The Magic Octagon

One of my favorite moments at NCTM was when I ran into David Masunaga and he showed me the magic octagon.

Take a minute to let the magic octagon blow your mind.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. suehellman

September 9, 2013 - 4:58 pm -

It would be interesting to poll observers on their predictions. I’m going to make one that’s transparent so I can understand the magic.

2. Andrew

September 10, 2013 - 6:03 am -

I agree with Dan (comment #1). In fact, we just started discussing rotations and reflections TODAY! How wonderfully timely.

(and I sympathize with Matt E. It “got” me. In fact, I had to watch the video three times before I figured out what was going on)

3. Cynthia Nicolson

September 10, 2013 - 7:48 am -

I’m wondering what mathematical experiences would help make this more intuitive. Playing around with a hexagonal pattern block for a couple of minutes helped me get a feel for it. Does this show the need for concrete as well as virtual materials – even at higher grades?

4. Andrew

September 10, 2013 - 11:07 am -

@James… I don’t think it is a joke, but I was perplexed a bit as to why the arrow didn’t do what I had originally predicted. I had missed that in the way that the octagon was spun at the beginning, it didn’t keep the arrow on the my left side when the back of the octagon.

So, when I saw that the answer was different than I had predicted, it took me a second to catch up. That’s what I was referring to.

5. James McKee

September 10, 2013 - 1:21 pm -

@Andrew – so most people don’t predict correctly? I guess I can see that. I guess most people would expect the arrow on the back to rotate the “same direction” (clockwise) as the arrow on the front, and don’t see what the reflection is going to do?

6. Jenny

September 10, 2013 - 5:31 pm -

I read the comments first so I knew what was going to happen and I still couldn’t wrap my head around it when I watched it. I’m curious now about the differences in the thinking between the handful of folks who knew what would happen and the majority of us who predicted wrong. Any thoughts on the reasons?

7. math teach

September 11, 2013 - 9:04 pm -

I had to watch it without the sound the second time to “see” it! You were a great magician, distracting me with your words! Great fun….great thinking for the students!

8. Eric Merryman

September 14, 2013 - 2:17 pm -

I like this, it is pretty simple, the trick to see past it is to understand that that when side A is at noon, side B without flipping it, is at 3 o’clock, when flipped, it is at the shown 9 o’clock. So when side A is at 2 o’clock, through the paper side B is at 5 o’clock, however when flipped it is at 7 o’clock(how it is shown, due to it being flipped), not the expected 11 o’clock. Side A isn’t flipped, side B is flipped(when shown), so both go in opposite directions.

9. Bruce James

September 14, 2013 - 10:03 pm -

I like starting at “12”, like Dan, then going counterclockwise to “11.” That’ll spook ’em!

10. K.Savage

September 18, 2013 - 11:35 am -

These are the kinds of things that I love. When you can get inside their mind, turn it around, and then let the students explore why it works, those lightbulb moments are brilliant. I’m just trying to think if I could use it in a lesson plan early in the year in a 3rd grade class? Possibly what it would tie into/connections to be made?

11. Eric I.

July 30, 2014 - 7:03 pm -

I think people are missing the true “magic” of this trick.

In the first position, as the octagon is rotated, the hands are a 90 degrees apart.

In the second position, as the octagon is rotated, the hands are 180 degrees apart.

And then at the third position, as the octagon is rotated, the hands are again 90 degrees apart.

How can this be?

To me it’s kind of like this question: How does a mirror know to reverses left and right but not up and down?

12. Billie Mitchell

August 6, 2014 - 4:17 pm -

Amazing. Found you through a TED Talk. I wish we had this perfectly simple way to learn and understand math in my day. Now I am a grandmother to a 3 year old and I hope that this is how we can teach him.