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Is the sum odd? (Posted on 2016-06-08) Difficulty: 3 of 5
Consider a set of all 9 digit numbers composed of 9 distinct non-zero digits (e.g. 546327891).
An unordered pair chosen from the above set may sum to 987654321.

Without performing exhaustive search, prove that the number of such pairs is odd.

No Solution Yet Submitted by Ady TZIDON    
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An idea Comment 1 of 1
I'm not sure this is a demostration:

S = [set of all the pairs of numbers composed by 9 distinct non-zero digits, whose sum is 987654321].

A= (abcdefghi) ,B= (jklmnopqr) belongs to S => i+r=h+q
Then for each A,B there will be another pair C,D con C= (abcdefgih), D= (jklmnoprq). Then C,D will also belong to S.

F. es: 123456789      123456798
         864197532      864197523
        -------------     -------------
         987654321       987654321

But if A and B have all their digits in the same order except for the last to digits, [A=(abcdefghi) B =(abcdefgih)], there is not a C, D, pair because the permutation of the last two digits is again the pair A,B. 

There is a case in which this happen:
   493827165
+ 493827156
----------------
   987654321

This case is unique. Then the number of elements of S will be odd. 

Edited on June 9, 2016, 11:56 am
  Posted by armando on 2016-06-09 10:46:56

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