Flexagons are flexible polygons folded from strips of paper (e.g. see post viewtopic.php?f=1&t=9831). They come in many different shapes and there are lots of different flexes you can do, where a flex is analogous to rotating a portion of a typical twisty puzzle. Think of them as a twisty puzzle you can fold up and put in your pocket, with similarities to folding a Rubik’s Magic. Figuring out new ways to solve one can involve discovering new flexes as well as new sequences of flexes. And to make matters more interesting, each flex changes the structure of the flexagon, making it so the set of possible flexes change dramatically with each move.

Flexagons have been around for quite awhile, but recently I’ve figured out a bunch of new ways to make interesting puzzles that are a lot like working with a polyhedral twisty puzzle. The most well known of the flexagons is the hexaflexagon, with six equilateral triangles arranged in a hexagon. But you can make a flexagon from any polygon and fold them into many different shapes. Only two sides are visible at any given time, but an arbitrary number of sides can be hidden. I’ve concentrated primarily on triangle flexagons because it’s easier to have five or more polygons per face, which makes for a large variety of flexes. Plus triangles allow such flexagons to lie flat in many configurations.

A sampling of triangle flexagons:


Puzzles on flexagons:


Some of the other possible flexagons:


To give you an idea of what it’s like to manipulate one of these puzzles, here are links to videos of several interesting flexagons from my youtube channel http://www.youtube.com/user/loki3dot:

• 5 sided hexaflexagon: http://www.youtube.com/watch?v=tGQ_I4_N3sE
• 6 sided enneaflexagon: http://www.youtube.com/watch?v=KVRqS4eDxnk
• 4 sided star dodecaflexagon: http://www.youtube.com/watch?v=k43niP5iWdg

For a collection of flexagon templates you can make yourself, a list of some of the flexes and description of the notation I use to describe flex sequences, you can go to my web site, http://loki3.com/flex/.

I know these are a bit different from the other puzzles here, but they have enough similarities that I thought this group would enjoy them.

- Scott Sherman

I will print many of these soon!
I like the folding "puzzle" which is sold on this site, but this adds an actual puzzle aspect to it in a very beautiful way.
The fact that you can just print them yourself makes it even better.

Thanks, these are rather nifty.

Have you seen this before? Hexagonal trihexaflexagon slipagon

Worth a look at the patent. Have a go at one of the designs in there!

sweet! I'll print some soon.

I wonder if you got this idea from the origami forums....

I'm not a member of them, I just looked at a few once.

Good link. I had seen reference to slipagons before (e.g. http://tech.groups.yahoo.com/group/Flex ... essage/380), but I didn't have enough detail on what they were to try making one. Sounded like you thought they were mildly interesting but a bit difficult. Some of the pictures from the patent application look pretty involved!

I like lots of different kinds of paper folding, but since making a flexagon involves cutting and pasting, it's not quite origami. I don't know if they'd accept something that involved cutting the paper .

I first learned about flexagons through Martin Gardner's book Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games. I learned about some of the other types of flexagons from Les Pook's Flexagons Inside Out. And some of the theory behind them from Conrad & Hartline's 1962 paper on flexagons (see http://erikdemaine.org/flexagons/Conrad ... xagon.html).

But I haven't seen any references to the flexes that make flexagons feel like a twisty puzzle to me. They talk about a simple flex like the pinch flex, which you can use to visit all the faces once you know the trick (the Tuckerman traverse). While that's interesting, it's not much of a puzzle. With these other flexes, it can be pretty tricky to figure out how to mess up a flexagon and get it back to its original state since each flex changes what's possible, sort of like the Square 1 type of puzzle, only more so.

Essentially yes. Maybe I'll try one of the simpler ones, the six component hexagonal trihexaflexagon slipagon was too stiff to slide and too complicated to flex. Maybe it would be better if some tolerances were built in and it was made of a decent material.

I think I'll have a go at four component octahedral one shown in fig 12 - "FIG. 12 shows 4 trihexaflexagons linked into a loop to form a half of an octahedron shape. This puzzle can be gotten into many different geometric forms."

This is great! Thanks I would still like the shape changing puzzle on this site though...

''Beautiful'' doesn't sum these up completely. I'm going to make as many as I can