Tricolor Triangle

Probability Level 5

An equilateral triangle with side length 3333 is divided into 33233^2 smaller unit equilateral triangles each with side 1, forming a triangular lattice. We color each segment of length 1 either Red, Blue or Green, subject to the condition that each small unit equilateral triangle has 3 sides with either 3 different colors or all the same color. If there are NN distinct ways to color this triangle, what is the value of log9N \lfloor \log_9 N \rfloor ?

This problem is proposed by Hendrata.

Details and assumptions

Two colorings are distinct if at least one segment is colored differently.
Rotations and reflections are considered distinct colorings.


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