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Video: Puzzle 0003

Tim Burnham

This video presents a puzzle in which we wrap a piece of string once around Earth so that it fits exactly. We then wrap a second piece of string, one meter above it, and ask for the difference in their lengths. We also present and explain a solution.

04:13

Video Transcript

Welcome to puzzle time! Puzzle number three. Here’s a picture of the Earth. Now for this puzzle, we’re gonna assume that the Earth is a perfect sphere. And we’re also gonna imagine that we can take a piece of string — a very long piece of string — and wrap it all the way around the Earth. And it’s a special string; it floats on the surface of the oceans and the seas. So it’s exactly fits the circumference of the Earth. And we tie it in a little knot at the top here. Now let’s say that that string is gonna be 𝐿 metres long. Now we’re gonna get loads and loads and loads of people to stand all the way round the earth. Yes, they can walk on the water in the oceans as well. And they’re gonna hold another piece of string, one metre, above the first piece of string.

The question is how much longer is the second piece of string than the first? Right so what I want you to do is pause the video now. I’m gonna wait three seconds. And then I’m gonna talk through how to solve the problem. Right so throughout this question, the units that we’re gonna be using are metres. So let’s say that the diameter of the Earth is 𝑑 metres. So I don’t actually know what that is, but 𝑑 metres. And since we’re assuming that the Earth is a sphere, that shape that this string makes is a circle. So the circumference of a circle is equal to 𝜋 times its diameter. So the length of the orange string is 𝜋 times 𝑑 or 𝜋𝑑 metres.

So what’s the diameter of the circle made by the green string? Well it’s the same diameter as the orange string, but I’ve got an extra bit here: an extra one metre here and an extra one metre here. So the green diameter is 𝑑 metres plus an extra metre plus an extra metre either side; so that’s 𝑑 plus two metres. So the length of the green string then is 𝜋 times this diameter, 𝑑 plus two. And we can use the distributive property of multiplication to re-express this as 𝜋 times 𝑑 plus 𝜋 times two; in other words 𝜋𝑑 plus two 𝜋. And in this format it’s easy to see that it’s 𝜋𝑑, the same as the orange string plus two 𝜋. So the extra bit of length is two 𝜋 metres. And when I work out the value of that, it’s about six point three metres to one decimal place.

Now it might surprise you to know that we didn’t actually need to know the diameter of the earth. We just called it 𝑑 and we did the calculation without knowing the actual number. But more importantly the difference between the two was two 𝜋. It didn’t depend on the value of 𝑑. So it doesn’t matter how big or small the Earth is. The extra length if you want to lift that string up an extra metre away from the surface of the Earth, it’s always gonna need to be six point three metres longer.

So let’s say we did the same thing for a tennis ball, which is between six and a half and six point eight, six point nine centimetres in diameter. So let’s say nought point nought six six metres. If I tied a piece of string around that and then tied another piece of string one metre further out from the surface of that tennis ball, the small circumference is 𝜋 times nought point nought six six metres and the big circumference is 𝜋 times nought point nought six six metre plus two 𝜋 metres. So the difference is still two 𝜋 metres. That’s an extra six point three metres of string needed.

Even if we went to the surface of Jupiter which is a hundred and thirty nine point eight million metres across and then we raised another string around the outside of metre off the surface of Jupiter, the difference would be two 𝜋 metres. That’s about an extra six point three metres.

We hope you enjoyed puzzle time.