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Make it equal (Posted on 2010-05-26) Difficulty: 4 of 5
Divide the set {1,2,3,4, ....,n} into three disjoint subsets A , B , C whose sums of elements are equal. For what values of n it is feasible?

  Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)
Solution: (Hide)
Let us show that once a solution exist for (1,2,3,... n), creating a 3E-Partition for (1,2,3,...n+2, n+3) is trivial .
My proof:
Let the 3E-Partition for sum(n) be S1, S2, S3.
We have to add 3 additional numbers n+1 , n+2 ,n+3 i.e. to increase each subset by n+2.
We will replace number 1 (present in one of the subsets, say S2) by n+3; the number n+2 will go to another subset (e.g. S1) without replacement.
Two unassigned numbers 1 and n+1 go to the last subset (S3 in our example).
Since existence of a solution for n=5 and 4=6 was shown de facto, the process of full induction is concluded.


Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: Full proof.Ady TZIDON2010-05-27 13:51:04
SolutionFull proof.Jer2010-05-27 11:32:44
re: No problemsAdy TZIDON2010-05-27 06:19:31
Some ThoughtsNo problemsSteve Herman2010-05-27 01:32:23
Hints/Tipsre: half of it SOME HINTS(a partial spoiler)Ady TZIDON2010-05-26 19:29:38
Some ThoughtsDoesn't work for n = 10Steve Herman2010-05-26 18:20:31
half of itDaniel2010-05-26 16:52:29
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