How many triangles ?

Question

You have 5 equally sided and sized triangles. How many triangles can you form with all 5 provided that no triangles overlap.

I should have asked what the maximum amount of triangles you can form without overlapping the triangles.

Answer 1

I can make 11.
With the first 4 triangles, align 2 on top and 2 on the bottom to form a regular octahedron (which is 3-dimensional.) This immediately forms 8 triangles which are the faces of the octahedron.
The 5th triangle must be placed inside the octahedron so that two of its verticies touch opposite edges of the octahedron and the third vertex touches a third edge. This adds 3 triangles.
For a picture see:

http://mudandmuck.com/5triangles.jpg

Sorry Michael M, but you cannot squeeze together a regular octahedron without deforming the size/shape of the triangular faces. Thats why its a unique shape, one of the five platonic solids: there is only one regular configuration for an octahedron. Therefore I still believe 11 is the maximum in 3-space.

This is of course assuming planar triangles. If non-planar triangles are permitted, then 455 (or even more) triangles are possible!
This is equivalent to the combination of 15 things taken 3 at a time. Take all 5 triangles and "bend" them so that their verticies all meet in one point. What is formed is a network of 15 lines that start and end at the same point. Then any three lines form a "bent" triangle with coincident verticies. The number of 3-line combinations is:
C(15,3) = 455.

Answer 2

11 is very impressive Scott, and I really like your visual. However I can get 12.

Do the same thing that Scott did, except we can move the black triangle into a position to make one more triangle. The octahedron can be compressed by squeezing the top and bottom points together without changing the shape of the equilateral triangles. If they are squeezed to just the right distance, the black triangle can touch the very top point, and two legs of the bottom triangles. The sides of the octahedron form 8 triangles, the black triangle = 1, and there are 3 triangles that are formed from using the black triangle and legs from the other red and blue triangles. I really which that I could show a visual of this.

Answer 3

9

Answer 4

Draw a square, now draw 2 lines that are perpendicular to each other, now draw 2 lines that connect 2 of the vertices outside the square. This will give you 5 triangles. Those 5 triangles will make 9 triangles.

ANS : 9

Answer 5

8

Answer 6

I GOT 9!!!

Answer 7

I love this question!!! :)