4N.  If the quotient was an integer then I would have K.  The missing pages would therefore be from K+1 to K+4N.



4N    N    8N^2+2N                           K            
4      1      10          9808    9798    2449.5    
8      2      36          9808    9772    1221.5    
12    3      78          9808    9730    810.8333    
16    4    136          9808    9672    604.5    
20    5    210          9808    9598    479.9    
24    6    300          9808    9508    396.1667    
28    7    406          9808    9402    335.7857    
32    8    528         9808    9280     290    
36    9    666         9808    9142     253.9444   


The integer K occurs for 4N=32 so the pages are from 291 to 322.


  Posted by brianjn on 2009-11-13 21:37:44
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