I suppose a road map is the most common form of folding puzzle. Imagine a road map with different picture fragments on each of the rectangles between the folds on both front and back. Then you have to fold it in such a way to produce one or more complete pictures. This is what most folding puzzles amount to.
In 1939, a Princeton University graduate student, A. H. Stone, discovered what became known as "hexaflexagons". They are interesting "six-sided" figures made from a strip about ten times longer than it is wide into a hexagon shaped figure. By "flexing" it, you can uncover six different "sides". I might prepare instructions for making one sometime in the future.

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