Hexaflexagons are basically folded paper hexagons, which, when, unfolded, reveal a surprising number of faces.
The flexagon was discovered by Arthur H. Stone in 1939, that figured out that polygons made out of folded strips of paper, can be flexed, or reversed, revealing more than 2 faces.
Stone was a student at Princeton University when he folded his first trihexaflexagon, a flexagon with three sides. One with six sides is called a hexaflexagon, and so on.
He allegedly used strips of paper obtained after cutting out his foolscap paper to convert it to letter size, so that it would fit in his binder, Wikipedia
Along with classmates and fellow mathematicians Bryant Tuckerman, Richard P. Feynman and John W. Tukey, he started the Princeton Flexagon Committee. His discovery was included in the "Mathematical Games" section of the Scientific American magazine, in 1956.