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 Make it equal (Posted on 2010-05-26)
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. Ans:5,6.8.9.11,12.....(3n-1),(3n),....

 Subject Author Date re: Full proof. Ady TZIDON 2010-05-27 13:51:04 Full proof. Jer 2010-05-27 11:32:44 re: No problems Ady TZIDON 2010-05-27 06:19:31 No problems Steve Herman 2010-05-27 01:32:23 re: half of it SOME HINTS(a partial spoiler) Ady TZIDON 2010-05-26 19:29:38 Doesn't work for n = 10 Steve Herman 2010-05-26 18:20:31 half of it Daniel 2010-05-26 16:52:29

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