I basically wrote brute force with cache. That was enough to pass hard.

The main idea is we don't re-throw dice in the turn if we still can get the answer. Then I just kept all groups I could have by current turn and iterated which number I get in current turn. Then honestly update vector, sort it and check that given one is still greater in each position.

Why that works fast? It seems we don't have that many states.

It seems the most challenging part was to understand the statements. Basically we need to sort all segments, then take one by one and greedy place robots (you actually use each of robots once, not reuse them). Then you might have some suffix left for the robot. We just take next segment and check if this robot can cover it as well. If it can, we do nothing. If it partially covers it, remove that prefix from the segment and then put new robot as described before.

Could you provide any insight or intuition on why the recursion depth is actually low?

Is this joke

Well, you have an exciting opportunity to learn it. Imo, you cannot find a truly involved problem in terms of probability because a few people would solve it. So you only have to understand basics from probability (and presumably a lot more at coding).

Try this may be it'll help