# Maximize/Minimize It

Let $$x_1,x_2,\cdots,x_{19},x_{20}$$ be permutations of numbers from 1 to 20. Then what will be the maximum value of the summation $\large x_1x_2+x_2x_3+\cdots+x_{19}x_{20}+x_{20}x_1$ At which values of $$x_1,x_2,\cdots,x_{19},x_{20}$$ will that maximum occur? And when will minimum occur?

Note by Vilakshan Gupta
10 months, 2 weeks ago

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I looked at this a little but I haven't done anything substantive. It seems like it should be possible to show that the minimum is obtained by alternating large and small numbers in a smart way.

E.g. for $$n=4$$ you get a minimum when you do something like $$3142.$$

For $$n=6$$ you do $$351624.$$

For $$n=8$$ you do $$53718264.$$

For $$n=10$$ you do $$57391(10)2846.$$ And so on.

Inductively, to get from one level to the next, it looks like you add two to the numbers larger than $$n/2,$$ and then on the outside you stick $$n/2$$ and $$n/2+1$$ in a smart way.

I don't have any kind of proof though, and I haven't thought about the maximum yet.

- 10 months, 1 week ago

For the maximum for $$n = N$$, going from $$N$$ down to $$1$$ while alternating from one side of the sequence to the other might be optimal. For $$n = 4$$ this would be $$4213$$, yielding a product-sum of $$25$$, (compared to your minimum of $$21$$), and for $$n = 6$$ it would be $$642135$$, yielding a product-sum of $$82$$, (compared to your minimum of $$58$$).

- 10 months, 1 week ago

Here's one idea I tried: for it to be a minimum (resp. maximum), the result has to be smaller (resp. bigger) than what you get after doing a transposition. For say $$x_2,x_3,$$ this ends up being the statement that $$(x_1-x_4)(x_2-x_3) < 0$$ (resp. $$>0$$). And so on for other pairs of adjacent indices. Your examples satisfy this criterion.

There is another condition you get if you transpose $$x_2,x_4$$ (say), and then you can look at a $$3$$-cycle too. Taken together, those give you pretty strong conditions on the permutation, but I don't know if they determine it uniquely.

- 10 months, 1 week ago

Please help me with this anyone. @Brian Charlesworth , @Chew-Seong Cheong , @Anirudh Sreekumar , @Sharky Kesa anyone?

- 10 months, 1 week ago

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- 10 months, 1 week ago