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Arrange $\{1,2,\cdots,n\}$ on a circle. What are the arrangements that minimize the maximal sum of all adjacent $k$ integers? Some nontrivial examples of $(n,k)$ are welcome. A random algorithm that gives a probabilistic bound would be great.

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  • $\begingroup$ It's not quite clear what you're asking. Perhaps a graphic illustrating how you're arranging the integers? How are you defining "maximal sum of all adjacent $k$ integers"? $\endgroup$ – Adrian Keister Jun 20 '18 at 19:23
  • $\begingroup$ @AdrianKeister: Given a circular arrangement of those $n$ natural numbers, consider the set $K$ of $k$ adjacent natural numbers on the circle. Take the set $S$ of the sums of those elements of $k$ adjacent natural numbers on the circle. The maximum $M$ of the element of $S$ is then a function of the circular arrangement. Now find the minimal $M$ amongst all the circular arrangements. Is this clear? I can certainly write the question in the formal language. But I thought it was clearer this way. Let me know what you think of it after my explanation. $\endgroup$ – Hans Jun 20 '18 at 19:52
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    $\begingroup$ You could formulate it as a mixed integer linear optimization problem to get some examples of optimal arrangements. $\endgroup$ – LinAlg Jun 21 '18 at 1:49
  • $\begingroup$ @LinAlg: Good idea. I need $n^2$ binary variables with associated $2n$ inequalities to express the distinctiveness of the $n$ variables to formulate the problem in the mixed integer linear programming, don't I? $\endgroup$ – Hans Jun 21 '18 at 3:44
  • $\begingroup$ Yes. Minimizing the maximum can be phrased with purely linear constraints. $\endgroup$ – LinAlg Jun 21 '18 at 11:29
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we don't want the two largest numbers in the set: which is n and n-1 to be adjacent to each other: You can quickly figure out the pattern that the lower bound of the maximum adjacent sum is n+2:

enter image description here

The general pattern is to start with n - the greatest number - on the very top position and make sure that n-1 isn't beside n, and then if you can n-2 isn't beside either n or n-1 and n-3 isn't beside any of those and so on. I placed them exactly 2 spots away in the clockwise direction (although you could use the Counter-Clockwise convention instead).

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  • $\begingroup$ No, it is not so simple. $\endgroup$ – Hans Jun 21 '18 at 3:08
  • $\begingroup$ @Hans Wait then what's wrong with my diagram? $\endgroup$ – ray lin Jun 21 '18 at 3:10
  • $\begingroup$ Your answer is very sloppy. You do not even specify what $k$ you are using. Apparently you set $k=2$. There is no proof 2even some kind of plausibility justification. $\endgroup$ – Hans Jun 21 '18 at 4:53
  • $\begingroup$ @ Hans I have a question, if my diagram is k=2 how would k = 3 work? Would I select 2 of the numbers to the CW position and 1 on the CCW position or something else? $\endgroup$ – ray lin Jun 21 '18 at 16:11
  • $\begingroup$ $k$ means the sum is over $k$ adjacent/consecutive natural numbers. $\endgroup$ – Hans Jun 21 '18 at 17:45

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