# geometry and combinatorics

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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.

• geometry and combinatorics -

297

• geometry and combinatorics -

297 is correct. how did you come up with that answer?

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