A strip of paper folded into a hexagon hides more faces than you'd expect
Take a long strip of paper, fold it into a ring of triangles, and glue the ends: you've built a hexaflexagon. Pinch three alternating corners together and it opens out flat again — but the face looking back at you isn't the one you started with. How many different paper surfaces can a simple six-triangle version like this actually hide?
Reveal the answer
Three — flexing repeatedly cycles a more elaborate 19-triangle version through as many as six faces, but the basic trihexaflexagon reveals exactly three distinct paper surfaces, each appearing and vanishing as you flex it. Princeton graduate students discovered flexagons in 1939 after an American notepad didn't fit an English binder and had to be trimmed down; Martin Gardner turned the toy into a sensation by making it the subject of his very first 'Mathematical Games' column for Scientific American, in December 1956.