Really cool idea for a puzzle game! The easier difficulties especially feels like a safe-cracking minigame
How did you go about calculating the lowest possible distance for each level? is there a formula or is it just done iteratively?
The solutions are brute-forced, in some cases (as the algorithm is vulnerable to local minima and it's got limited degrees of freedom) where you can get over 100%. The highest I've seen was 108%, but there's something nice about beating the algorithm.
Here's the algorithm as code, the step variable determines how many steps it takes to do a spin of a ring, then it counts through, scoring each change, and finds the best one it can in two passes through all the rings. More passes could do better, but on secret expert mode (holding 'g' will unlock super secret expert mode) it's already really demanding to compute the problem space, and it gets slower and less accurate. If I were to do a daily puzzle for instance, with say 30 rings, I would solve it locally and store the solution, but since these puzzles are procedurally generated the solutions need to be done at the time of generation. 
I guess the short answer is that it's done iteratively. I'm mixed on the 100+% thing, and for psychological reasons, it may be better to cap it at 100, and solutions over 100 would still just display as 100 as has been suggested to me in the past.
Thank you for playing!