I wonder if the authors can explain the aparent inconsistency between what we now know about R1 and their statement “They don’t engage in logical reasoning” from the first lesson. My simple-minded view of logical reasoning by LLMs is that the hard question (say a math puzzle) has a verifiable answer that is hard to produce and is easy to verify, yet within the realm of knowledge of humans or the LLM itself, so the “thought” stream allows the LLM to increase its confidence by a self-discovered process that resembles human reasoning, before starting to write the answer stream. Much of the thought process that these LLMs use looks like conventional reasoning and logic, or more generally higher level algorithms to gain confidence in an answer, and other parts are not possible for humans to understand (yet?) despite the best efforts by DeepSeek. When combined with tools for the boring parts, these “reasoning” approaches can start to resemble human research processes as per the Deep Research by OpenAI.