# geometry and combinatorics

posted by
**sanji**
.

An equilateral triangle with side length 33 is divided into 33^2 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 N distinct ways to color this triangle, what is the value of ⌊log9N⌋?

Details and assumptions:

-Two colorings are distinct if at least one segment is colored differently.

-Rotations and reflections are considered distinct colorings.

-Given a single unit equilateral triangle, there are 9 ways that it can be colored.