Problem 3

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\le n\le 3018

, such that there exists a collection of

n

squares that is tri-connected?

Step 4 of 4: Count the even n in range

Previous step Next step

|\{n\in\mathbb Z: 2018\le n\le 3018,\ n\ \text{even}\}|=\frac{3018-2018}{2}+1=501

Detailed analysis

Every $n$ with $2018\le n\le 3018$ satisfies $n\ge 38$ , so by steps 1 and 3 a tri-connected collection of $n$ squares exists exactly when $n$ is even. The even integers from $2018$ to $3018$ inclusive number $\frac{3018-2018}{2}+1=501$ .