Infinity is weird. It behaves unlike any other number, in weird and wonderful ways. It can flatten other numbers into insignificance, but at the same time can reveal unique and interesting patterns that would otherwise be hidden. There are just two important things to remember when dealing with infinity.

Firstly, infinity is not a number. The most common misconception with infinity is that it is a really big number. Infinity is more accurately described as an amount of something – or a way of measuring something which can't be measured with any number. For example, when saying how many whole numbers there are, it's infinite – if I picked any number, it would be too small. Because infinity is not a number, it cannot and should not be used in sums or as the answer to sums. Just to be clear, one divided by zero is not infinity.

Secondly, infinity is not intuitive. I will demonstrate some specific examples below – but I will say that infinity definitely does some strange things which at first glance seem impossible. However, if we are willing to accept the insights provided to us by the rigour of maths, we can make some sense of these ridiculous results. What's even more surprising is that some of these absurd results play an important part in physics.

The Ball Game

This game is exciting, fun and the best ball game there is - things like football just seem inferior in comparison, they don't teach you anything about infinity. To play this game, you are going to need a large bucket and a lot of balls, I find old ping pong balls work well. We number each ball starting at 1, then 2 and so on, all the way to infinity (really this just means we need a ball for every number, as I mentioned before infinity is not a number so there isn't a ball with infinity on it). We put each ball in the bucket one by one and if its square root (that is the number that when multiplied by itself makes what you just put in) is in the bucket, we will take that ball out.

First we put in the ball numbered 1. As 1 is its own square root (1x1=1) we take this ball straight back out. We then put in balls 2 and 3, neither of their square roots are in the bucket so we don't take anything out. When we put in ball 4 we must take ball 2 out (2x2=4). We then don't take any more out until we reach 9 (3x3=9). As you can see, this is a really fun game for parties, you just don't see it much because no one can find enough balls to play it for as long as they would like.

I did promise we would talk about infinity rather than just putting lots of balls in buckets and taking them back out again. So here's the question: after I have played this game with all the infinite balls for every positive whole number, how many balls are left in the bucket?

If we think about how many balls are in the bucket as we go along, it is always getting larger. At each step we put in one ball, and occasionally we take one out. It looks we'll need a rather large bucket. However, to get an exact number we have to think about which balls will never be removed. These are the balls which are not a square root, but every whole number when multiplied by itself gives another whole number so when that other number goes in it will come out. Every ball has a 'remover' which will get used at some point so every ball will also come out eventually.

This means that after we've been through all the infinite balls the bucket is empty! If we stop at a finite number we get an increasingly large number of balls but at infinity, the pattern is broken – all the balls disappear!

Another Ball Game

This game is, I promise, even more exciting to play than the previous one. Imagine you are at a party and have just wowed your friends by putting an infinite number of balls in a bucket and making them all disappear. They now all want a ball for themselves to remember how cool the party was. The only problem is that there are an infinite number of people at this party (it is pretty exciting). To make sure everyone gets a ball just get them to line up and walk along the line giving the first person ball 1, the second ball 2 and so on. Everyone will get a ball and they are all happy.

Unfortunately, the host was in the bathroom whilst everyone was performing this epic ball heist. The host decides that they would also like a ball, but you've just given them all away! However, there is a solution – just get everyone to pass their ball down one space: the person with ball 1 passes it to the host, the person with ball 2 passes it to person 1 and so on. As there are an infinite number of people, there is no one on the end who has given away their ball and not got another one. Everyone is happy again.

However, your ball distributing skills have made you instantly famous and now an infinite number of other people have gate crashed the party all hoping for a ball of their own. You had worked out how to do it for a finite number of people (just repeat what you did with the host) but this is an entirely different challenge. At first you despair, not wanting to turn them all away, but then you have an idea! You get everyone with a ball to pass it to the person who did have double their number: the person with ball 1 passes it to person 2, the person with ball 2 passes it to person 4 and so on. You now have an infinite number of spare balls, the odd numbers, which you can hand out to all the newcomers.

So without ever changing the balls you were using or adding new balls, you've managed to satisfy an infinite number of people, then one more. Finally, you managed to somehow split it into two infinities the same size as the first! You can take infinity, add it to itself or any finite number and it's still exactly the same infinity.