This is not the easiest thing in the world to explain, but let me try:
In 4th grade, we learned to make these flat hexagonal items out of folded paper (regular old copy paper IIRC). Once finished, you could draw a picture on the front and back of the hexagon. THEN you could fold the hexagon INSIDE OUT to show a third BLANK side (IIRC, one of the other two sides would then be on the reverse side jumbled up).
Does anyone know what I am talking about and how to make one? I have wanted to relearn to make the hexagon for quite some time.
NOTE: Just to make clear, this is most definitely NOT one of those 4-sided paper “fortune-tellers” that slipped over the forefingers & thumbs.
More specifically, they are called Hexaflexagons. One source of instructions for making them is chapter 1 of ‘Mathematical Puzzles and Diversions’ by Martin Gardner. By the way, the actual hexaflexagon (which is hexagon shaped, unsurprisingly) has SIX faces, and can accommodate SIX pictures, not just three. Perhaps the ones you learned at school were simpler.
A flexagon which has six edges is called a hexaflexagon; I don’t know if anyone’s done research into non-hexagonal flexagons. A hexaflexagon with three sides, such as the one mentioned in the OP, is called a trihexaflexagon. The more common (and advanced) flexagon, with six sides, is called a hexahexaflexagon. (I think the first mathematicians working on these gave it such a silly name on purpose.) I have also made a dodecahexaflexagon, which has 12 sides, but it didn’t turn out well. I don’t know if there’s an upper limit to the number of sides you can have in one, but the biggest one I have heard of had 21 sides. Most of what I know about these I learned from the book Hexaflexagons and Other Mathematical Diversions by Martin Gardner, which seemed to be a compilation of articles, so it probably has exactly the same material in it as ianzin’s book. I would have to say, from personal experience, that the funnest thing about hexaflexagons is coming up with patterns you can color the sides with and still have it look good after it’s folded.