How Many Triangles Do You See?

Determine your answer, then look below for the answer.

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In our case, = 4, and p = 3. Any triangle we find in the drawing should have one top vertex and two others on the same horizontal line, so for each horizontal line, the number of triangles with two vertices on that line is equal to the number of ways we can choose these vertices, Bonahon says—namely the number of ways we can choose two distinct points out of n, or “choose 2.”

Remember high school math? That’s n(n-1)/2. And since there are horizontal lines, says Bonahan, this gives p n(n-1)/2 possible triangles. In our case, that’s 3×4(4-1)/ 2=18.

Here’s a handy breakdown of how to find each possible triangle:

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KORY KENNEDY

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