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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 month 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.

Patrick Corn - 4 weeks, 1 day ago

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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 - 4 weeks, 1 day ago

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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 - 4 weeks, 1 day ago

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

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