So we’re gonna do one of the famous puzzles that you can play with a chess board. It’s not chess, itself, but it’s a puzzle you can play. The standard 8 by 8 chess board, and queens. So a queen, if you don’t know, a queen is the strongest; it’s the best piece on a chess board ‘Cause it can move any number of squares to the left and the right. It can move any number of squares up and down, and it can move diagonally as well, any number of squares. Oh! If there’s another piece in its way, it takes it. Aha! Gotcha! Right, so that’s how a queen moves: diagonally, up and down, left and right. Now the puzzle is: Can you place 8 queens on a chess board so that none of the queens can attack each other? So all of the queens are safe. So this wouldn’t work as a placement, because we’ve got one queen here that can take another. They are under attack. But maybe something like this would work just fine. Now I’ve only got 2 queens here. All queens can take each other queen, so the colors don’t matter. The question is, can we place 8 queens on the board? Can we? Well, how many ways can we place 8 queens on the board? First of all, how many ways are there to do that? Let’s look at that first. Right, which does mean that we’ve got 56 blanks? I think we can do this calculation. I think even Brady can do this calculation. How many ways can we arrange 64 objects, Brady? Brady: 64 factorial. James: Yeah, 64 factorial. If you have 64 objects, there are 64 factorial ways to do that: 64 times 63 times 62… all the way down to 1. So, there are 64 objects, 8 queens and 56 blanks. But you can rearrange the blanks between themselves, and it doesn’t change the position on the board. So, we’re going to divide by how many blanks there are: 56 factorial ways to arrange the blanks. So we have to consider that the queens are the same. I can move the queens between each other, I can commute the queens between each other And it doesn’t affect the solution; the positions they are on the board. So for that reason, we are also going to divide through by 8 factorial. There are 8 queens, and they can be swapped around between each other. So we’ll divide through by 8 factorial. So the number of ways you can arrange those 8 queens naively is 4,426,165,368. So there are over 4 billion ways you can place your 8 queens on the board, but that’s naively doing it. That’s not considering how the queens attack each other, so you’re going to get silly solutions Like this one, or this, or this. These are no good; these aren’t solutions. So, how many solutions are there, out of that 4 billion figure that do work, so the queens don’t attack? What do you think? It is in fact only a small fraction of this 4 billion. There are 92 distinct solutions. Now that 92 out of 4 billion as a percentage is… tiny! Unbelievably tiny. So here is one of the solutions you could place. A queen here, I can put one here, they don’t attack each other. They’re not on the same row, they’re not on the same column, they’re not on the same diagonal. I can put a queen here. Again, not in the same row, column or diagonal. I could put one I think down here. One here. And maybe I’ll put one here. And here, and… where shall I put it? Here? Yeah. And I think I got away with that. So that would work – No queens are attacking each other. Now, there are 92 distinct solutions you could have. That includes rotating the board, and reflecting the board as well. So they’re not all going to be really different. Some of them are just turning this 90 degrees. That would count as a solution. Now, there are 8 ways that you could rotate the board and reflect it. 4 rotations, and 4 reflections. So, how many actual individual ways are there to do it? There’s 12. There’s only 12 fundamentally different solutions. 11 of them have 8 reflections and rotations, which takes us up to 88. 1 of them only has 4 rotations and reflections And it’s actually this one I drew. This only has 4 different solutions, because if I rotate it 180, it’s symmetric. So that only has 4, that has half as many. But the others have 8. So that’s 92 distinct solutions. 12 fundamentally different ones. So you can do this – you can do this with other pieces. Can you place 8 knights on the board without attacking each other? I think that would be quite easy. Could you place 8 bishops on the board without attacking each other? I think that’s quite an easy problem as well. Maybe 8 rooks? Again, I think these are easier problems. Although how MANY ways you can do it – that’s a much more interesting question. How many different ways can you do it? Brady: We’d like to that lynda.com for supporting this video. lynda.com is a great go-to resource for learning about, well, all sorts of stuff! You can watch video courses, and tutorials put together by top experts in all sorts of fields: Creative, Technology, Business. I use lynda myself, especially for photo shopping . Any sort of trick, or thing I can’t quite figure out how to do – lynda’s always got a top tutorial that teaches me how to do it. Funnily enough, I was just browsing the site this week and found an amazing coincidence: They have a course called “Code Clinics” that happens to use the 8 queens problem. It seems this problem might be quite old, but it’s a popular aid when teaching people about computer programming. So if you’re a coder, why not check that one out? Now, with over 3000 video courses, and 100,000 tutorials in the vault, lynda’s a real treasure chest of skills and information. And you can sign up now, and get free access to all of it for 10 days By going to lynda.com/numberphile Oh, and there’s also a link in the video description. Our thanks to lynda for their support of Numberphile.