There’s naught quite favor a maddening mathematics problem, mind-bending optical illusion, or twisty reasonable puzzle to halt all efficiency in the Popular Mechanics office. We’re curious people by nature, but we additionally collectively share a stubborn insistence the we’re right, dammit, and so we often tend to throw work by the wayside whenever us come upon a difficulty with several seemingly feasible solutions.

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This triangle mind teaser isn’t new—shoutout to Popsugar for unearthing the a pair years ago—but based upon some shady web magic, the tweet below reappeared in my feed today and kick-started a brand-new debate on ours staff-wide Slack channel, a location traditionally booked for workshopping ideas, but instead mainly used for yelling around other stuff the we occasionally turn right into content.

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Because I’m a masochist, I attracted the triangle again and asked everyone on employee to promptly drop what they to be doing and also attempt to deal with the an easy question: How plenty of triangles deserve to you find?

I’ll preventive you the full conversation—trust me, nobody desires to see that—but the team’s responses ranged almost everywhere the place. Part editors saw 4 triangles. Others observed 12. A few saw 6, 16, 22. Even more saw 18. One wiseguy counted the triangle in the A’s in the question itself, while an additional seemed to be having an gift crisis: “None of this lines room truly straight, simply curves—thus you cannot define any kind of of them as a triangle,” the said. “There room no triangle in this photo. Life has no meaning.”

We then posed the trouble to ours Instagram followers, whose replies likewise ran the gamut, native 5 come 14 come 37. When we recognize the high probability the trolling here, it’s clean that human being respond come the problem plenty of different ways.

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“It’s convenient to generalize to the instance where there space n lines passing v the top vertex, and p horizontal lines,” states Francis Bonahon, Ph.D., a professor of mathematics at the university of southerly California.

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

Remember high school math? That’s n(n-1)/2. And also since there are p horizontal lines, claims Bonahan, this gives p n(n-1)/2 possible triangles. In ours case, that’s 3x4(4-1)/ 2=18.

Here’s a handy break down of just how to uncover each possible triangle:


Johanna Mangahas, Ph.D., one assistant math professor at the university at Buffalo, additionally came come 18—first through an easy brute-force counting, then with the exact same crafty combinatorics as above—but admits ours triangle mind teaser isn’t quite as cool together this one native Po-Shen Loh, Ph.D., a mathematics professor at Carnegie Mellon college in Pittsburgh, together featured in the New York times last year:


This one has a slicker math answer, she says, due to the fact that here, counting triangles is the exact same thing as counting combine of three lines preferred out of 6 <6-choose-3 = (6*5*4)/(3*2*1)>.