http://loki3.com/flex/
“Flexagons”, shown above, are a little like origami – they’re folded pieces of paper that you “flex” (manipulate) to reveal different “faces” or views. Unlike some of the other puzzles here, they appeal particularly to people who like to work with objects and like to feel the textures of things. When you first make a flexagon, the faces don’t slide easily and they have a kind of awkward, new, glossy feel….but after you fold them a few times and work with them, they become like soft cotton or denim - a little worn, with a cozy feel! AND, you can personalize them for kids, so that they can discover a hidden picture, drawing or message from you!
As you can see from the picture above, flexagons come in many different forms. The diagram below shows how to create a “hexa-tetraflexagon” (sounds like a joke name, doesn’t it?). “hexa” in the name signifies that the flexagon has six faces that can be revealed. “tetra” means the flexagon has four sides. The diagram and directions are from Martin Gardner’s Mathematical Puzzles column in Scientific American back in the 1950’s, which started Flexagon Mania.
Begin with the square-shaped strip shown in 3a (front) and 3b (back). Crease along each internal line in 3a so that each line is the trough of a valley, flatten the strip again, then fold on the four lines marked with arrows. (The folds will be easier if performed in the order indicated by the red numbers). All folds are made to conform with the way the lines were originally creased. The strip now looks like 3c. Fold on the three lines marked with arrows to form a square Hexagon. Overlap the ends so that all the "2" squares are uppermost (3d). Attach a piece of transparent tape to the edge of the square at upper left, then bend it back to overlap the edge of a ''1'' square on the opposite side. This diagram, together with the video linked below, will help you fold the flexagon correctly!
Below are links to templates for printing flexagons with numbers, plain blank hexagons for customizing, and Valentine’s Day or Happy Birthday flexagons with some blank faces that can be personalized. The link to a Youtube video (below) shows how to create these flexagons and also how to fold them to reveal their hidden faces.
Blank grid template for making customized flexagons: blank grid
Templates for plain flexagons with numbers: Front face Back face
Valentine’s Day flexagon for personalizing: Front Face Back face
Birthday flexagon: Front face Back face
A very helpful printable version of the Directions for folding the flexagon
Video showing construction and “flexing” of the flexagons is below
