# Minimizing maximal adjacent integer sum on a circle

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.

• 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"? – Adrian Keister Jun 20 '18 at 19:23
• @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. – Hans Jun 20 '18 at 19:52
• You could formulate it as a mixed integer linear optimization problem to get some examples of optimal arrangements. – LinAlg Jun 21 '18 at 1:49
• @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? – Hans Jun 21 '18 at 3:44
• Yes. Minimizing the maximum can be phrased with purely linear constraints. – LinAlg Jun 21 '18 at 11:29

• 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. – Hans Jun 21 '18 at 4:53
• $k$ means the sum is over $k$ adjacent/consecutive natural numbers. – Hans Jun 21 '18 at 17:45