A flexagon is something like what's explained in this article. You can find another type of flexagon here .

Generalize and develop a (complete) theory of flexagons.

Construct, with paper, scissors and tape, these following surfaces: (Go through all possible combinations)
Clarification
Example

(1) 2 Side, 1 Edge, (a) Simple Closed Curve/Knotted
(2) 1 Side, 2 Edges (a) Both Simple Closed Curve / Both knotted/ One Simple, One knotted (b) Linked/Unlinked
(3) 2 Side, 2 Edges (a) Both Simple Closed Curve / Both knotted/ One Simple, One knotted (b) Linked/Unlinked

A difference triangle is a triangle with elements such that an element is the positive difference of the two just above it. Example

Develop a (complete ?) theory of a difference triangle. As an warmup, construct a difference triangle with elements from .

(If you don't know programming, learn it by all means ! Also, try to do it from scratch without resorting to external black-boxed modules) (a) Write a programme to help investigating into the previous puzzle.

(b) You start at a square in a chessboard, and at each step, you take a king move, with the restriction you don't go in the same direction consecutively. For example, if you go up in the first move, you can't go up in the second move, though you can go up in the third move. Oh, and you keep tracing a path joining the center of the squares you step on, and you wan't to minimize the number of intersection of the path with itself.

Write a programme to investigate this "Lost kings tour" of a rectangle.

Go to this Youtube Video featuring Tadashi Tokieda, and watch until 1:06 (don't progress further, also don't hover to see the preview pictures; that will spoil the fun). Try to develope a theory of what will happen, and generalize a bit further.

Go to this Youtube Video , featuring the same guy,
and don't watch further 1:08. Believe me, there's no dumb trickery/sleigh of hand involved. Can you predict how the trick works ?

(Sources: )