How do you solve the Math Puzzle of the Hexagonal Pattern "The disguised triangles?"?

Question

I am looking for how to prove the correct answer.

Answer 1

I'm assuming you are referring to the disguised triangles in the attached link... (link 1). I'm sure there are some rigorous mathematical proofs you could supply, or automated ways to count the triangles, but I basically decided to just use the "brute force" method of counting. The hard part is recognizing all the different triangles while not double counting...

What I finally did was to create several copies of the diagram and tried to color the triangles. You can color the individual triangles (24), right off the bat. Now if you try to pair up triangles, you can pair them one way (12) or another using the flipped image (12). Grouping them further in groups of three or four triangles, I counted another set of (7) and the flip for another (7). Then I noticed four other ways to group four triangles (3), (3), (3), (3). Lastly, there are two large triangles composed of the inner sixteen triangles (1) and (1).

Altogether I got seventy (76) triangles. You can double check my work using the attached image (link 2). I think I got them all and didn't double count...

Answer 2

I count 58. Hard to prove. You could start with each corner and count how many eminate from there, but puzzling is right, you have to avoide double counting.