Consider an equilateral triangle in which straight lines are drawn from each vertex to the middle of the opposite side, such as in the size 1 triangle in the sketch below.
<imgclass="img-responsive center-block"alt="triangles with size 1 and size 2"src="https://cdn.freecodecamp.org/curriculum/project-euler/cross-hatched-triangles.gif"style="background-color: white; padding: 10px;">
Sixteen triangles of either different shape or size or orientation or location can now be observed in that triangle. Using size 1 triangles as building blocks, larger triangles can be formed, such as the size 2 triangle in the above sketch. One-hundred and four triangles of either different shape or size or orientation or location can now be observed in that size 2 triangle.
It can be observed that the size 2 triangle contains 4 size 1 triangle building blocks. A size 3 triangle would contain 9 size 1 triangle building blocks and a size $n$ triangle would thus contain $n^2$ size 1 triangle building blocks.
If we denote $T(n)$ as the number of triangles present in a triangle of size $n$, then