A Flexagon for Martin Gardner

Hello, everyone. Yesterday, on the 22nd of May, 2010, Martin Gardner died at the age of 95. Martin Gardner was an American puzzle expert, magician, and sceptic. For many years, he had a column in Scientific American called “Mathematical Games.” And he’s the author of over 100 books. He’s a large figure in the popularisation of mathematics, and a big influence on a lot of today’s mathematicians. I urge you to go and seek out his name. Go search for his name on the internet, go buy his books. But in honour of Martin Gardner, today I thought I’d show you one of the things that he helped popularise. So today I’m going to be showing you how to make a cyclic hexatetraflexagon. Let me show you. It’s this little toy here, and we’ve got number one on the outside, but I can turn this inside out and reveal number two hidden underneath. I can do this again. I can turn it inside out, and we get number three. If we do it again, and there’s number four. And I can keep going. There’s number one again. If I keep going, there’s a fifth hidden face, and if I keep going again, there’s a sixth hidden face. And, in fact, there are six hidden faces inside this little toy here. Now, I’m going to show you how to make this. All you need is a piece of paper, and perhaps a pair of scissors. So first of all, you need to make a square. So you can do this with any piece of paper. You just give it a fold just like that, and to make it a square, let’s use our scissors. I’ll just give it a cut up here. Now, in a second, we are going to fold this in half. So we can open it up, fold it in half–there we go. Fold it away from you, just like that. That’s fine. And then fold it in half again, so it looks like this. Now we can open it up, give it a quarter turn, and give it another fold in half, and fold it in half one more time– like this. Now I’m going to open this up and what we’re going to do is we’re going to cut out the middle bit, here. So I’ll just take my scissors again, and I’ll cut here and here, and just cut out the middle bit, like so. And if I open it up, this is what we have. Now, again, in my best Blue Peter, I’m going to show you one I’ve made earlier. OK, here it is. Now, first of all, I’m going to be folding from the left, like this. Now I fold from the top, and then fold from the right. These folds are going to go in both directions, so it might be worth working them a bit, so that they fold in both directions. And then finally, we fold from the bottom. However, this five, and this five on the outside, on this outside edge, I’m going to fold those inwards so that they now slip into the inside of the flexagon. And this is what we get. And now we can start labeling the faces. This is face one. There you go, there’s face two. Open it up, there’s face three. and there’s face four. You go back to one again, but I can keep going. There’s face five, and there is face six. So there you go! This is your cyclic hexatetraflexagon. Enjoy!