How Many Triangles Do You See?

A puzzle shows a graphic which includes several triangles and we are asked to count how many triangles we see. What is your answer?

How many triangles?

A popular version of this puzzle states that there are 4 possible answers: 7, 9, 13, and 21.

Let’s take a look at the graphic in question:

There is a minimum of 6 small red triangles, though that isn’t always given as an option. We will count these 6 small red triangles for use in all possible solutions:

After this, there are a variety of ways you can include more triangles. One way is to count larger triangles. This may include 3 “medium” triangles – which include three red and one white triangle – along with a large triangle which encompasses the entire illustration, as shown below:

 

Finally, we might want to count the three small white triangles.

There are a variety of solutions, depending on how you look at it.

We never found 7 or 21 triangles, but we did find several possible solutions:

• 6 (Only small red triangles)
• 9 (6 small red + 6 small white) or (6 small red + 3 medium,  but no large)
• 10 (6 small red + 3 medium + 1 large)
• 12 ( 6 small red + 3 medium + 3 small white)
• 13 (6 small red + 3 medium + 3 small white + 1 large)

How many triangles did you count?