How many times?

Last night, I read this posting from John Baez‘s blog and found a problem stated in that posting. The reason why I am reposting this problem is because this problem is another one that shows mathematical problem is sometimes is just a matter of viewpoint. From a certain viewpoint, there is a very elegant and trivial solution to this problem:

For starters, consider an ordinary ball rolling on another ordinary ball that’s the same size. How many times does the rolling ball turn as it makes a round trip around the stationary one? If you watch this you can see the answer:

Follow the line drawn on the little ball. It turns around not once, but twice!

Next, consider one ball rolling on another whose radius is 2 times as big. How many times does the rolling ball turn as it makes a round trip?

It turns around 3 times.

And this pattern continues! I don’t have animations proving it, so either take my word for it, read our paper, or show it yourself.

So, the assertion is that if a circle is rolling over another circle which radius is n times longer than the first circle, then the first circle will turns around (n+1) times to make a round trip. Try to prove it yourself! 🙂