Exact minimal square-container side length for packing n unit squares

Establish the exact minimal side length L of an axis-aligned square container required to pack n non-overlapping unit squares (with arbitrary rotations allowed) for general n > 10, i.e., determine L(n) in cases where the minimum is currently unknown.

Background

The square-packing task seeks the smallest axis-aligned square (side length L) that can contain n unit squares without overlap, allowing rotations. Validity is easy to verify, and known results exist for n ≤ 100 (some proven optimal). The benchmark sets human references and bounds to enable partial scoring while leaving room for improvement.

The paper explicitly notes that for n > 10, most exact minimal L values remain open, motivating formal determination of L(n) beyond small n where optimal packings are known.