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It's cheating because it has memorized the answer to the puzzle instead of using logic to solve it.

Your concept of cheating is simply how LLMs work.

It is not. LLMs do not just memorize; they also extrapolate, otherwise they would be useless. Just like any ML model.

I thought that is essentially what LLM's do? They learn what words/topics are associated with each other and then stream a response.

In some ways, this is proof that Gemini isn't cheating... It is just doing typical LLM hallucination

Llm's can also do some exploring based on combinatoric play of learned elements.