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Title: Algorithmic Thinking Theory for Foundation Models Abstract: The last few months have witnessed tremendous advances on Large Language Model (LLM) reasoning capabilities with Gemini and GPT winning a gold medal at the International Mathematical Olympiad (IMO) [1] and International Collegiate Programming Contest (ICPC) [2], and more recently to prove new theoretical computer science results [3]. Several papers have shown that inference scaling techniques significantly improve the reasoning performances of the LLM, in particular for the IMO [4]. We will discuss these results and how one can frame the problem as an optimization problem, relate it to empirical results shown in [4], and derive optimal (algorithmic) thinking strategies. We will also discuss avenues for refining the model and improving inference scaling methods. Joint work with Mohammad Hossein Bateni, Yuzhou Gu, Silvio Lattanzi, Simon Meierhans, and Christopher Mohri [1] https://deepmind.google/discover/blog... [2] https://deepmind.google/discover/blog... [3] https://arxiv.org/abs/2602.03837 [4] https://arxiv.org/abs/2507.15855