Happy pi day 2016!

Happy pi day 2016!

It’s March 14th, which in the US is written as 3/14 — and since 3.14 are the first three digits of that most famous of mathematical constants, $\pi $, today is celebrated internationally as pi day.

Coin rolling

How many revolutions will the smaller circle make when rolling around the bigger one?

The number $\pi $ is the ratio of the circumference of a circle to its diameter. To celebrate this lovely number, here’s a little puzzle to ponder. Imagine a circle with radius 1 cm rolling completely along the circumference of a circle with radius 4 cm. How many rotations does the smaller circle make?

The circumference of a circle with radius $r$ is $2\pi r$, so the circumference of a circle with radius $4r$ would be $8\pi r$. Since

$8\pi r \div 2\pi r = 4,$

it seems the answer must be four revolutions. But that’s not true! The answer is actually 5! Can you figure out why?

We found out about this curious question from Yutaka Nishiyama, a mathematical friend in Japan. You can read about the answer in his article Circles rolling on circles. Happy puzzling!

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