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).
I wonder if I'm alone in thinking that the wording of this question is very confusing. For example thisOriginally Posted by wonderboy1953
"Now lay down the cylinder where the lowest part meets the first cardboard piece from where it starts along the long side..."
Really I think that wording can be improved. And this
"How far has the circumference of the cylinder moved..."
The circumference is a measure (a distance/length/real number), and it can't "move".
"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).
Okay I'll demonstrate the ambiguity. A wheel with radius 1 is rolled along the floor from point A to point B, and the distance from A to B is 2*pi. (A and B are points on the floor.) There are a few different questions we can ask
(1) How far does a point on the surface of the wheel travel? (The point is distance 1 from center.)
(2) How far does the center of the wheel travel?
(3) How many rotations of the wheel occur?
Asking how far the circumference moves, in my opinion, gives us no clue which one of these quantities is desired.
There is a difference between being complicated and being unclear!
That said, I think I understand the problem now. I don't have a solution* but maybe can clarify for others reading this thread:
Scenario 1: A wheel is rolled along the floor from point A to B where distance from A to B is circumference of wheel. (A and B are points on the floor.)
Scenario 2: Begin with two equivalent wheels in the same plane and position them so they are tangent; keep one of them stationary, and roll the other one around it until it gets to the starting place.
Scenario 3: Begin with two equivalent wheels oriented orthogonally and tangent, as in: let the radius be r, let the first circle be in the xy plane with center at (r,0,0) and let the second circle be in the yz plane with center at (0,0,r). Keep one stationary and roll the other one along it until reaching the starting point.
In all these scenarios we are interested in the number of rotations the moving wheel performs.
Scenario 4: Multiply the circumference of the wheels by a constant and determine the ratio of (previous number of rotations) : (new number of rotations) for scenarios 1-3.
* Actually I don't see why the answer isn't 1 rotation in all cases, and ratio of 1 : 1 for scenario 4.
Scenario 2: Wolfram Demonstrations Project: Generating a Cardioid I: One Circle Rolling around Another
Scenarios 1-3: If it were not 1 rotation, then these would be pretty useless wouldn't they? Amazon.com - Measuring Wheels
Obviously not rigorous proofs, but still..
"Actually I don't see why the answer isn't 1 rotation in all cases, and ratio of 1 : 1 for scenario 4." To the best of my memory I recall that the number of rotations goes up when you go around the wheel as opposed to a straight line according to the puzzle that was given.
I'll await further responses.
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