### APMO 2018 Problem 3

Posted:

**Thu Jan 10, 2019 11:05 pm**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?

(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?