2018 APMO 第 3 题

A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied:
(i) All the squares are congruent.
(ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares.
(iii) Each square touches exactly three other squares.

How many positive integers $n$ are there with $2018 \leq n \leq 3018$, such that there exists a collection of $n$ squares that is tri-connected?

Answer: 501

如果满足以下条件，平面上 $n$ 个正方形的集合称为三连通的：

(i) 所有的正方形都是全等的。

(ii) 如果两个正方形有一个公共点 $P$，则 $P$ 是每个正方形的顶点。

(iii) 每个方格恰好接触其他三个方格。

$2018 \leq n \leq 3018$ 有多少个正整数 $n$，使得存在一个由 $n$ 个三元连通的正方形组成的集合？

答案：501