Question: What's maths in juggling

Suzanne McEndoo answered on 13 Mar 2012:

Hi Anna,

So, the mathematics of juggling. A lot of mathematics is about recognising patterns in the world around us, figuring out how to write it down and then using this to discover new things. If you just watched someone juggle, it would be quite hard to say exactly what’s happening. In my friend’s talk on juggling he shows us how we can draw a picture of juggling by saying which ball is in which hand and when.

So say we have a green ball, an orange ball and a pink ball. My right hand throws the green ball into my left. Then it throws the pink, and then the orange. At this stage my left hand has been throwing the balls back, so my right hand can throw the green ball again. So my right hand is always throwing green, pink, orange, green, pink, orange, and so on.

We know that everything has to be symmetric (because you have two hands working, and you always have to get the balls back in the right order), so my left hand is also throwing green, pink, orange, green, pink, orange, only it’s happening a bit later than the right hand.

In the juggling talk (I’ll add the links below), you can see this written down as green, pink and orange lines bouncing back and forth through time.

Why bother with all of this? Once you know how it works, you can start playing games and inventing tricks. Like how does four balls look written down? (Actually, for four balls, you juggling two in each hand, and they never switch hands). How about 5? Five looks like three again, with the balls passing from one hand to another. Are there different rules for odd and even numbers of balls? If you could juggle 21 balls, what would that look like?

Then you can mix things up even further. If you have five balls, you have to throw them higher than three balls, as they need to be in the air for longer. But if you throw one ball really high, how many other throws can you fit in in between? How do you throw the balls so that they don’t all land on your head at once? It’s all about making and playing with patterns.

The best way to really see what’s going on though is to check out some videos so you can see the juggling in motion.