2007 USAMO 第 4 题

An animal with $n$ cells is a connected figure consisting of $n$ equal-sized square cells.${}^1$ The figure below shows an 8-cell animal.

A dinosaur is an animal with at least 2007 cells. It is said to be primitive if its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.

${}^1$Animals are also called polyominoes. They can be defined inductively. Two cells are adjacent if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.

具有 $n$ 个细胞的动物是由 $n$ 个大小相等的方形细胞组成的连通图形。${}^1$ 下图显示了一个 8 细胞的动物。

恐龙是一种至少有 2007 个细胞的动物。如果它的细胞不能分成两个或多个恐龙，那么它就被认为是原始的。找到并证明原始恐龙细胞的最大数量。

${}^1$动物也称为多骨牌。它们可以归纳地定义。如果两个单元共享完整的边缘，则它们是相邻的。单个细胞是一种动物，给定一只具有 $n$ 个细胞的动物，通过使新细胞与一个或多个现有细胞相邻来邻接一个新细胞，即可获得具有 $n+1$ 个细胞的动物。