Problem 3
A collection of 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 in common, then is a vertex of each of the squares. (iii) Each square touches exactly three other squares. How many positive integers are there with , such that there exists a collection of squares that is tri-connected?
Step 2 of 4: Build a chainable local unit
Detailed analysis
One exhibits two small configurations of congruent squares meeting only at vertices: a strip of arbitrary (even) length in which every square touches three neighbours except the two end squares, which touch two; and a fixed corner block, several copies of which can be chained together to turn a corner while keeping every internal square at exactly three touches. Both building blocks only ever meet other squares at vertices, satisfying condition (ii).