2026/07/17 by Diego Lago Gómez
Mathematics · #math.CO
Let T(n,k) be the minimum total surface area of k axis-aligned boxes with integer sides and pairwise distinct dimension multisets whose union is the cube [0,n]3. We determine the column k=6 completely: T(n,6)=8n2+2n+12 for 5≤ n≤ 9, and T(n,6)=8n2+2n+6 for all n≥ 10, together with the exceptional values T(3,6)=100 and T(4,6)=148. The threshold n=10 equals 1+2+3+4, the least possible sum of four distinct stick lengths, and the general law holds: for every k≥ 4 and every n≥ (k-2)(k-1)/2, T(n,k)=8n2+2n+2(k-3), with thresholds at the triangular numbers. Three structural results support and extend these values. First, a fault-free law: the minimum internal interface of a partition of the cube into six boxes with no fault plane is exactly 2n2+n for all n≥ 3 (OEIS A014105), proved by an exact accounting of spanning pieces, floating pieces and cube corners. Second, a reduction theorem: within an explicit range, the three-dimensional problem collapses to a two-dimensional one, I(n,k)=n2+W^*(n,k-1), where W^*(n,m) is the minimum internal wall of a tiling of the n× n square by m rectangles of pairwise distinct dimensions; the key ingredient is an unconditional slab lemma. Third, a doubling law in the middle regime of the 2D problem: W^*(n,4)=n+4 for n=4,5 and W^*(n,5)=n+6 for 4≤ n≤ 9, proved by finite case trees; via the reduction theorem this gives computer-free proofs of the middle regimes of the columns k=5 and k=6. The lower bound for the main family does not use the distinctness of the pieces.