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For which values of positive integer k is it possible to divide the first 3k positive integers into three groups with the same sum?

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User

( 4 months ago )

I'm on a GCSE-a level syllabus currently, and I can't seem to think of any algebraic equation that I could comprise to solve this (with the GCSE/early a level syllabus). The question in full is

For which values of positive integer k is it possible to divide the first 3k positive integers into three groups with the same sum? (e.g. if k = 3, then the first 3k integers are 1,2,3,4,5,6,7,8,9. You can split these into 3 groups of 15, for example {{1,2,3,4,5},{7,8},{6,9}}. so it is possible for k=3)

Any help would be appreciated. Thanks

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User

( 4 months ago )

When I were a lad... we did lots of combinatorics at A-level. Shame it's gone, it's good grounding for a lot of university maths.

This is a nice example of applying algebra to combinatorics. Thank you for the question!

As a starter, consider the case where kk is even. In this case you can pair off numbers so they sum to k+1k+1 (and recall the story of Gauss being asked to sum the numbers 1 to 100). The number of pairs is still a multiple of 3 so group them any way you like!

Next consider the case k=3k=3. You have a solution to that in your question, but there are lots of other solutions. Ever seen the magic square?

 

672159834618753294
There are 2 solutions here, grouping by row or column. Each row has exactly one number from each of {1,2,3}{1,2,3}{4,5,6}{4,5,6} and {7,8,9}{7,8,9}. Also, it has exactly one number of each remainder modulo 3. Same goes for the columns. This guarantees that the sums are the same.

 

So you can see there are various approaches to this sort of problem. It's good practice to explore!

Ok, for the general case where kk is odd. Let's do an example, with k=7k=7 to illustrate.

Write all your numbers in 3 rows.

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