The “One Cut” Problem

What can you do with this?

Today, we did what I thought was one of the coolest explorations I have seen in a while. It is called the “One Cut” or “Fold and Cut” problem, with inspiration from Patrick Honner (@MrHonner, this part of his website). The premise is simple: you have a shape drawn on a piece of paper. How can you fold the paper so that you can make one straight snip with the scissors and cut the shape out?


Weirdly enough, there is a theorem that says that no matter what shape, this is possible (concave, convex, numerous closed figures, as long as the sides are all straight) . Which totally blows my mind. Because it’s really friggin’ hard in practice (though, the star above is pretty easy). Check out some patterns for some other shapes here.

photo (4)With one group today, we started doing this problem as a way to see what vocabulary they knew with polygons and to talk a bit about symmetry. I printed out little shapes on 1/4 sheets of paper and the levels went as follows, getting a bit harder as the levels go up:

Equilateral Triangle –> Square –> Isosceles Triangle –> Rectangle –> Regular Pentagon –> Regular 5 pointed Star –> Scalene Triangle –> Arbitrary Quadrilateral

They were TOTALLY hooked. Every single kid was working on their own and kept either having the “YAYY, TEACHER LOOK!” reaction or laughing hilariously at the silly shapes that they made by accident. The kid in the picture with his hand over his face kept yelling out “AGHHH, TRICKY TRIANGLE, TRICKY TRIANGLE” because he couldn’t do the scalene triangle. But he wouldn’t accept a hint from me because he wanted to find out on his own!

45 minutes later, the classroom was a total mess, and I was wondering where all the time had gone…

Now, working with the regular polygons you might get duped into thinking it’s pretty easy. But try a scalene triangle. Or a non-special quadrilateral (i.e. most quadrilaterals). It’s actually VERY difficult. And I think the solution is pretty fascinating, because my solution to it (which I got after about 10 triangles and an hour) heavily involved the triangle’s incenter (the intersection of the three angle bisectors). Which made me think that this would be a super cool thing to do in a geometry class when talking about angle bisectors. Are there multiple “incenters” on shapes with more sides, even if you have to define the idea in a different way?

I won’t share the solution but just post a picture to show you that YEAHHH it’s possible.

picstitch (1)Geometry or other math teachers… where else could this go in the classroom?

(If you haven’t heard me bemoaning how much energy it takes to do math for 2.5 hours with 11 and 12 year olds at a summer camp, well then FYI: I am currently teaching at a summer camp for 6th and 7th graders from China who are here doing an academic program that is half ESL, half math. It’s getting more and more fun, but certainly has been an adjustment from teaching seniors in high school for only 45 minutes at a time).


Posted on July 11, 2013, in Math Circle. Bookmark the permalink. 18 Comments.

  1. Wait, whoa, what?! Am I right in seeing that the triangle one is the incenter of the triangle, the intersection of the three angle bisectors? Makes sense, cause folding an angle in half will lay the two sides atop each other. I could do something with this in the geometry class. Gears are turning. Thanks!

    • yeah!! cool huh? And the way I solved the scalene triangle one, the incenter became very important. in fact you can see it in the picture of me cutting… it’s the little tip on the right

  2. Awesome! This makes me wish I still taught geometry.

  3. I’m thinking that this will make a great opening day(s) activity for my honors geometry class. I can put them in groups/pairs and get them going for a day or two. Maybe have them solve a harder one or make their own for homework. I can revisit the scalene when we get to inscribed/circumscribed and triangle centers. Sounds fun and messy- my kind of project. It’s open ended and should let me see how they work and require some thinking on their part. It might even lead to a writing prompt—something I almost never do.

  4. Oh wow! This looks like a great challenge! Hmm, maybe one or two for a couple of days, before discussing bisectors…?

  5. Nice tricks on geometry! Neat and awesome. Thanks for sharing!

  6. Awesome trick Bowman. Thanks for sharing.

  7. Thanks for this trick. Included in in the MTAP 65.

  8. Nice article. But I’m glad I didn’t scroll down after your “tricky triangle” paragraph, because your third picture is a dead giveaway. So it might be a good idea to put in a “spoiler alert” right after you state the problem.

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