Flexagons are nifty little mathematical toys if you're into that sort of thing. They are paper Möbius strips folded into polygonal shapes (most commonly hexagonal). These paper shapes may be 'flexed' - folded and 'opened up' to reveal hidden faces. Thus, what appears to be an innocent hexagon, one side colored blue and one side colored red, may be flexed to reveal a previously-hidden green side.

Flexagons were first invented in 1939 by Arthur H. Stone, a 23-year-old graduate student. He folded a strip of paper into a hexagon and then put three corners together and watched it fold open to reveal a hidden side. This was a trihexaflexagon. He managed to create a hexahexaflexagon the next day, having 6 faces, four of which are hidden at any time. (The nomenclature is simple: the first prefix is the number of faces, the second the number of sides on the polygon.)

To make a trihexaflexagon, cut the following shape out of paper:

   ______________________________
  /\    /\    /\    /\    /\    /
 /  \  /  \  /  \  /  \  /  \  /
/____\/____\/____\/____\/____\/

Those are ten equilateral triangles in a row. Each step below will involve a fold (specifically, a mountain fold). Fold so that the triangle labeled "dn" ends up facing down and the triangle labeled 'up' stays facing up.

       ______________________________
      /\    /\    /\    /\    /\    /
1)   /  \  /up\dn/  \  /  \  /  \  /
    /____\/____\/____\/____\/____\/

       ______
      /\    /\
2)   /  \  /  \
    /____\/____\
         /\    /
        /up\  /
       /____\/
      /\    /
     /  \dn/
    /____\/
   /\    /
  /  \  /
 /____\/

    ____________
    \  2 /\    /\    In the place I've marked '1', there
3)   \  /1 \  /  \   are two triangles, one on top of the
      \/____\/____\  other. The one on top should be the
       \    /\    /  one directly attached to the one
        \  /  \  /   marked '2' and the other should be
         \/____\/    tucked underneath.

    ____________
    \    /\    /\    The opposite face of the one marked
4)   \dn/up\  /  \   'dn' is flat against another tri-
      \/____\/____\  angle. Glue this junction together.
       \    /\    /  You are left with a complete tri-
        \  /  \  /   hexaflexagon, which is also a triple
         \/____\/    Möbius strip.

Lovely. Now, before you begin flexing it rashly, fold it in half each of the three different ways (along the edges of the triangles, dummy!) To flex it, 'pinch' two triangles as though you were folding in in half along one of the radial lines. The opposite vertex should be pushed inward so that you have three corners meeting at the bottom and three others on the outside. Now you may open it up - the former center of the flexagon will open up to yield a different face.

Once you've gotten it together, you'll see that there are 9 separate triangles, or 18 'pats' as they're sometimes called. When they're folded, the rhombi created by pairs of joined triangles end up on the opposite side, reversed, meaning that the vertex of the rhombus that had been at the outside is now at the inside. If you affix a paperclip to a triangle, it will move around the hexagon as its flexed, even though the hexagon itself isn't rotated. It will take 18 iterations before the paperclip returns to its original position.

Instructions can be easily found to make a four, five, or six-faced flexagon. I won't try to illustrate them here; the painstaking ASCII art I've just undergone was enough. They're all fairly simple to make, assuming you are accurate with a protracter and ruler (or really good at guesstimating!) There are also square flexagons (quadraflexagons?) out there.