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⌋?

Question

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⌋?

Answer 1

To solve this problem, let's start by considering a single small unit equilateral triangle in the triangular lattice. There are only two possibilities for the coloring of the segments in this triangle:

1. All three segments have the same color.
2. All three segments have different colors.

Now, let's count the number of ways to color the entire equilateral triangle with side length 33 using these two possibilities.

For the first type of coloring, since all three segments in a small unit equilateral triangle have the same color, there are only three possible choices for the color (Red, Blue, or Green) for each triangle. Therefore, there are 3^(33^2) ways to color the triangle using this type of coloring.

For the second type of coloring, since all three segments in a small unit equilateral triangle must have different colors, there are three possible choices for the first segment, two possible choices for the second segment (since it cannot have the same color as the first segment), and only one possible choice for the third segment (since it cannot have the same color as the first two segments). Therefore, there are 3 * 2 * 1 = 6 ways to color each small unit equilateral triangle of this type. As there are 33^2 small unit equilateral triangles in the entire triangle, the total number of ways to color the triangle using this type of coloring is 6^(33^2).

Therefore, the total number of distinct ways to color the equilateral triangle is 3^(33^2) + 6^(33^2).

To find the value of ⌊log9N⌋, we need to calculate log9(N).

Taking the logarithm base 9 of the total number of colored triangles, we get:

log9(3^(33^2) + 6^(33^2)).

Putting it all together, the value of ⌊log9N⌋ is the floor of log9(3^(33^2) + 6^(33^2)).