1
$\begingroup$

I was able to find (and prove) arrangements that would result in the sum of the products of adjacent pairs attain the maximum.

I am able to conjecture that the arrangement that would result in the minimum sum of products of adjacent pairs: begin with 1 and 2, squeeze n between the two. We can have 1 in the clockwise direction and 2 in the anticlockwise direction of n. We can then add n-1 traversing in clockwise direction of 1. we can then add 3 (clockwise from n-1), then add ( n-2 clockwise from 3),...

I have tried a few different ways of trying to prove that this arrangement is the desired minimal arrangement but all proofs have broken down. Would be great if someone could throw some light.

Improve this question
$\endgroup$
3
  • 7
    $\begingroup$ This is essentially problem 10725 from the American Mathematical Monthly. You can look up the solution here jstor.org/stable/2695399 $\endgroup$
    – Gjergji Zaimi
    Dec 28, 2020 at 18:36
  • $\begingroup$ That was quite helpful. Thank you! $\endgroup$
    – David
    Dec 29, 2020 at 9:27
  • $\begingroup$ Thanks for the solution. I noticed, however, that the minimal arrangement is different from the one I conjectured. I essentially looked at a few cases and generaized. I wonder if there is some transparent process through which one might arrive at the arrangement proposed in the solution. As it stands, it seems a bit opaque how one might arrive at the proposed arrangement, $\endgroup$
    – David
    Dec 29, 2020 at 9:43

0

Reset to default