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How Many Triangles Do You See?

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?

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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:

triangles1

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:

triangles6

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:

triangles4

 

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

triangles3

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?

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  • Richard J

    I counted 13 first try. I cannot see how to come up with 21. I do see how it is possible to come up with 7. You would have to not count the white areas, leaving six small red and than the one large which could be counted as red.

  • Timothy Campbell

    I got 13 on my first count.

    I cannot see 7 or 21, either. If there IS a way to come up with 7, then somebody might consider rotating the largest triangle to its three possible orientations for a total of 3 x 7 = 21. But that doesn’t explain how you can get 7 in the first place.

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