A fascinating math paradox. How can 2 wheels of different diameter both turning one complete revolution travel exactly the same distance?? This puzzle is nicely demonstrated using this wood model. The Objective is to explain the Paradox. Made in our Hudson, Florida wood shop. Measures approx 11 x 3.5"
Not everyone agrees that Aristotle invented this little paradox, but everyone agrees that it would be just like him to come up with something like this. The paradox involves two different-sized wheels, one inside another. The two rotate in sync and you can see as you turn the wheels to travel the length of the opening in the puzzle that the etched line on the wheels turns one time, thus the circumference of each wheel has rotated over an equal length.
Both wheels use exactly one circumference to trace the same amount of distance - the length of the space in the puzzle. But, clearly one circumference is smaller than the other. Either that means that the wheels have the same circumference, which they don't, or that different circumferences "unroll" to the same length, which they can't.
That can't possibly be right.... The smaller radius can't possibly be equal to the larger one, so what's going on?
To further illustrate the paradox, I have included a separate copy of the inner wheel. We know that when attached to the bigger wheel that it turns one rotation to cover the length of the opening starting at the hash mark. But now take the small wheel and position it at the same starting point and rotate one revolution.... it does not travel the same distance! Such a dilemma.... Can you explain it?? Great fun for thinkers!
Level of Difficulty: Level 3