Henry Ernest Dudeney/Modern Puzzles/209 - A Wheel Fallacy/Solution

Modern Puzzles by Henry Ernest Dudeney: $209$

A Wheel Fallacy

Here is a curious fallacy that I have found to be very perplexing to many people.

The wheel shown in the diagram makes one complete revolution in passing from $A$ to $B$.

It is therefore obvious that the line $AB$ is exactly equal in length to the circumference of the wheel.

Now the inner circle (the large hub in the diagram) also makes one complete revolution along the dotted line $CD$ and,

since the line CD is equal to the line $AB$, the circumference of the larger and smaller circles are the same.

This is clearly not true.

Wherein lies the fallacy?

Solution

Fairly obviously, the inner wheel does not only roll along $CD$ but slides along it as well.

Suppose the inner wheel rolls along $CD$ without sliding.

Then the bottom of the upper wheel move backwards in the same way as the bottom of the flange on a railway wheel, as analysed in $211$ - Another Wheel Paradox in this collection.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $209$. -- A Wheel Fallacy
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $293$. A Wheel Fallacy