Answer
The answer is as follows. 
Katrielle and the Millionaires' Conspiracy walkthrough
Puzzle Location: Conundrum Kings' Cloakroom
When the star moves, it leaves a trail in its wake.
Can you fill the whole night sky with its trail?
The mysterious objects floating in the sky are warp holes.
They are paired according to their appearance, and when the star goes into one of them, it reappears out of the other.
Touch a square to move to it, and make sure the star moves through all of the squares.
You are not allowed to go through a square that already has a trail on it.
There is more than one route the star can take, but let's look at an example.
First, move to the bottom, then go around the closest warp hole in order to enter it from the right.
Continuing on from hint one...
After warping to the top-right, move right and then down to the next warp hole.
Continuing on from hint two...
After warping to the top, aim for the warp hole on the left-hand edge.
Enter the warp hole from the top-left corner, leaving a trail all the way to the corner.
Continuing on from hint three...
After the final warp, move right to leave a trail in the corner, and then simply travel through the remaining empty squares.
The answer is as follows.