Fold a paper strip into a hexagon, and it hides more faces than it should have
Take a long strip of paper, fold it into a flat hexagon, and glue the ends. It looks like it has just two faces. But pinch three alternating corners together and fold it open from the centre, and a third face appears, one that wasn't visible a moment ago. Keep flexing it the right way and you can cycle through even more hidden faces. How many faces can a hexagon built this way actually have?
Reveal the answer
A basic "trihexaflexagon" has three faces; more elaborate versions built from longer strips can hide six, nine or more. The shapes were discovered in 1939 by British student Arthur H. Stone, who was trimming American paper to fit English binders and started folding the offcuts. He and fellow Princeton students Bryant Tuckerman, Richard Feynman and John Tukey formed an informal "Flexagon Committee" to study them. Martin Gardner's very first Mathematical Games column for Scientific American, December 1956, was about hexaflexagons — its popularity launched his 25-year run writing the column.
— Martin Gardner, Flexagons — Scientific American, December 1956