The recent advancements in the field of large language models (LLMs) have significantly enhanced their capabilities in complex reasoning tasks, particularly in mathematical and logical domains. Researchers are exploring innovative methods to optimize LLMs for problem-solving by categorizing problems, leveraging backward planning, and combining induction with transduction. These approaches aim to mitigate issues such as hallucination and representation collapse, which have been identified as critical limitations in LLM performance. Notably, the development of specialized algorithms and frameworks, such as those focusing on state-transition reasoning and neuroscientific approaches to identify task-relevant units, are pushing the boundaries of what LLMs can achieve. These advancements are not only improving the accuracy and efficiency of LLMs but also providing deeper insights into their internal mechanisms and functional organization. The introduction of benchmarks like FrontierMath underscores the growing need for rigorous evaluation tools to measure and advance AI's mathematical reasoning abilities. Overall, the field is moving towards more sophisticated and specialized applications of LLMs, with a strong emphasis on enhancing their reasoning and problem-solving capabilities.