MathNet 2024-part-01 — 168 problems (EN–VI) Question 138

Given a positive integer $n$, a large equilateral triangle of side length $3n$ is divided into $9n^2$ small equilateral triangles of side length 1. Each small triangle is colored in one of three colors: red, yellow, or blue, such that there are $3n^2$ triangles of each color. A trapezoid consisting of 3 small triangles is called a *standard trapezoid*. For example, in the case of $n = 2$ shown below, the two shaded areas are standard trapezoids. A standard trapezoid is called a *colorful trapezoid* if the three small triangles it contains are all of different colors.
Find the maximum possible number of colorful trapezoids. (These colorful trapezoids do not need to be non-overlapping.)