Birth of the flexagon
You might think a simple strip of paper is an ordinary and boring thing. If you are a Geocacher, you would see it as useful as a micro-log, but not much else. But if you were a bored mathematician, you might discover that a single strip of paper can be transformed into something incredible. This is exactly what mathematician Arthur Stone did one day in 1939. A brit studying at Princton, Arthur found himself having to trim American leaf paper to fit into his British-style binders, This left many thin strips of paper. Instead of tossing these out, he found himself playing with them, folding them into shapes, twisting them into mobius strips.... what any self-respecting mathematician would do. One of his paper-strip creations was particularly unusual. It was a rather simple looking hexagon, but by pinching and folding it, it could be revealed to have 3 different "sides", not just a regular "front and back". The hexaflexagon was born. With friends like Richard Feynman and Bryant Tuckerman, this curious folded shape was soon explored in depth, with interesting mathematical properties, and a whole family of fun variants. To this day, people are still playing with and discovering the joys of flexagons. Easy to make, surprising to discover.
Your first hexaflexagon
The image below can be used to make a tri-hexaflexagon. This is one of the simplest types of flexagons, and a good one to start with.
Click image for larger version suitable for printing
Careful cutting and folding is key. When you have properly constructed your hexagon each face should be a different color (white, green and blue. Note that on the blue face, all the circles should be concentric. If they are not, the maze will not work properly and you should adjust your creases accordingly. Your goal is to draw a path from the 
to the 
. Encountering a letter transports you to the other instance of that letter.
Be prepared to spend some time at both stage 1 and the Final. Just signing your name to the logbook may become a challenge.