# Rolling one coin round another

 Think about this puzzle first, before trying it. You have two coins of the same size touching. Keep the right-hand coin still (put a finger on it, perhaps). Roll the left hand round the right one, keeping them touching all the time, until it ends up on the other side, on the right. One coin hasn't moved. Which way up will the other coin be? You could reason like this: The moving coin travels round half of circumference of the still one. This means that it also travels half its own circumference, so will end up upside down. Is this true? Try it! Click on 'Roll coin'. No, it isn't! The moving coin ends up the same way up it started. Why? There are several ways to look at this.

 The picture on the right is a single coin rolling in a straight line for the same distance. Trying rolling it, then click on 'Path of coin' for both pictures. You can see how different the paths are. With two coins, the path is a semi-circle, and you could think of it as flipping the coin over. The 'flip' makes it the original way up again. You get fooled because the moving coin is actually moving relative to the still coin. Try clicking of 'Both paths'. You could move the coins so they both moved (imagine them as cogwheels for example). Then if they moved the same amount, they would both end up upside down. But one stays still. The movement is transferred to the other one, so it ends up doing two semi-circles, making it end up the same way up.

Another way to look at it is to think of the moving coin relative to the still one. Go back to the first picture and click on 'Before'. There is an arrow on the moving coin showing its oriantation relative to the moving one, pointing towards it. Now click on 'After'. The same arrow is now pointing away from the still coin. We think that the moving coin is the same way up, but the still coin 'thinks' that the moving coin is now upside down.

An intuitive grasp of how the world works is very useful for mathematics, but it does have its limits! We make assumptions and some of those assumptions are wrong. Here we are making a judgement based on straight lines when we are dealing with curves. Or we are seeing things from our point of view when we should be looking from the coin's point of view.

Of course, you could just look at this as a puzzle for fooling other people! You don't need a webpage, just a couple of coins the same size.