## Cross-hatched triangles

### Problem 163

Published on Saturday, 13th October 2007, 02:00 am; Solved by 1133; Difficulty rating: 70%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.

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

T(1) = 16

T(2) = 104

Find T(36).