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.
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:
- The Star Guardian moves at a constant speed of $g$, in any direction she wants.
- She is only able to carry one Star at a time.
- Once left alone, the Stars always travel in their eponymous directions at speed 1.
- If only two or three Stars meet, they will just pass through each other without any interaction.
Let $G=\{g\vert$ the speed $g$ Guardian can reunite the Stars in finite time$\}$. Define $g^*=\inf G$.
Question 1: What is the value of $g^*$?
Question 2: if we can put the four Stars in any other initial positions, and give them preferred directions other than the four cardinal directions, what is the maximum $g^*$ we can get?
Source: I originally created this as a puzzle for the site puzzling.stackexchange.com, but the dynamics turned out to be more complex than I intended for puzzle site. For progress on question 1, the smallest $g$ found so far is $10/3$, but no one knows whether it could be even smaller. The puzzle was also previously posted to MathOverflow, but math.se may be a more appropriate site for it.