The pentaflexagon has 5 triangles meeting in the center.

At first glance, you may expect the pentaflexagon to be fairly uninteresting, given that it only has 5-fold symmetry. But there’s actually a surprising range of things you can do with this flexagon, from the simple pocket flex available on all variants to the slot flex and pyramid shuffle on some of the variants with more sides.

The first couple examples on this page demonstrate the pocket flex which is similar to what you can do on a equilateral triangle tetraflexagon. A series of pocket flexes can rearrange things slightly, but always keeps A’s on and side and B’s on the other. In contrast, you can do the slot flex and pyramid shuffle on the final example, both of which allow you to arrive back at a pentagon after rearranging the leaves. It makes for an interesting puzzle to figure out what these flexes can do.

To make each of the flexagons, cut out the strip and pre-crease all the edges. Copy the small letters and numbers on to the back. For this first one, fold 1 on 1, 2 on 2, 3 on 3, 4 on 4 and 5 on 5. Tape the tabs onto the appropriate faces. You should have five A’s on one side and all the B’s on the other. You should be able to do a single pinch flex at any point on either side of the flexagon to open up two pairs of numbers.

For this next one, fold 1 on 1, 2 on 2 and 3 on 3. Tape the tabs onto the appropriate faces, giving you A’s on one side and B’s on the other. On side B you should be able to find a place where you can pinch and open up the pockets on both sides of the pinch at the same time.

You can do more flexes on this next flexagon. Using a series of slot flexes and pyramid shuffles, you can switch from having all 1’s on one side and 2’s on the other to having all 3’s on one side and 4’s on the other. It’s a challenging puzzle to figure out how to do this traversal.

To create the flexagon, start by folding 8a onto 4a and 5a onto 3a. Next fold 8b on 4b, 5b on 3b and 7b on 6b. Continue doing the same with c, d and e - 8 on 4, 5 on 3 and 7 on 6. It should end up turning into a pentagon. Finish by tucking the final flaps under each other so 7a and 6a are face to face with all 1’s on one side and all 2’s on the other. Tape together the edges of the first and last leaves in the strip to complete the flexagon.

To do a slot flex from this initial arrangement, start by folding together two adjacent 2’s. This should allow you to do a pocket flex from side 1. The inside of the pocket will have 6’s and 7’s. A second pocket flex will reveal 3’s and 5’s. A third pocket flex will reveal 4’s and 8’s. You should now be able to slide one of the 8’s through the slot between the 2 and 4, turning the pyramid into a tetrahedron and the tetrahedron into a pyramid. The smooth curve along the 8 allows the slot flex to work without requiring you to bend the leaves. You can now perform the pocket flex three times on the pyramid to arrive back at a pentagon. If you pay attention to how you finished this series of flexes, you should be able to reverse the moves to come back to the original position.

To do a pyramid shuffle from the initial arrangement, fold together any pair of 2’s, then do a pocket flex from side 1. The inside of this pocket will have 6’s and 7’s. Now do a second pocket flex, but rather than opening it from the outside to reveal 3’s and 5’s, open it from the inside to reveal 3, 5, 6 and 7. This next step requires bending the leaves. The 1-3 pat and 1-5 pat need to bend so you can stuff them into the tetrahedron, with the top tip being shoved all the way down into the bottom tip of the tetrahedron. You will find that this arrangement opens back up into a pentagon. Folding together the 8 and 4 and repeating these steps will take you back to the original position.

A combination of a single slot flex and a pair of pyramid shuffles changes a single leaf on each side. Done in the proper sequence, this can take you from the original arrangement of 1’s and 2’s to one with 3’s on one side and 4’s on the other. It’s a good challenge to figure out how to do this

Next: The Hexaflexagon

Some of the possible flexes

Return to the Triangle Flexagon Bestiary