When n = 10, the answer is 16 by any of the correct solutions here. However, the example in the question clearly gives a scenario that you have to pay 21 to win the game. If you only have 16, how is it possible to guarantee a win?

I see a post said this problem is essentially "Best strategy and worst luck". However, assume there is a person coming into play, how is it possible that he knows the best strategy? If not, he is very likely to make some bad moves. If a win needs to be guaranteed, then should we consider the "worst strategy"?

So my question is, why each time we minimize the max cost?