# How many triangles?

It’s been quiet on the blog, but a lot has been happening. University classes in Adelaide have just resumed after a two week mid-semester break. To warm up, I gave my MATH 1070 students the following problem. I found it via Tanya Khovanova who states that it was an entrance problem for the 2016 MIT PRIMES STEP Program. (Read more on Tanya’s blog.)

I drew several triangles on a piece of paper. First I showed the paper to Lev and asked him how many triangles there were. Lev said 5 and he was right. Then I showed the paper to Sasha and asked him how many triangles there were. Sasha said 3 and he was right. How many triangles are there on the paper? Explain.

Here are some solutions from my students, all considered to be correct. The ones in blue originally appeared in Tanya’s blog post (although several students came up with these too). Additional ideas are shown in red below. The black rectangle shows the piece of paper. Two of the rectangles contain instructions instead of diagrams.

Update (8 October 2017): A few new ideas from this week’s classes are shown in purple.

I loved this as an opener to encourage creative problem solving. Thanks Tanya!

## 3 thoughts on “How many triangles?”

1. Frank Wilhoit says:

How did Sasha see three triangles (really, three of anything) in the pentagram?

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• Anthony Anguiano says:

Frank, this photo shows a triangle in the pentagram which is repeated 3 times symmetrically:

If you only saw triangles of this form and ignored those on the edges of each point, you would find three triangles.

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2. Albert Putnam says:

There are five of those triangles too

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