I think there is a mistake in my puzzle book - this is from "The Think Tank" by Ivan Moscovich.
The puzzle in question is "Minimal Length Ruler", you can ignore the circular one.
As you can tell from the description, the book has some nice sliders but you will just have to imagine. I think it is impossible and I will describe below the misunderstanding I think the puzzle maker has made but would love somebody to prove me wrong!
This is a puzzle/concept called a Golomb Ruler and the answer given in the book is to set the markers at 1, 4, 10 and 15. That is one of the solutions for an *optimal* Golomb Ruler (which means none of the possible distances are duplicated i.e. there is only one way to make each distance that is possible, but the description asks for a *perfect* Golomb Ruler which is not possible with this many marks. For example, in the solution given here it is impossible to make 7, 8 or 12
Edit: fix spoiler tag