# 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
1 year, 8 months 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.

- 1 year, 8 months 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 $58$).

- 1 year, 8 months 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.

- 1 year, 8 months ago

explain this

- 1 year, 8 months ago

again

- 1 year, 8 months ago

wc

- 1 year, 8 months ago

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

- 1 year, 8 months ago