Since there is no indication of orientation on the middle coin, it could also be turning? Otherwise the coin on top would only make half of a rotation.

Pull out two coins and do it.

I feel bad posting an explanation of why this worksâ€¦ is â€śspoilersâ€ť taken for granted in a thread on a problem like this?

Maybe just try and help someone along, like this:

Imagine the red lines were lengths of string. What would happen if you flattened out the string from the top set of coins?

My weightlifter bro explained it to me this way:

7x in 2 weeks = 3.5 times a week, genius.

And yeah, 3x a week, full body workouts are good.

Well the answer is right there in the headline: itâ€™s a paradox. These coins defy logic - humanity will never explain them.

The coin has moved only half as far in the top illustration, and in fact *relative to the line* has only completed half a revolution. Relative to the line, in the first position the arrow is pointing to the right, and in the second position is pointing to the left. To end up in the same orientation relative to the surface of the coin it needs to complete one revolution, i.e. travel the same distance as the bottome illustration.

Damn it, whatâ€™s the notation for spoilers, again?

What? No, in both cases the coin has done a full revolution relative to the line, as shown by the orientation of the arrow.

The truth of the matter is that the coin hasnâ€™t actually traveled any less distance in the top illustration. You can see that by looking at its actual center. So itâ€™s just that with the semicircular path, the point of contact now traces half the distance, and thatâ€™s because itâ€™s twice as close to the center of the semicircle.

The top coin rotates twice as fast. It rolls along the edge of the stationary coin at the same time it rotates around the center of the stationary coin.

Shut it down, you win the thread.

No need for further discussion.

Slam dunk; brass ring; potroast and cola on me.

Alright, if we are spoiling it, here it is:

The inner edge moved less distance, the outer edge moved more, the center moved the same.

Whatâ€™s interesting to me is whether or not this â€śparadoxâ€ť relies on using the inner edge of the rolling coin. I have a suspicion that it does. Imagine the same problem were posed but pointing out that the outer edge moved much further. Somehow itâ€™s obvious that you arenâ€™t considering the inner edge. I think weâ€™re very obsessed with things rubbing against one another.

The distance isnâ€™t being measured the same way between the two examples.

For the coins on top: if the coin that starts on the left rolls around the coin on the right, then the point where the coinâ€™s touch goes from the **right side** of the rolling coin, to the **left side** of the rolling coin.

Whereas the bottom coin is being measured in the same spot. Both coins roll the same distance, but the measurement is wrong.

I think the fairer measurement would be to consider the coin that doesnâ€™t move in the top example. Clearly only have of that coinâ€™s circumference is traveled.

It might have been more accurate of me to say that the coin has only *rolled* half as far. Imagine that the coin being moved around is a world and youâ€™re walking beside the moving coin. At position one, relative to your orientation the arrow is not pointing up, itâ€™s pointing forwards, i.e. towards the direction of movement. It rolls halfway around the equator to position two. The arrow is now pointing backwards, i.e. from where it came. Ergo, the coin has only rolled half as far, and turned 180Â°.

For an interesting thought experiment, trace the line the point of the arrow moves. How long is that line?

These puzzle things arenâ€™t as fun as they used to be.

Canâ€™t we have one like this again?

Or roll out a Monty Hall Problem thread, thatâ€™s always amusing.

Note that while the blue dots in the bottom diagram are at the same point on the coin, the blue dots on the top diagram are not; in fact, they are on opposite sides of the moving coin. So the first blue dot ends up at the end point of your outer semi-circle, and the second dot is originally at the beginning of your outer semicircle. Neither blue dot follows the short semi-circular path, but one that is twice the radius.

(I think I might have proved my first explanation wrong there. *Câ€™est la vie mathematique.*)

Follow the distances travelled by the centers of the coins, and youâ€™ll see that theyâ€™ve travelled the same distance (one coin circumference)â€¦

Huh? No matter what the coin rolls the length of the diameter (2 pi R). The first picture illustrates a complete rotation of the coin; the line, however, is measuring only half the distance (pi R) because it connects the nearest points on the edge of the circle.