Could a human code the solution if they didn't learn to code from someone else? No. Could they do it if someone didn't tell them the rules of towers of hanoi? No.

That doesn't mean much.

It does since humans where able to invent a programming language.

A human can learn and understand the rules, an LLM never could. LLMs have famously been incapable of beating humans in chess, a seemingly simple thing to learn, because LLMs can't learn - they just predict the next word and that isn't helpful in solving actual problems, or playing simple games.