Improving 25 square packing upper bounds through evolutionary LLM program search (slightlysupervised.substack.com)

A recent breakthrough in square packing has achieved record-breaking configurations for packing unit squares into the smallest possible square containers using an evolutionary approach powered by large language models (LLMs). Notable records include optimal packings for n=51, n=103, and n=105, accomplished in just two days with over 7,500 CPU hours. This work builds on traditional methods but leverages the creativity of LLMs to explore program generation and optimization, breaking records that had not changed in decades. The significance of this advancement lies in its innovative use of LLMs in a program search context, similar to techniques like AlphaEvolve. By employing a strategy that iteratively refined packing algorithms with novel techniques—such as rounding and rearranging squares, and optimizing initialization patterns—the researchers not only improved packing efficiency but also demonstrated the potential for AI to contribute to solving complex mathematical problems. The findings highlight how combining human insight with AI can lead to unexpected and faster solutions in mathematical research, marking a notable step forward for the AI/ML community.