The Greatest Geometric Puzzle Ever

Let's say you have two rectangular pieces of cardboard 3 feet long. You lay down the first cardboard on a table and tape the second one so that the two short sides meet one another forming a cylinder.

Now lay down the cylinder where the lowest part meets the first cardboard piece from where it starts along the long side and roll the cylinder over to the other side (along the long, 3-foot side of the laid down cardboard piece). How far has the circumference of the cylinder moved and how many times has it rolled to get to the other side?

Take the first cardboard piece and tape up its two short sides to make another cylinder. Now roll the second cylinder around again back to its starting point. How long has the circumference rolled and how many time did it roll?

Now take the second cylinder and stand it up straight (the circular part) perpendicular to the first cylinder and repeat the rolling procedure around the circumference of the first cylinder and the same questions: how far has the circumference of the second cylinder rolled and how many times.

Repeat the rolls, but assume the second cylinder's circumference shrunk down to two feet. Now what happens? How about a foot?

(I don't know the answers myself and this was inspired by a puzzle I read quite a few moons ago - some of this I made up myself).

For migraine sufferers: take two aspirin and call me in the morning.

"The circumference is a measure (a distance/length/real number), and it can't "move"." Yet a hula hoop can move along the floor.

It's interesting to note that Galois was far more complicated and he was deciphered.

I'll simplify the puzzle. Let's start with a rope ring (which can hold it's shape) that's 3 feet along the circumference and roll it along a straight line on the floor which is also 3 feet from end to end. If one starts at one end of the line and rolls it to the other end, one would expect the ring to actually roll the 3 feet along its circumference (but I've never seen a proof for this btw).

Let's say that a 3-foot rope was laid out along that straight line and we reshape it into a rope ring and lay it out along the floor and take the first rope ring and lay it down on the floor too so that the two rope rings touch each other at a point. Now roll the outer rope ring around the other rope ring and the question is how many rolls does the outer rope ring need to make to get back to the starting point and what distance has the circumference of the outer rope ring actually traveled? (this was the original puzzle I read about - the rest is mine going forward).

Suppose we take the outer rope ring and stand it up perpendicular to the floor and repeat rolling it around the inner rope ring and the same question: how many rotations does it need to make to get back to the starting point of the inner rope ring and what distance has the circumference of the outer rope ring traveled? Finally what happens when we reduce the circumference of the outer rope ring down to two feet and then to a foot? (just picture this in your minds).