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On an infinite plane, the Prime Star has disintegrated into four constituent stars, the North Star, the South Star, the East Star and the West Star, each traveling at a constant speed of $1$ in their eponymous directions.

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The Star Guardian at the center wants to reunite the four Stars back into the Prime Star again, which can only be achieved if the four Stars meet at a single point in spacetime. Furthermore:

  1. The Star Guardian moves at a constant speed of $g$, in any direction she wants.
  2. She is only able to carry one Star at a time.
  3. Once left alone, the four Stars always travel in their eponymous directions at speed $1$.
  4. If only two or three Stars meet, they will just pass through each other without any interaction.

Question: what is the minimum (or infimum) value of $g$ (denoted $g^*$) for her to be able to reunite the Stars in finite time?

Generalization 1: suppose the Prime Star breaks into $N$ Sub Stars symmetrically, is $g^*(N)$ a simple function of $N$? Do we have, for instance, $g^*(N)=N-1$?

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Generalization 2: suppose we break the symmetry by putting the four Stars in some other initial positions, and giving them some other desired directions. Do we always have $g^*_{other}\leq g^*$?

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Note: I originally created this as a puzzle for the site puzzling.stackexchange.com, where an amazing solution with $g=10/3$ was found. It was conjectured that $g^*=3$, but no one could prove it.

Edit: Previously I tried to generalize the problem to higher dimensions but encountered some difficulties, as pointed out by Roland Bacher in the comment. So I now reserve my generalization for 2 dimensions.

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  • $\begingroup$ This problem is linear: The minimal time needed is proportional to the distance to the origin of a star at the beginning. (I think you forgot to mention that all four stars are at distance $1$ when the Star Guardian realizes that they are gone. Another point: There is no "even distribution of $n$ points" on an $m$-dimensional sphere for $m\geq 3$. Evenly distributed makes only sense on a one-dimensional circle (and is well defined up to rotation), I think. $\endgroup$
    – Roland Bacher
    Aug 20, 2021 at 13:16
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    $\begingroup$ @RolandBacher I'm asking for the minimum speed, not time, so the initial distance is irrelevant. As for the even distribution, I'm being a bit lax there. What I had in mind is an even distribution on a sphere of unit distance from the origin in m dimensional space. I was thinking of them as vertices of a regular polyhedron. So yes, I just realized that there're only finitely many of them for $m\geq 3$. I'll think about a better way to generalize. $\endgroup$
    – Eric
    Aug 20, 2021 at 14:38
  • $\begingroup$ Correct. By the way, I think there is only an infimum: time needed to do the job will get huger and huger if the Guardian goes slowlier. $\endgroup$
    – Roland Bacher
    Aug 20, 2021 at 15:17
  • $\begingroup$ @RolandBacher Actually, a natural way to generalize this to higher dimension $m$ is to have $2m$ stars evenly distributed on the unit circle centered at the origin. Place the stars at (1,0,0,...,0), (-1,0,0,...,0),(0,1,0,...,0),(0,-1,0,...,0) etc.. $\endgroup$
    – Eric
    Aug 21, 2021 at 10:30
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    $\begingroup$ Now posted to m.se, math.stackexchange.com/questions/4230133/… without informing either site. That's an abuse. $\endgroup$
    – Gerry Myerson
    Aug 22, 2021 at 9:27

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