Problem 163
Cross-hatched triangles
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 n2 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).
纵横交错的三角形
考虑一个等边三角形，从三角形的每个顶点向对边的中点引一条线段，构成如下图所示的1级三角形。
我们可以从这个三角形中数出16个不同形状、不同大小、不同方向、不同位置的三角形。使用1级三角形作为材料，我们可以构成更大的三角形，比如右边的2级三角形。在2级三角形中可以输出一百零四个不同形状、不同大小、不同方向、不同位置的三角形。
可以看出一个2级三角形包含有4个1级三角形。一个3级三角形包含有9个1级三角形，而一个n级三角形包含有n2个1级三角形。
如果我们用T(n)表示n级三角形中能够数出的三角形个数，那么
T(1) = 16
T(2) = 104
求T(36)。