Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Can you tell me how many triangles are in this picture? I've counted 96, not sure that I got right answer.

enter image description here

share|improve this question
3  
Perhaps you could try counting them by using the symmetry of the shape (if you haven't already). –  Shaun Nov 21 '13 at 13:50
    
Clearly there are 19 unique points. Number of triangles that can be formed from 19 points taking 3 at a time=$19\choose3$. Also note that some of the points lie on a straight line. When we took $19\choose3$ we included the ones that lie on a straight line. Perhaps finding the number of lines and subtracting that from $19\choose3$ will yield the answer. –  GTX OC Nov 21 '13 at 14:05
    
@GTXOC I see 19 unique points... –  anorton Nov 21 '13 at 14:08
    
Alright I missed the ones in the middle of the base. –  GTX OC Nov 21 '13 at 14:09
    
@All: Not all triangles are valid even if the points could form one, consider lines between "star-corners". –  AlexR Nov 21 '13 at 14:11

2 Answers 2

Let's see! We'll break it up into the following cases (where an apex means one of the six points of the star):

A: Triangles that contain three apexes
There are 2 of these.

B: Triangles that contain a pair of opposite apexes
For each such pair, there are 2 of these, so 6 in total.

C: Triangles that contain a pair of non-opposite apexes
For each non-adjacent apex pair, there are three of these, so 18 in total.

D: Triangles that contain exactly one apex
For each apex, I count 10 such triangles, so 60 in total.

E: Triangles that contain no apex
For each edge of the internal hexagon, there are 3 triangles, so 18 in total.

I make that 2 + 6 + 18 + 60 + 18 = 104.

share|improve this answer
    
I verified and this seems to be correct. –  AlexR Nov 21 '13 at 14:16
    
Look OK to me too, though categories $D$ and $E$ could be somewhat refined. –  Marc van Leeuwen Nov 21 '13 at 14:24
    
@Marc: I agree with you about category D. –  TonyK Nov 21 '13 at 14:29
    
A small reality check: If you dismiss the two big equilateral triangles, all other triangles belong to orbits of $6$ under rotations by $2\pi/6$, so $T-2$ must be divisible by $6$, which $T=104$ is but $T=96$ (the OP's original count) is not. –  Barry Cipra Nov 21 '13 at 14:54
    
For Category $D$, you can take the apex at the very top and count triangles that are and aren't symmetric across the "$y$"-axis. There are $2$ that are and $4$ pair that aren't. –  Barry Cipra Nov 21 '13 at 15:13

I only get 98. I am missing the 3rd triangle under section C. I only see 2 triangles that contain a pair of non-opposite apexes.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.