# 2 circle with different radii covers same path on same number of turns... crack the fallacy

I found this puzzle from an old issue of a science magazine জ্ঞান ও বিজ্ঞান ("Knowledge and Science")(August, 1977); submitted by Amal Dash, a teacher of Scottish Church school, Kolkata; as mentioned in the magazine.

Incidentally; the answer wasn't published there.

The original publication did not contained any sketch. I've added a few to make it easily readable.

A wheel has an outer radius r1 and an inner radius r2. The wheel is rolling on a floor.

Normally, r1 > r2.

Before the wheel was being rolled, the topmost (from ground) point of outer margin (dot A) and inner margin (dot B) was marked with color dots.

After rolling; the wheel was stopped in a situation, when the 2 colored dots came upwards. So, in the mean-path; the wheel turned complete (integer) turns.

Since the both circles on the wheel revolved n-complete turns; then point A (on big circle) should have been shifted 2nπr1 and similarly the point B (on small circle) should have been shifted 2nπr2.

But if so; since we can see after complete turns the point A and B came to similar orientation as initial, and so dots A and B shows same shift from their own initial positions in horizontal direction; so 2nπr1 should be = 2nπr2; ie. r1 = r2 . And we know very well that r1 ≠ r2.

So, where lies the fallacy? what was going inside the circle?

This may be a better question for the Math SE. Having said that ....

The fallacy is that the inner circle is not tracing out its own perimeter - it's along for the ride. You can see how quickly this breaks down if you consider the limit as $r_{\small_2}\to0$ that if both circles were independently moving $2n\pi r$ for their respective radii, as suggested in the fallacious argument, that this would leave the center-point of the wheel essentially motionless while the outer rim of the wheel moved as expected—a clearly nonsensical situation.

Since the inner surface of the wheel is not independent of the outer surface of the wheel, they both rotate together, but only point A on the outer circle is tracing out the perimeter of its $2n\pi r_{\small_1}$ circumference.

• IMO it suits better with puzzling because its main focus is a fallacy. As well overlap of physics, mathematics etc. wit puzzling is pretty common; including Carroll's classic 'Monkey on a Rope' problem; which are frequently discussed puzzle related literature. Jan 28 '17 at 18:33

This is better understood if you first consider what is happening to the centre (I'm British, hence the original spelling) of the circle. Applying the $2 \pi r$ approach clearly doesn't work here as the circumference of a circle with no radius is zero. However, the axle of a wheel on a vehicle moves without describing any curved path.