The towers of Hanoi one is kind of weird, the prompt asks for a complete move by move solution and the 15 or 20 disk version (where reasoning models fail) means the result is unreasonably long and very repetitive. Likely as not it's just running into some training or sampler quirk discouraging the model to just dump huge amounts of low-entropy text.

I don't have a Claude in front of me -- if you just give it the algorithm to produce the answer and ask it to give you the huge output for n=20, will it even do that?