用\(k\)个不等盒子平铺立方体的最小表面积：尖锐阈值、无故障定律以及降维到二维
The minimum surface area of $k$ unequal boxes tiling a cube: sharp thresholds, a fault-free law, and a reduction to two dimensions
浏览论文内容
中文总结 AI 辅助
研究用\(k\)个不等盒子平铺立方体的最小表面积问题，通过确定\(k = 6\)时的情况及规律，利用无故障定律、降维定理和加倍定律等方法，得到相关结果并给出特定区域无需计算机的证明。
中文摘要 AI 辅助
设\(T(n,k)\)为\(k\)个具有整数边长且维度多重集两两不同的轴对齐盒子的最小总表面积，这些盒子的并集为立方体\([0,n]^3\)。我们完全确定了\(k = 6\)这一列：当\(5\leq n\leq9\)时，\(T(n,6)=8n^2 + 2n + 12\)；对于所有\(n\geq10\)，\(T(n,6)=8n^2 + 2n + 6\)，还有特殊值\(T(3,6)=100\)和\(T(4,6)=148\)。阈值\(n = 10\)等于\(1 + 2 + 3 + 4\)，即四个不同棍长的最小可能和，一般规律成立：对于每个\(k\geq4\)和每个\(n\geq(k - 2)(k - 1)/2\)，\(T(n,k)=8n^2 + 2n + 2(k - 3)\)，阈值为三角形数。三个结构结果支持并扩展了这些值。首先，一个无故障定律：对于所有\(n\geq3\)，将立方体划分为六个无故障平面的盒子的最小内部界面恰好为\(2n^2 + n\)（OEIS A014105），通过对跨越块、浮动块和立方体角的精确计算证明。其次，一个降维定理：在明确范围内，三维问题可简化为二维问题，\(I(n,k)=n^2 + W^*(n,k - 1)\)，其中\(W^*(n,m)\)是用\(m\)个维度两两不同的矩形平铺\(n×n\)正方形的最小内壁；关键要素是一个无条件平板引理。第三，二维问题中间区域的加倍定律：对于\(n = 4,5\)，\(W^*(n,4)=n + 4\)；对于\(4\leq n\leq9\)，\(W^*(n,5)=n + 6\)，通过有限情况树证明；通过降维定理，这给出了\(k = 5\)和\(k = 6\)列中间区域的无需计算机的证明。主要族的下界不使用块的不同性。
英文摘要
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: $T(n,6)=8n^2+2n+12$ for $5\le n\le 9$, and $T(n,6)=8n^2+2n+6$ for all $n\ge 10$, 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 $k\ge 4$ and every $n\ge (k-2)(k-1)/2$, $T(n,k)=8n^2+2n+2(k-3)$, 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 $2n^2+n$ for all $n\ge 3$ (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, $I(n,k)=n^2+W^*(n,k-1)$, where $W^*(n,m)$ is the minimum internal wall of a tiling of the $n\times n$ 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 $4\le n\le 9$, 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.
