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If you click on the "puzzle" button here:

https://www.thesyntacticretina.com/

you will see a SEEMINGLY "mysterious" linear-algebraic "puzzle" for which John Burkardt has kindly provided the solution (to see this solution, click on the "solution" button.)

Question:

Are there any cases of this result other than those arising from "N-free" permutations of (0,...,n-1), or from "N-free" posets corresponding to "N-free" permutations?

David Halitsky
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  • The solution says so right in it. The only thing that is necessary of the vectors is that they have the same norm. Two permutations of $1$ to $n$ is just one way of guaranteeing it. And although I haven't translated the solution to check it, I suspect that it works by doubling that 45 degree rotation. – Paul Sinclair Nov 25 '17 at 00:36
  • @PaulSinclair - Thanks for taking the time to respond, and I am sorry that my question was unclear. What I am looking for is another case where the "same norm" condition is satisfied by some set of vectors that DON'T arise from a permutation, but DO arise in some already known pure or applied context not related in any way to permuatations. If you can think of any such case, I would be very grateful to know of it. – David Halitsky Nov 25 '17 at 00:54
  • Getting two vectors with the same norm is very easy. Just pick two non-zero vectors at random and divide each by its norm. All unit vectors have norm $1$. Vectors with the same norm arise in any context where the norm of a vector is defined. – Paul Sinclair Nov 25 '17 at 01:17
  • Can you think of a case where you don't divide each by its norm in order to satisfy the condition trivially ? (Again, a case that arises in some known context . . . ) Thank you for your patience here - I hope I'm not irritating you at this point . . . – David Halitsky Nov 25 '17 at 01:36
  • Usually when you have multiple vectors, you want them to be the same length. Converting vectors to be the same length is an easy and normal thing to do. Anywhere you see norms, you will find vectors of the same length commonly used. It is not some odd or mysterious or unusual thing. It is quite common. – Paul Sinclair Nov 25 '17 at 02:00
  • So for example, scaling one vector up or down to match the length of another (instead of scaling both to 1)? – David Halitsky Nov 25 '17 at 02:03
  • Let us continue this discussion in chat. – David Halitsky Nov 25 '17 at 02:09

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