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 3 of 4: Chain the units into a closed loop

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n\ \text{even},\ n\ge 38\ \Longrightarrow\ \exists\ \text{tri-connected collection of}\ n\ \text{squares}

Detailed analysis

Combining several corner blocks with strips of adjustable even length, one closes the chain into a loop so that every square, including those where strips meet corner blocks, touches exactly three others (the corner blocks absorb the two-touch ends of the strips). Adjusting the strip lengths realizes every even total count $n\ge 38$ , giving a tri-connected collection of exactly $n$ squares.