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?

#NumberTheory

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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.

Patrick Corn - 3 years, 7 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 $58$ ).

Brian Charlesworth - 3 years, 7 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.

Patrick Corn - 3 years, 7 months ago

explain this

Ayush Jain - 3 years, 7 months ago

again

Ayush Jain - 3 years, 7 months ago

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

Vilakshan Gupta - 3 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$