How Many Triangles?

The following problem was at a 2016 entrance test for the MIT PRIMES STEP program.

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.

The intended answer was 8: there were 5 triangles on one side of the paper and 3 on the other.

Most of the students didn’t think that the paper might be two-sided, but they came up with other inventive ideas. Below are some of their pictures, and I leave it to you to explain why they work. All the students who submitted these pictures got a full credit for this problem on the test.

Example 7Example 5Example 4Example 3Example 1Example 2Example 6
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One Comment

  1. Amie:

    I loved this problem; thanks Tanya!

    I gave it to my university pre-service teachers this week as a warm-up exercise. I wrote about some of their ideas at my blog: https://amiealbrecht.com/2016/10/05/how-many-triangles/

    I really enjoy your blog, and get lots of good ideas. Thanks again.