发表机构
Rotman School of Management, University of Toronto（多伦多大学罗特曼管理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明用单一$2\ imes4$乐高砖重建任意多砖模型需四倍放大且为最优，并发现忽略连通性时最小缩放比例恒为1、2或4，绝不为3。
AI 中文摘要
能否仅使用常见的$2\ imes4$砖块重新搭建由多种矩形砖块构成的乐高模型？将每个维度放大四倍可使砖块契合，但它们还必须相互连接。我们给出一种四层排列方式，利用每个源立方体八块竖立砖块，连接所有有限面连通的格立方体并集，并证明不存在更小的通用放大倍数。第二个问题有一个出人意料的答案：若忽略连通性，最小整数缩放比例始终为$1$、$2$或$4$，绝不可能是$3$。缺失的值源于在每个放大单元内部单独应用的棋盘格论证。
英文摘要
Can a LEGO model built from many kinds of rectangular bricks be rebuilt using only the familiar $2\times4$ brick? Enlarging every dimension by four makes the pieces fit, but they must also connect. We give a four-layer arrangement that joins every finite face-connected union of lattice cubes, using eight upright bricks per source cube, and show that no smaller universal enlargement works. A second question has an unexpected answer: if connectivity is set aside, the minimum integer scale is always $1$, $2$, or $4$, never $3$. The missing value follows from a checkerboard argument applied separately inside each enlarged cell.
Comments6 pages, 4 figures
