Improved Upper Bounds for Slicing the Hypercube
This study addresses the problem of determining the minimum number of hyperplanes, denoted $S(n)$, required to slice all edges of an $n$-dimensional hypercube. By integrating reasoning large language models with CPro1—a tool featuring automated hyperparameter tuning—we design an efficient search algorithm that yields novel constructive solutions, including an 8-hyperplane slicing scheme for the 10-dimensional hypercube ($Q_{10}$). Our main contributions are a significant improvement of the upper bound on $S(n)$ from $\lceil 5n/6 \rceil$ to $\lceil 4n/5 \rceil$, with a refined bound of $4n/5 + 1$ when $n$ is an odd multiple of 5, and the first non-trivial lower bound on the number of edges that can be sliced by fewer than $n$ hyperplanes. These results substantially advance the theoretical understanding of this classical problem in combinatorial geometry.