{"version":"1.0.0","segments":[{"startTime":5.845,"endTime":10.107,"body":"The Gresham College lecture that you're listening to right now is giving you knowledge and"},{"startTime":10.107,"endTime":12.665,"body":"insight from one of the world's leading academic experts."},{"startTime":13.445,"endTime":14.825,"body":"Making it takes a lot of time."},{"startTime":15.285,"endTime":18.416,"body":"But because we want to encourage a love of learning, we think it's well worth"},{"startTime":18.416,"endTime":18.625,"body":"it."},{"startTime":19.645,"endTime":22.905,"body":"We never make you pay for lectures, although donations are needed."},{"startTime":23.405,"endTime":24.745,"body":"All we ask in return is this."},{"startTime":25.295,"endTime":27.585,"body":"Send a link to this lecture to someone you think would benefit."},{"startTime":27.965,"endTime":31.848,"body":"And if you haven't already, click the follow or subscribe button from wherever you are"},{"startTime":31.848,"endTime":32.625,"body":"listening right now."},{"startTime":33.645,"endTime":35.105,"body":"Now let's get back to the lecture."},{"startTime":36.645,"endTime":37.905,"body":"So thank you very much for coming."},{"startTime":37.965,"endTime":41.065,"body":"We are gonna talk about the mathematics behind Sudoku."},{"startTime":41.605,"endTime":44.585,"body":"On the way we're gonna meet a sacred Chinese turtle."},{"startTime":45.125,"endTime":49.745,"body":"Uh, we're gonna visit a bleak Welsh hillside and a Parisian apartment block."},{"startTime":50.235,"endTime":54.575,"body":"We're gonna encounter different kinds of number squares like Latin squares that we'll, we'll meet"},{"startTime":54.575,"endTime":54.865,"body":"soon."},{"startTime":55.125,"endTime":56.185,"body":"But let's just begin."},{"startTime":57.305,"endTime":61.252,"body":"Probably everyone in the room knows what a Sudoku is, but just in case, let's"},{"startTime":61.252,"endTime":62.305,"body":"establish the ground rules."},{"startTime":62.605,"endTime":65.144,"body":"So a Sudoku grid, this is a completed one."},{"startTime":65.685,"endTime":68.025,"body":"Uh, it's uh, a square of side nine."},{"startTime":68.245,"endTime":71.665,"body":"And it features the numbers one to nine with the following rules."},{"startTime":71.875,"endTime":76.251,"body":"These numbers have to appear exactly once in each row, in each column, and in"},{"startTime":76.251,"endTime":78.585,"body":"each of these little three by three blocks."},{"startTime":79.365,"endTime":81.465,"body":"So that's what a completed Sudoku grid looks like."},{"startTime":81.685,"endTime":83.785,"body":"But that's not what we see when we start the puzzle."},{"startTime":84.205,"endTime":87.345,"body":"We see a partially completed grid, something like this."},{"startTime":87.845,"endTime":91.895,"body":"And then our job is to use just the rules that we know to complete"},{"startTime":91.895,"endTime":93.785,"body":"fill in the rest of the numbers."},{"startTime":94.565,"endTime":100.675,"body":"Um, you may not have noticed this, but by convention, the squares that are filled"},{"startTime":100.675,"endTime":102.305,"body":"are usually, um, symmetrical."},{"startTime":102.305,"endTime":103.705,"body":"They have a rotational symmetry."},{"startTime":103.925,"endTime":105.985,"body":"Um, we do 180 degree rotation."},{"startTime":106.125,"endTime":109.305,"body":"The numbers aren't gonna match, but the squares that are filled have that pattern."},{"startTime":110.205,"endTime":115.833,"body":"And if we are trying to create a Sudoku puzzle, it must be the case"},{"startTime":115.833,"endTime":119.585,"body":"that this grid can be completed in exactly one way."},{"startTime":119.605,"endTime":120.985,"body":"So there's a unique solution."},{"startTime":121.075,"endTime":124.385,"body":"There is a solution, and that's got to be unique, only one solution."},{"startTime":124.845,"endTime":127.305,"body":"And it should be obtainable without guesswork."},{"startTime":127.605,"endTime":132.198,"body":"In an ideal world, uh, well \u003claugh\u003e in an ideal world, so it should be"},{"startTime":132.198,"endTime":132.505,"body":"obtainable."},{"startTime":133.305,"endTime":137.305,"body":"Occasionally, one or two of us may, uh, resort to a guesses on occasion."},{"startTime":137.645,"endTime":139.825,"body":"Uh, but let's, let's, let's pretend we never do that."},{"startTime":140.485,"endTime":144.913,"body":"So this is not, by the way, a masterclass on how to solve Sudoku because"},{"startTime":144.913,"endTime":147.865,"body":"there's lots of resources about that that you could find."},{"startTime":147.865,"endTime":149.345,"body":"There are books, there are websites."},{"startTime":149.725,"endTime":153.649,"body":"But just to sort of give a basic idea of the sort of thing you"},{"startTime":153.649,"endTime":156.265,"body":"might do, um, we filled in a few entries already."},{"startTime":156.605,"endTime":160.345,"body":"So at this point, of course, you are gonna use these rules constantly."},{"startTime":160.485,"endTime":163.772,"body":"So for example, you could say that, look in this row, there's only one number"},{"startTime":163.772,"endTime":165.745,"body":"that we haven't got, and there's only one space."},{"startTime":166.085,"endTime":168.145,"body":"So we've got to put a one in that space."},{"startTime":168.555,"endTime":169.265,"body":"There it goes."},{"startTime":169.725,"endTime":173.376,"body":"Now we've got a column that only has one number missing the numbers one to"},{"startTime":173.376,"endTime":176.785,"body":"nine must appear exactly once in each row, in each column, in each block."},{"startTime":177.005,"endTime":180.09,"body":"So in this one we can see that there's, what is it, a three missing"},{"startTime":180.09,"endTime":181.325,"body":"and we can put that in."},{"startTime":181.585,"endTime":185.056,"body":"And then now look in this central block, we've now arranged things so that there's"},{"startTime":185.056,"endTime":186.445,"body":"only one gap there as well."},{"startTime":186.825,"endTime":188.525,"body":"And we can put a six in that gap."},{"startTime":188.825,"endTime":190.125,"body":"So we can proceed like this."},{"startTime":190.665,"endTime":192.485,"body":"We can get slightly more sophisticated."},{"startTime":192.785,"endTime":196.925,"body":"So there's one technique which is very useful, um, which is often called slicing."},{"startTime":197.145,"endTime":203.045,"body":"So imagine cutting your Sudoku into three thick slices, each of which contains three rows,"},{"startTime":203.045,"endTime":205.405,"body":"and it also contains three blocks."},{"startTime":205.625,"endTime":208.925,"body":"So you can see I've highlighted the middle slice of my Sudoku cake there."},{"startTime":209.505,"endTime":214.79,"body":"Now that's useful to notice because it means in that slice every number is gonna"},{"startTime":214.79,"endTime":215.142,"body":"show up exactly three times, once in each row, once in each each block, but"},{"startTime":215.142,"endTime":221.485,"body":"three times in total."},{"startTime":221.985,"endTime":225.201,"body":"So if we look at, say the sixes here, there are two sixes so far"},{"startTime":225.201,"endTime":225.845,"body":"in that block."},{"startTime":226.465,"endTime":230.336,"body":"Um, we can eliminate lots of possibilities for where the remaining six, there must be"},{"startTime":230.336,"endTime":231.885,"body":"another six where it could go."},{"startTime":232.345,"endTime":235.445,"body":"So it can't go in either of the rows that contain a six."},{"startTime":235.445,"endTime":236.485,"body":"So it's not in that row."},{"startTime":236.675,"endTime":237.565,"body":"It's not in that row."},{"startTime":238.065,"endTime":241.045,"body":"It can't go in either of the blocks that already contain a six."},{"startTime":241.225,"endTime":242.605,"body":"So it's not in the left hand block."},{"startTime":242.715,"endTime":243.765,"body":"It's not in the middle block."},{"startTime":243.945,"endTime":247.257,"body":"And we can see there's only one place it can go 'cause it must go"},{"startTime":247.257,"endTime":249.245,"body":"in the top row of the right hand block."},{"startTime":250.065,"endTime":251.885,"body":"And there's only one empty space in that row."},{"startTime":252.385,"endTime":253.005,"body":"So that's great."},{"startTime":253.005,"endTime":255.085,"body":"So we can pop our final six in there."},{"startTime":255.945,"endTime":260.178,"body":"And of course, well, it's slightly unconventional with cake, but you can slice vertically as"},{"startTime":260.178,"endTime":263.565,"body":"well and do the same thing with vertical slices of your sudoku."},{"startTime":264.435,"endTime":270.35,"body":"Another option available, which is useful sometimes is even if you can't absolutely pinpoint where"},{"startTime":270.35,"endTime":270.745,"body":"uh, numbers go, sometimes you can narrow it down and you can get pairs of"},{"startTime":270.745,"endTime":280.605,"body":"numbers that have to go in some order into two spaces."},{"startTime":281.385,"endTime":285.085,"body":"So for instance, if you look at this vertical slice here, look at the nines."},{"startTime":285.985,"endTime":288.125,"body":"Uh, you've got two nines, where's the third one gonna go?"},{"startTime":288.315,"endTime":290.925,"body":"It's got to go in the right hand column."},{"startTime":291.525,"endTime":294.245,"body":"'cause the other two columns have a nine and the top block."},{"startTime":294.785,"endTime":297.685,"body":"So there's two vacant cells where it could possibly go."},{"startTime":297.685,"endTime":301.845,"body":"There's nine, we don't know which one it goes in, but there's also threes."},{"startTime":302.585,"endTime":307.202,"body":"And by exactly the same reasoning, um, the other three that is in that vertical"},{"startTime":307.202,"endTime":311.205,"body":"slice must go in one of those two uh, squares that are highlighted."},{"startTime":312.025,"endTime":315.301,"body":"Now we don't know which way around they go, but we know that those two"},{"startTime":315.301,"endTime":317.485,"body":"squares contain a three and a nine in some order."},{"startTime":317.985,"endTime":321.485,"body":"So no other squares in that little block can contain a three or a nine."},{"startTime":322.065,"endTime":325.933,"body":"So even though we don't know which is which at this point, that can help"},{"startTime":325.933,"endTime":326.965,"body":"us with other entries."},{"startTime":327.145,"endTime":332.565,"body":"For instance, in this uh, row up here, there's gotta be a three somewhere."},{"startTime":332.565,"endTime":336.381,"body":"We haven't got one yet, but it can't be in either of those, uh, little"},{"startTime":336.381,"endTime":336.635,"body":"empty spaces on the right hand, the end of that row because we know that"},{"startTime":336.635,"endTime":336.889,"body":"the three that must be in that block, um, is in the top or the"},{"startTime":336.889,"endTime":345.285,"body":"bottom square that we've highlighted."},{"startTime":345.585,"endTime":349.8,"body":"It doesn't matter which one it is for us to deduce, but it's not in"},{"startTime":349.8,"endTime":351.205,"body":"that cent middle row there."},{"startTime":351.385,"endTime":355.206,"body":"So we know that the three has to go in, in on the left hand"},{"startTime":355.206,"endTime":357.245,"body":"end of this row where I've put it."},{"startTime":357.825,"endTime":362.606,"body":"So these kind of ideas and some others can help you to complete the Sudoku"},{"startTime":362.606,"endTime":362.925,"body":"grid."},{"startTime":363.105,"endTime":367.925,"body":"And at each stage you are proceeding using just, you know, pure logic."},{"startTime":367.945,"endTime":369.445,"body":"You're not guessing at any point."},{"startTime":370.185,"endTime":373.537,"body":"Uh, as I've said, lots of resources you can find if you want to, to"},{"startTime":373.537,"endTime":375.325,"body":"learn lots more tips and tricks about Sudoku."},{"startTime":375.785,"endTime":379.44,"body":"Uh, if you want to be loyal to Gresham College, you can seek out the"},{"startTime":379.44,"endTime":382.365,"body":"book on Sudoku by our former Gresham professor of Geometry, Robin Wilson."},{"startTime":382.825,"endTime":385.445,"body":"Um, but you know, that's, that's, that's my recommendation anyway."},{"startTime":386.705,"endTime":388.925,"body":"But mathematically speaking, what have we done?"},{"startTime":388.945,"endTime":392.005,"body":"We haven't actually had to do any arithmetic with the numbers."},{"startTime":392.145,"endTime":397.496,"body":"So if people say correctly that Sudoku is mathematical, it, it sometimes there's a misapprehension"},{"startTime":397.496,"endTime":397.853,"body":"that it's mathematical because it's got numbers in it, but that isn't the reason it's"},{"startTime":397.853,"endTime":403.205,"body":"mathematical."},{"startTime":403.505,"endTime":407.745,"body":"The reason it's it's mathematical, it's because we're using the structure and we are thinking"},{"startTime":407.745,"endTime":409.725,"body":"logically about how to complete the puzzle."},{"startTime":410.625,"endTime":414.765,"body":"And there are also mathematical questions we can ask about, about Sudoku."},{"startTime":415.025,"endTime":419.285,"body":"For example, how many possible complete Sudoku grids are there?"},{"startTime":419.745,"endTime":420.445,"body":"Can we count them?"},{"startTime":420.505,"endTime":421.885,"body":"Can we work out, classify them?"},{"startTime":422.665,"endTime":424.045,"body":"How many puzzles are there?"},{"startTime":424.045,"endTime":425.005,"body":"That's slightly different."},{"startTime":425.285,"endTime":429.455,"body":"'cause a puzzle is a partially completed grid, which has a unique solution that can"},{"startTime":429.455,"endTime":430.845,"body":"be arrived at without guesswork."},{"startTime":431.505,"endTime":435.717,"body":"So how many of those and can we make those and related, how many clues"},{"startTime":435.717,"endTime":438.525,"body":"do we actually need to be able to do this?"},{"startTime":438.545,"endTime":441.833,"body":"So if you just put one number into a empty grid, of course you're not"},{"startTime":441.833,"endTime":444.245,"body":"going to be able to find a unique solution to that."},{"startTime":444.825,"endTime":448.743,"body":"What's the minimum number you need in order to be able to actually finish that"},{"startTime":448.743,"endTime":449.005,"body":"puzzle?"},{"startTime":449.105,"endTime":451.765,"body":"It might be very hard, but what's the smallest number you need?"},{"startTime":452.545,"endTime":460.42,"body":"But let's begin by saying Sudoku is in the UK at least fairly recent, um,"},{"startTime":460.42,"endTime":463.045,"body":"in, you know, geological timescales."},{"startTime":463.185,"endTime":468.06,"body":"The first Sudoku ever in the UK printed in a national newspaper was in November,"},{"startTime":468.06,"endTime":469.685,"body":"2004 in the Times newspaper."},{"startTime":469.905,"endTime":473.445,"body":"So it's fairly recent, um, coming to, to the uk."},{"startTime":473.945,"endTime":478.909,"body":"But you know, there's, we sort of all feel that like Sudoku is this ancient"},{"startTime":478.909,"endTime":479.24,"body":"Japanese thing that, you know, imagine the Sudoku master crafting this perfect puzzle on the"},{"startTime":479.24,"endTime":487.845,"body":"slopes of Mount Fuji or something like this that is complete nonsense."},{"startTime":488.245,"endTime":489.365,"body":"\u003claugh\u003e, I mean stand line."},{"startTime":489.395,"endTime":492.565,"body":"This picture I've just made, that's just as a massive lie."},{"startTime":493.185,"endTime":495.645,"body":"Sudoku was invented in America in the 1970s."},{"startTime":496.265,"endTime":497.525,"body":"Uh, it wasn't called Sudoku then."},{"startTime":497.525,"endTime":499.085,"body":"It was called something like number place."},{"startTime":499.345,"endTime":503.903,"body":"And there were these puzzles that were like Sudoku and then they got picked up"},{"startTime":503.903,"endTime":504.206,"body":"by a Japanese, uh, puzzle magazine who started printing them, um, about a few years"},{"startTime":504.206,"endTime":508.765,"body":"later."},{"startTime":509.105,"endTime":512.874,"body":"And they really took off in Japan and then they made their way to, to"},{"startTime":512.874,"endTime":514.885,"body":"the uk and you need back to America."},{"startTime":515.585,"endTime":520.005,"body":"But these puzzles, this kind of number grid is less than 50 years old."},{"startTime":520.225,"endTime":525.608,"body":"And in the grand scale of mathematical ideas, that's very recent, there are much older"},{"startTime":525.608,"endTime":526.685,"body":"squares of numbers."},{"startTime":527.145,"endTime":530.982,"body":"And we're gonna go now on a little journey through the history of these, some"},{"startTime":530.982,"endTime":534.565,"body":"of these squares of numbers, partly because they'll feed into our thinking about Sudoku."},{"startTime":534.745,"endTime":537.965,"body":"Um, but partly 'cause they're just super interesting already in their own right."},{"startTime":538.185,"endTime":541.445,"body":"And we're going to begin with the oldest kind of number square."},{"startTime":542.665,"endTime":543.965,"body":"And that's the magic square."},{"startTime":544.305,"endTime":549.347,"body":"So this is a picture of um, what's believed to have been the oldest magic"},{"startTime":549.347,"endTime":551.365,"body":"square, uh, that we know of."},{"startTime":552.385,"endTime":553.965,"body":"And it's called the low shoe square."},{"startTime":554.145,"endTime":556.245,"body":"So this is a picture from a Tibetan scroll."},{"startTime":556.245,"endTime":559.085,"body":"This is some centuries after the, the square probably originated."},{"startTime":559.535,"endTime":562.045,"body":"Let's put the numbers into modern format."},{"startTime":562.195,"endTime":562.805,"body":"Here they are."},{"startTime":563.145,"endTime":565.205,"body":"And this shows you this is a magic square."},{"startTime":565.205,"endTime":566.325,"body":"So what's a magic square?"},{"startTime":566.835,"endTime":571.57,"body":"This time we do need to do some arithmetic with the numbers because a magic"},{"startTime":571.57,"endTime":571.886,"body":"square is one where the sum of the numbers in every row, every column and"},{"startTime":571.886,"endTime":577.885,"body":"every diagonal is the same."},{"startTime":577.945,"endTime":579.205,"body":"We call it the magic number."},{"startTime":579.625,"endTime":583.525,"body":"So if you pick a row or a column or diagonal vis and everyone picks"},{"startTime":583.525,"endTime":587.165,"body":"one, and you should find that the sum of the numbers is 15 always."},{"startTime":587.665,"endTime":588.845,"body":"So this is a magic square."},{"startTime":589.475,"endTime":594.565,"body":"This square, um, is reputed to have originated in the following way."},{"startTime":594.975,"endTime":599.261,"body":"There was a big flood in China and the people were trying to appease the"},{"startTime":599.261,"endTime":602.405,"body":"river God by making sacrifices to say, please stop the flooding."},{"startTime":602.865,"endTime":607.783,"body":"Um, and as a sign that their prayers were going to be answered, that God"},{"startTime":607.783,"endTime":608.111,"body":"sent a sacred turtle and he crawled out of the river and on his back"},{"startTime":608.111,"endTime":615.325,"body":"into his shell was carved this magic square."},{"startTime":615.595,"endTime":615.885,"body":"Okay?"},{"startTime":615.905,"endTime":617.405,"body":"So that, that's the origin."},{"startTime":617.945,"endTime":619.805,"body":"It is claimed of this magic square."},{"startTime":620.435,"endTime":622.285,"body":"It's a particular kind of magic square."},{"startTime":622.585,"endTime":624.045,"body":"It features the numbers one to nine."},{"startTime":624.665,"endTime":626.965,"body":"Um, so that, you know, you can have other ones which don't do that."},{"startTime":627.945,"endTime":631.388,"body":"If you start with the first, you know, however many numbers that you need in"},{"startTime":631.388,"endTime":631.617,"body":"this case, if it's an order three square, so it's three by three square, then"},{"startTime":631.617,"endTime":638.045,"body":"you need to, to fill it, you need the numbers one up to nine."},{"startTime":638.465,"endTime":641.085,"body":"Um, that's called a classical magic square."},{"startTime":641.195,"endTime":641.485,"body":"Okay?"},{"startTime":641.785,"endTime":643.965,"body":"But you don't, they don't all have to be classical."},{"startTime":644.265,"endTime":649.165,"body":"So the oldest order, four magic square that we know about, um, is this one."},{"startTime":649.275,"endTime":652.65,"body":"This is not a classical magic square 'cause that would have to contain the numbers"},{"startTime":652.65,"endTime":653.325,"body":"one to 16."},{"startTime":653.835,"endTime":655.645,"body":"This doesn't, but it's still a magic square."},{"startTime":655.665,"endTime":660.033,"body":"The sum of the numbers in each row column diagonal, uh, is, is a, is"},{"startTime":660.033,"endTime":660.325,"body":"constant."},{"startTime":660.785,"endTime":665.365,"body":"And so this comes from a book, uh, an old Indian book by uh, Rahmi."},{"startTime":666.105,"endTime":671.16,"body":"And its purpose stated in the book was, um, to give you different recipes for"},{"startTime":671.16,"endTime":672.845,"body":"making perfume with four ingredients."},{"startTime":673.225,"endTime":674.125,"body":"So here's what you do."},{"startTime":674.265,"endTime":677.981,"body":"You get your four ingredients, A, B, C, and D, and you pick a row"},{"startTime":677.981,"endTime":678.725,"body":"column or diagonal."},{"startTime":678.725,"endTime":679.965,"body":"Let's say we pick the first row."},{"startTime":680.425,"endTime":685.782,"body":"And that tells you to use two quantities of ingredient a, three of ingredient B,"},{"startTime":685.782,"endTime":687.925,"body":"five of C, eight of D."},{"startTime":688.305,"endTime":692.708,"body":"And the total amount, because it's a magic square, whichever row column or diagonal you"},{"startTime":692.708,"endTime":696.525,"body":"pick the total volume of the perfume that produce will be the same."},{"startTime":697.025,"endTime":699.805,"body":"So that's, this is what was stated the purpose of this square."},{"startTime":700.225,"endTime":702.165,"body":"So that's an order four magic square."},{"startTime":702.705,"endTime":707.045,"body":"Now Sudoku grids, um, let's have a mini Sudoku."},{"startTime":707.345,"endTime":708.605,"body":"That's where you can see what it is."},{"startTime":708.605,"endTime":710.845,"body":"It's got the numbers one to four instead of the numbers one to nine."},{"startTime":710.845,"endTime":712.205,"body":"But the same principles apply."},{"startTime":712.705,"endTime":717.992,"body":"Uh, those are not magic squares necessarily because um, the sums along the diagonals do"},{"startTime":717.992,"endTime":721.165,"body":"not have to uh, fit in with this rule."},{"startTime":721.235,"endTime":724.685,"body":"They're called semi magic squares, an example of a semi magic square."},{"startTime":725.385,"endTime":729.285,"body":"And so a semi magic square has the same sum of the rows and the"},{"startTime":729.285,"endTime":731.365,"body":"columns, but we don't mind about the diagonals."},{"startTime":731.625,"endTime":734.165,"body":"Now Sudoku sort of gets into that classification."},{"startTime":734.675,"endTime":738.279,"body":"It's not cheating exactly, but of course, because this one has the numbers 1, 2,"},{"startTime":738.279,"endTime":738.519,"body":"3, and four in every row and in every column obviously the sum of those"},{"startTime":738.519,"endTime":743.565,"body":"numbers is always gonna be the same."},{"startTime":743.565,"endTime":744.485,"body":"They're the same numbers."},{"startTime":745.065,"endTime":747.085,"body":"So Sudoku is a semi magic square."},{"startTime":747.085,"endTime":747.645,"body":"So-called."},{"startTime":748.825,"endTime":755.689,"body":"Now the most famous magic square in western art is in this picture melancholia by"},{"startTime":755.689,"endTime":756.605,"body":"albe dura."},{"startTime":757.345,"endTime":760.354,"body":"You might not be able to see it so clearly because it's in the top"},{"startTime":760.354,"endTime":760.554,"body":"right hand end of that picture, but I've, I've enlarged it so you can see"},{"startTime":760.554,"endTime":763.965,"body":"it, uh, better."},{"startTime":764.265,"endTime":769.458,"body":"So this is a classical, it's got the numbers one to 16 order four magic"},{"startTime":769.458,"endTime":769.805,"body":"square."},{"startTime":771.345,"endTime":773.925,"body":"The picture was made in the year 15, 14."},{"startTime":773.945,"endTime":777.938,"body":"And you can see 1514 is those two numbers in the middle of the bottom"},{"startTime":777.938,"endTime":778.205,"body":"row."},{"startTime":779.185,"endTime":784.405,"body":"Uh, 1514 was the year that JIRA's mother died, hence the melancholia."},{"startTime":784.905,"endTime":788.405,"body":"But there's lots of other symbolism in this picture, for example."},{"startTime":788.945,"endTime":795.405,"body":"Um, at that time different planets were associated to different characteristics of various kinds."},{"startTime":796.105,"endTime":798.645,"body":"The planet's SA was associated with melancholy."},{"startTime":799.265,"endTime":804.125,"body":"It was also associated completely independently, I should say with geometry."},{"startTime":804.205,"endTime":805.765,"body":"Geometry does not make you sad."},{"startTime":806.125,"endTime":807.045,"body":"\u003claugh\u003e makes you happy."},{"startTime":807.345,"endTime":810.476,"body":"Um, but that's why there's a lot of geometry going on in this picture as"},{"startTime":810.476,"endTime":810.685,"body":"well."},{"startTime":811.905,"endTime":816.914,"body":"Now magic squares of different sizes were also associated to different planets, but the size"},{"startTime":816.914,"endTime":817.248,"body":"of magic square that's associated to Saturn is, if I remember rightly, order six six"},{"startTime":817.248,"endTime":822.925,"body":"by six square."},{"startTime":823.265,"endTime":825.605,"body":"So what's this four by four square doing in there?"},{"startTime":826.155,"endTime":831.647,"body":"Well, if you are feeling melancholy, an antidote to that might be to do something"},{"startTime":831.647,"endTime":833.845,"body":"a bit more jovial and jovial."},{"startTime":833.845,"endTime":838.085,"body":"Things are associated with the planet Jupiter, 'cause JoVE is another word for Jupiter."},{"startTime":838.385,"endTime":842.205,"body":"So four by four magic squares are associated with Jupiter."},{"startTime":842.865,"endTime":845.285,"body":"But these, this is an antidote your melancholy."},{"startTime":845.585,"endTime":847.445,"body":"So that's what that square is doing in there."},{"startTime":848.545,"endTime":853.363,"body":"Now we could work out the magic number of this particular order for classical magic"},{"startTime":853.363,"endTime":853.685,"body":"square."},{"startTime":854.025,"endTime":858.28,"body":"We can just add up the numbers in one of the rose cons or diagonals,"},{"startTime":858.28,"endTime":858.563,"body":"but we don't have to because it turns out that every classical magic square of"},{"startTime":858.563,"endTime":864.805,"body":"a given order has the same magic number."},{"startTime":864.825,"endTime":868.685,"body":"Here's that dura one again reproduced here just just for a note."},{"startTime":869.705,"endTime":873.323,"body":"We can prove this and we do something that mathematicians quite like to do, which"},{"startTime":873.323,"endTime":873.564,"body":"is to work out the same thing in two different ways and then say, oh"},{"startTime":873.564,"endTime":873.805,"body":"those, then these things, uh, they're both equal to this middle thing, so they're the"},{"startTime":873.805,"endTime":881.525,"body":"same as each other."},{"startTime":882.305,"endTime":885.342,"body":"And the thing we're going to do, the thing we're going to work out is"},{"startTime":885.342,"endTime":887.165,"body":"the sum of all the numbers in that square."},{"startTime":887.265,"endTime":889.005,"body":"And you'll, you'll see why this helps us."},{"startTime":889.665,"endTime":892.509,"body":"So if you want to work out the sum of all the numbers in that"},{"startTime":892.509,"endTime":894.405,"body":"square, well we just have to add up the numbers."},{"startTime":894.465,"endTime":895.725,"body":"One to 16, fine."},{"startTime":896.225,"endTime":899.056,"body":"Now there's a cute little trick to doing this without actually having to do any"},{"startTime":899.056,"endTime":899.245,"body":"work."},{"startTime":899.315,"endTime":903.682,"body":"It's another thing mathematicians like, um, so we can also write that s that sum,"},{"startTime":903.682,"endTime":906.885,"body":"we can just add them up in the other direction, right?"},{"startTime":906.885,"endTime":908.485,"body":"16 down to one."},{"startTime":909.025,"endTime":913.813,"body":"And the reason that's useful is if we now add those two equations together on"},{"startTime":913.813,"endTime":917.645,"body":"the left, we've got twice the sum the thing we're interested in."},{"startTime":917.905,"endTime":921.085,"body":"On the right, you can see all those pairs add up to the same thing."},{"startTime":921.485,"endTime":923.845,"body":"17 one plus 16, two plus 15 all the way down."},{"startTime":924.265,"endTime":928.365,"body":"So on the right we've got 16 lots of 17 on the left, we've got"},{"startTime":928.365,"endTime":930.005,"body":"twice the thing we're interested in."},{"startTime":930.225,"endTime":935.009,"body":"So if we just halve everything, we now find that our sum S is half"},{"startTime":935.009,"endTime":936.285,"body":"of 16 times 17."},{"startTime":936.905,"endTime":940.227,"body":"We don't need to work that out yet or indeed ever because there's another way"},{"startTime":940.227,"endTime":942.885,"body":"to work out the sum of all the numbers in the square."},{"startTime":943.345,"endTime":947.89,"body":"And that is to notice every row sums two the same thing, the magic number"},{"startTime":947.89,"endTime":949.405,"body":"that we're trying to find."},{"startTime":949.575,"endTime":951.125,"body":"Let's call that magic number M."},{"startTime":951.755,"endTime":954.325,"body":"Then we know that there's sum of all the numbers in the square."},{"startTime":954.395,"endTime":957.085,"body":"It's just four lots of the magic number 'cause we've got four rows."},{"startTime":958.215,"endTime":958.565,"body":"Great."},{"startTime":958.585,"endTime":962.339,"body":"So now we've got two different expressions for our s make them equal to each"},{"startTime":962.339,"endTime":962.589,"body":"other and we can find out m we don't even need to find s not"},{"startTime":962.589,"endTime":966.845,"body":"interested in that."},{"startTime":967.065,"endTime":968.325,"body":"We can find M directly."},{"startTime":968.595,"endTime":971.645,"body":"It's one eighth of 16 times 17."},{"startTime":971.865,"endTime":975.723,"body":"And we're at the point now where even I can do that with a live"},{"startTime":975.723,"endTime":979.325,"body":"audience \u003claugh\u003e, an eighth of 16 is two and two times 17 is 34."},{"startTime":979.665,"endTime":983.077,"body":"So the magic number, and you can check this for the juris square, but it'll"},{"startTime":983.077,"endTime":985.125,"body":"be true for any order for classical magic square."},{"startTime":985.425,"endTime":991.505,"body":"The magic number is 34 has to be, and really the four, the fact of"},{"startTime":991.505,"endTime":995.965,"body":"it being four was not involved in any of this stuff."},{"startTime":996.305,"endTime":998.045,"body":"We didn't need it to be four that wasn't special."},{"startTime":998.665,"endTime":1003.376,"body":"And you can actually use this same process to work out the magic number of"},{"startTime":1003.376,"endTime":1004.005,"body":"any order."},{"startTime":1004.325,"endTime":1005.925,"body":"NN is of course our favorite number."},{"startTime":1006.345,"endTime":1010.482,"body":"Um, any classical magic square of order n so an n by n square, it"},{"startTime":1010.482,"endTime":1012.965,"body":"will contain the numbers one up to n squared."},{"startTime":1013.185,"endTime":1017.207,"body":"So if you just replace all the fours in what I did before with Ns,"},{"startTime":1017.207,"endTime":1019.085,"body":"um, you know, the same reasoning applies."},{"startTime":1019.105,"endTime":1023.262,"body":"And you get this formula for the magic square, uh, the magic number of a"},{"startTime":1023.262,"endTime":1024.925,"body":"classical magic square of order M."},{"startTime":1025.425,"endTime":1027.885,"body":"So we can work out for example, when n equals three."},{"startTime":1028.145,"endTime":1028.765,"body":"What's that?"},{"startTime":1028.785,"endTime":1030.365,"body":"Uh, the low shoe square."},{"startTime":1030.385,"endTime":1031.925,"body":"The magic turtle square."},{"startTime":1032.545,"endTime":1036.626,"body":"Um, if N is three, you can put in that into this uh, this equation"},{"startTime":1036.626,"endTime":1036.898,"body":"and you find that the magic number is 50 and here is the square and"},{"startTime":1036.898,"endTime":1042.885,"body":"you can check yes indeed, that that's true."},{"startTime":1044.385,"endTime":1048.925,"body":"It also shows you that there cannot be a classical magic square of order two."},{"startTime":1050.065,"endTime":1053.595,"body":"And the reason for that is if you put n equals two into that little"},{"startTime":1053.595,"endTime":1057.125,"body":"formula there, uh, two squared plus one is five and half times two is one."},{"startTime":1057.125,"endTime":1059.485,"body":"So you get five, that would be the magic number."},{"startTime":1060.345,"endTime":1065.445,"body":"But if you try and actually make five, what's got to go with one to"},{"startTime":1065.445,"endTime":1066.125,"body":"make five?"},{"startTime":1066.345,"endTime":1069.04,"body":"You know, this, this square would have to have the numbers 1, 2, 3, and"},{"startTime":1069.04,"endTime":1069.58,"body":"four in it."},{"startTime":1070.265,"endTime":1072.205,"body":"But one would always have to be paired with four."},{"startTime":1072.905,"endTime":1076.033,"body":"So four would have to go actually in every single one of those gaps to"},{"startTime":1076.033,"endTime":1077.285,"body":"make this be a magic square."},{"startTime":1077.385,"endTime":1078.285,"body":"And that's not allowed."},{"startTime":1078.285,"endTime":1079.405,"body":"Where are two and three gonna go?"},{"startTime":1079.585,"endTime":1086.072,"body":"So there are no classical magic squares of order two, but it turns out there"},{"startTime":1086.072,"endTime":1089.965,"body":"are definitely classical magic squares of every other order."},{"startTime":1091.545,"endTime":1092.645,"body":"So two doesn't work."},{"startTime":1092.645,"endTime":1098.769,"body":"Everything else you can do and methods for finding these things were being developed in,"},{"startTime":1098.769,"endTime":1099.177,"body":"in, you know, from hundreds of years ago, one of the most interesting writers and"},{"startTime":1099.177,"endTime":1099.585,"body":"kinda mathematicians working on magic squares was um, Mohamed ib Mohammed Alani Al Kwi who"},{"startTime":1099.585,"endTime":1116.325,"body":"was born in, I put this sort of red, red uh, blob there."},{"startTime":1116.325,"endTime":1120.775,"body":"He was born in what's now northern Nigeria and then he moved north and he"},{"startTime":1120.775,"endTime":1123.445,"body":"lived and worked uh, mostly in Cairo in Egypt."},{"startTime":1123.905,"endTime":1128.445,"body":"And it was there when in 1732 he wrote a treatise on magic squares."},{"startTime":1128.585,"endTime":1130.045,"body":"And in that he did loads of stuff."},{"startTime":1130.585,"endTime":1136.797,"body":"He classified, he found all the order three classical magic squares and he gave away"},{"startTime":1136.797,"endTime":1140.525,"body":"of producing a magic square of any odd order."},{"startTime":1140.705,"endTime":1143.005,"body":"So order five, order seven and so on."},{"startTime":1143.665,"endTime":1146.885,"body":"And that there are algorithms that produce all other sizes as well."},{"startTime":1147.305,"endTime":1148.285,"body":"I'm gonna show you his method."},{"startTime":1148.285,"endTime":1151.694,"body":"You not gonna try it for like making a seven by seven magic square on"},{"startTime":1151.694,"endTime":1153.285,"body":"your way home, the train or something."},{"startTime":1153.385,"endTime":1156.222,"body":"So here let's make a five by five one and it's the same rule for"},{"startTime":1156.222,"endTime":1157.925,"body":"any size as long as it's an odd number."},{"startTime":1158.665,"endTime":1163.015,"body":"So you put a one in the middle of the top row and then the"},{"startTime":1163.015,"endTime":1166.205,"body":"basic principle is you move upwards and diagonally to the right."},{"startTime":1166.225,"endTime":1168.125,"body":"So up one square and right one square."},{"startTime":1168.465,"endTime":1171.681,"body":"Now already you are thinking, but Sarah, you can't move up one square, you're already"},{"startTime":1171.681,"endTime":1172.325,"body":"at the top."},{"startTime":1172.545,"endTime":1175.561,"body":"So what you do then you imagine you're playing like an old fashioned game of"},{"startTime":1175.561,"endTime":1176.165,"body":"S space invaders."},{"startTime":1176.355,"endTime":1180.744,"body":"This may be you guys in the olden days there are these games where the"},{"startTime":1180.744,"endTime":1183.085,"body":"memory on the computer wasn't, wasn't very high."},{"startTime":1183.265,"endTime":1185.428,"body":"So when you went off the top of the screen, you just reappeared at the"},{"startTime":1185.428,"endTime":1186.005,"body":"bottom of the screen."},{"startTime":1186.035,"endTime":1187.645,"body":"Kind of saves memory in producing the game."},{"startTime":1187.835,"endTime":1190.643,"body":"When you go off the right hand side of the screen, you reappear on the"},{"startTime":1190.643,"endTime":1191.205,"body":"left hand side."},{"startTime":1191.305,"endTime":1193.325,"body":"So you, that's exactly what you do with this method."},{"startTime":1193.425,"endTime":1196.787,"body":"So with this one, if we go up one that brings us, actually we reappear"},{"startTime":1196.787,"endTime":1199.925,"body":"at the bottom of the square and then you go one to the right."},{"startTime":1199.925,"endTime":1201.005,"body":"So you move in diagonally."},{"startTime":1201.105,"endTime":1205.989,"body":"So your two goes there and then you move diagonally up one, right, one, three"},{"startTime":1205.989,"endTime":1206.315,"body":"up, one right one where we loop back to the left hand side four, move"},{"startTime":1206.315,"endTime":1211.525,"body":"diagonally again."},{"startTime":1211.985,"endTime":1214.245,"body":"Now six, we want to put where that one already is."},{"startTime":1214.545,"endTime":1217.885,"body":"The rule in that situation is you just move down a square instead."},{"startTime":1217.985,"endTime":1223.943,"body":"So we get six here, then we're moving diagonally again, 7, 8, 9, 10, oh"},{"startTime":1223.943,"endTime":1224.34,"body":"our way is blocked, move down then diagonally uh, from 11, 12, 13, 14, 15"},{"startTime":1224.34,"endTime":1232.285,"body":"our way will be blocked again."},{"startTime":1232.465,"endTime":1234.205,"body":"We want to put 16 where the 11 is."},{"startTime":1234.705,"endTime":1239.888,"body":"No worries, we just moved down a square 16 and then we can finish 17,"},{"startTime":1239.888,"endTime":1240.925,"body":"18, 19, 20."},{"startTime":1241.745,"endTime":1246.205,"body":"Oh we blocked, moved down diagonally, diagonally, diagonally."},{"startTime":1246.305,"endTime":1247.005,"body":"And there we are."},{"startTime":1247.345,"endTime":1250.963,"body":"So that, I mean you can check, I'm not gonna do all the additions right"},{"startTime":1250.963,"endTime":1251.205,"body":"now."},{"startTime":1251.435,"endTime":1254.685,"body":"That indeed is a classical magic square of order five."},{"startTime":1254.865,"endTime":1258.553,"body":"So that's the method and that will allow you to create one of, of any"},{"startTime":1258.553,"endTime":1259.045,"body":"odd order."},{"startTime":1260.435,"endTime":1260.725,"body":"Okay?"},{"startTime":1260.985,"endTime":1265.327,"body":"So the great thing is once you've got one magic square of a given size,"},{"startTime":1265.327,"endTime":1266.485,"body":"you can make more."},{"startTime":1266.955,"endTime":1270.496,"body":"This is again, mathematicians really like making use of something we already have and using"},{"startTime":1270.496,"endTime":1271.205,"body":"it many times."},{"startTime":1272.625,"endTime":1277.267,"body":"So here is our low shoe square sacred turtle sitting on the top with that"},{"startTime":1277.267,"endTime":1279.125,"body":"square we can make another one."},{"startTime":1279.145,"endTime":1284.218,"body":"And what we do is we can rotate that square 90 degrees to the right"},{"startTime":1284.218,"endTime":1286.925,"body":"and what you get is another magic square."},{"startTime":1286.985,"endTime":1289.325,"body":"The reason is that the rows have become columns."},{"startTime":1289.325,"endTime":1290.565,"body":"The columns have become rows."},{"startTime":1290.785,"endTime":1293.725,"body":"The diagonals are still diagonals but it doesn't affect the sums."},{"startTime":1294.145,"endTime":1296.445,"body":"So you still have this same magic property."},{"startTime":1297.345,"endTime":1299.045,"body":"So that's one of the symmetries of a square."},{"startTime":1300.045,"endTime":1303.845,"body":"Actually any of the symmetries of a square will still give you a magic square."},{"startTime":1304.425,"endTime":1306.205,"body":"You're not affecting that property."},{"startTime":1306.615,"endTime":1309.733,"body":"There are eight symmetries of a square, including the one where you just sit there"},{"startTime":1309.733,"endTime":1310.565,"body":"and don't do anything."},{"startTime":1310.985,"endTime":1314.716,"body":"Um, there are then there are three kind of rotations you can do apart from"},{"startTime":1314.716,"endTime":1314.965,"body":"that."},{"startTime":1315.425,"endTime":1318.085,"body":"And then there are reflections in the, in the various symmetry lines."},{"startTime":1318.625,"endTime":1324.285,"body":"So from one starting square that actually gives you a set of eight magic squares."},{"startTime":1324.665,"endTime":1326.845,"body":"So eight for the price of one is a pretty good deal."},{"startTime":1327.535,"endTime":1331.525,"body":"Those eight squares, we call those the orbit of the original square."},{"startTime":1332.625,"endTime":1336.375,"body":"Uh, because the orbit is everywhere you can get to and we are getting there"},{"startTime":1336.375,"endTime":1338.125,"body":"by symmetries, which are of course wonderful."},{"startTime":1338.505,"endTime":1341.605,"body":"So our original low shoe square could give us eight squares."},{"startTime":1342.185,"endTime":1347.885,"body":"We can also use this idea to find actually all of the order eight of"},{"startTime":1347.885,"endTime":1350.925,"body":"no of the order three classical magic squares."},{"startTime":1350.945,"endTime":1352.845,"body":"So these three by three ones, here's what we do."},{"startTime":1352.845,"endTime":1354.165,"body":"We say start with a blank square."},{"startTime":1355.115,"endTime":1356.525,"body":"Where's one going to go?"},{"startTime":1357.135,"endTime":1358.205,"body":"Where can we put one?"},{"startTime":1358.205,"endTime":1359.005,"body":"Well are the options."},{"startTime":1360.305,"endTime":1362.845,"body":"Now remember the magic number here is 15."},{"startTime":1363.025,"endTime":1365.485,"body":"So the sum of every row column and diagonal is 15."},{"startTime":1365.865,"endTime":1370.913,"body":"So wherever the number one goes in its row, in its column and if it's"},{"startTime":1370.913,"endTime":1371.25,"body":"on a diagonal, you've got to have two other numbers and the sum of the"},{"startTime":1371.25,"endTime":1377.645,"body":"three of them is 15."},{"startTime":1377.705,"endTime":1380.325,"body":"So the sum of the two mystery numbers is 14."},{"startTime":1380.905,"endTime":1382.645,"body":"And there's actually only two ways to do that."},{"startTime":1383.105,"endTime":1386.005,"body":"You can have nine plus five or eight plus six."},{"startTime":1386.345,"endTime":1388.485,"body":"You can't have seven plus seven, you've only got one seven."},{"startTime":1389.145,"endTime":1393.645,"body":"So there's two possibilities for one, um, to, to be able to, to go with"},{"startTime":1393.645,"endTime":1396.045,"body":"one to make a row column or diagonal."},{"startTime":1396.425,"endTime":1400.412,"body":"So one cannot go in the center because then it's in four, it's involved in"},{"startTime":1400.412,"endTime":1402.805,"body":"four sums, a row, a column, and two diagonals."},{"startTime":1403.505,"endTime":1406.94,"body":"It can't go, it's a corner 'cause then it's involved in three sums, a row,"},{"startTime":1406.94,"endTime":1408.085,"body":"a column, and a diagonal."},{"startTime":1408.585,"endTime":1410.925,"body":"So it must be in one of these off diagonal places."},{"startTime":1411.185,"endTime":1412.205,"body":"That's the only option."},{"startTime":1413.145,"endTime":1417.436,"body":"We dunno which one, but what we're gonna do is we're gonna say, well I'll"},{"startTime":1417.436,"endTime":1419.725,"body":"just rotate it until it's on the left."},{"startTime":1420.465,"endTime":1424.624,"body":"And that's, I haven't lost anything because when I'm done with this and I've got"},{"startTime":1424.624,"endTime":1424.901,"body":"all the possibilities I can, can then reproduce all the ones I I rotated by"},{"startTime":1424.901,"endTime":1430.725,"body":"just working out, you know, these orbits."},{"startTime":1431.385,"endTime":1434.473,"body":"So let, we can assume without any worrying that one is on the left hand"},{"startTime":1434.473,"endTime":1434.885,"body":"side there."},{"startTime":1435.505,"endTime":1436.485,"body":"Now where is nine?"},{"startTime":1436.675,"endTime":1438.485,"body":"Nine has the same problem as one."},{"startTime":1438.785,"endTime":1442.365,"body":"It only can fit with two different pairs of numbers to make 50."},{"startTime":1442.995,"endTime":1443.565,"body":"Here they are."},{"startTime":1443.835,"endTime":1446.525,"body":"They can go with one and five or it can go with two and four."},{"startTime":1446.865,"endTime":1448.045,"body":"That's the only possibilities."},{"startTime":1448.665,"endTime":1453.085,"body":"So again, nine cannot go in any, any square where there's a diagonal."},{"startTime":1453.085,"endTime":1456.145,"body":"It can't go in any of these three places because then it would be involved"},{"startTime":1456.145,"endTime":1457.165,"body":"in at least three sums."},{"startTime":1458.145,"endTime":1461.925,"body":"We know that nine must go with one in either its row or its column."},{"startTime":1462.345,"endTime":1464.165,"body":"So there's only one one place left."},{"startTime":1464.625,"endTime":1465.445,"body":"It must be like this."},{"startTime":1465.465,"endTime":1466.765,"body":"And then five must be in the middle."},{"startTime":1467.985,"endTime":1471.849,"body":"And the column containing one must have a six and an eight in it in"},{"startTime":1471.849,"endTime":1472.365,"body":"some order."},{"startTime":1472.745,"endTime":1476.778,"body":"But again, we don't really care which, which way it is in the particular square"},{"startTime":1476.778,"endTime":1477.047,"body":"we're thinking of because we can just reflect it and then assume that six is"},{"startTime":1477.047,"endTime":1484.845,"body":"at the top and we'll do the orbit and get everything back when we're done."},{"startTime":1485.225,"endTime":1489.87,"body":"Now this is great 'cause at this point we now know we have got actually"},{"startTime":1489.87,"endTime":1490.18,"body":"everything that allows us to complete the square because the two diagonals have two of"},{"startTime":1490.18,"endTime":1495.445,"body":"the three numbers."},{"startTime":1495.995,"endTime":1498.245,"body":"That means there's only one option for the remaining ones."},{"startTime":1498.585,"endTime":1500.765,"body":"And then we can fill in those last two squares."},{"startTime":1500.935,"endTime":1501.965,"body":"We've got our square."},{"startTime":1503.305,"endTime":1507.82,"body":"So what we've shown is that every single order, three classical magic square is in"},{"startTime":1507.82,"endTime":1509.325,"body":"the orbit of this guy."},{"startTime":1509.745,"endTime":1512.205,"body":"So we can just reconstruct all of them there."},{"startTime":1512.205,"endTime":1512.605,"body":"They all are."},{"startTime":1512.605,"endTime":1514.285,"body":"Those are the ones you can get by rotations."},{"startTime":1514.695,"endTime":1516.565,"body":"These are the ones you can get by reflections."},{"startTime":1516.955,"endTime":1517.565,"body":"That is it."},{"startTime":1517.685,"endTime":1518.605,"body":"That's the full list."},{"startTime":1519.225,"endTime":1521.285,"body":"So the low shoes square must be somewhere in there."},{"startTime":1521.485,"endTime":1522.565,"body":"I dunno if you spotted it."},{"startTime":1522.875,"endTime":1523.485,"body":"Here it is."},{"startTime":1524.265,"endTime":1524.805,"body":"That's it."},{"startTime":1524.805,"endTime":1525.365,"body":"There are eight."},{"startTime":1527.145,"endTime":1531.058,"body":"Now you could say there are somehow really there are one, there's one but it's"},{"startTime":1531.058,"endTime":1532.885,"body":"just one and then all it's symmetries."},{"startTime":1533.265,"endTime":1537.61,"body":"So if you want to say that, and that's often what mathematicians will do if"},{"startTime":1537.61,"endTime":1537.9,"body":"we're trying to work out how many of these magic squares there are of different"},{"startTime":1537.9,"endTime":1543.405,"body":"sizes by that same token."},{"startTime":1543.405,"endTime":1547.645,"body":"So if we're saying somehow there's really just one of order three, then here are"},{"startTime":1547.645,"endTime":1551.885,"body":"the numbers which grow absolutely vast of how many there are of bigger, bigger orders."},{"startTime":1552.265,"endTime":1556.602,"body":"And that order six, which is a ridiculously large number, it's got a little question"},{"startTime":1556.602,"endTime":1559.205,"body":"mark over it because it's still not absolutely confirmed."},{"startTime":1559.635,"endTime":1563.378,"body":"Even though magic squares have been studied for thousands of years, it's still not absolutely"},{"startTime":1563.378,"endTime":1565.125,"body":"confirmed that this is the exact number."},{"startTime":1565.465,"endTime":1567.885,"body":"Um, there's lots of computer checking that's being done."},{"startTime":1568.195,"endTime":1570.725,"body":"This is the latest as at September this year."},{"startTime":1571.675,"endTime":1573.965,"body":"They're pretty sure, but they're not absolutely certain."},{"startTime":1574.465,"endTime":1575.485,"body":"So vast numbers."},{"startTime":1575.585,"endTime":1579.136,"body":"And we have, I mean if you think about order nine, which is kind of"},{"startTime":1579.136,"endTime":1579.373,"body":"comparing to Sudoku, we don't even know at all how many there are of that"},{"startTime":1579.373,"endTime":1582.925,"body":"size."},{"startTime":1582.985,"endTime":1586.005,"body":"That's how huge this, this search space is."},{"startTime":1586.865,"endTime":1591.645,"body":"But this is for classical ones that involve the numbers one up to n squared."},{"startTime":1592.225,"endTime":1595.245,"body":"If you're allowed to put any numbers you like in there are infinitely many."},{"startTime":1595.585,"endTime":1599.365,"body":"And the reason is once you've got one is that perfume one from before."},{"startTime":1599.715,"endTime":1602.325,"body":"What you could do is double all the numbers in that square."},{"startTime":1602.475,"endTime":1605.565,"body":"That'll still be a magic square or treble or multiply by anything."},{"startTime":1605.865,"endTime":1607.325,"body":"And you'll still get another magic square."},{"startTime":1607.325,"endTime":1611.495,"body":"If you multiply all of the entries by the same fixed constant, the answer will"},{"startTime":1611.495,"endTime":1612.885,"body":"still be a magic square."},{"startTime":1613.625,"endTime":1618.406,"body":"And if you've got several different magic squares, you can add them together and make"},{"startTime":1618.406,"endTime":1618.725,"body":"more."},{"startTime":1619.345,"endTime":1622.285,"body":"So once you've got one, you can get infinitely many."},{"startTime":1623.825,"endTime":1627.65,"body":"So that gives you, you know, if, if you remove the classical requirement, you can"},{"startTime":1627.65,"endTime":1628.925,"body":"get infinitely manu magic squares."},{"startTime":1630.305,"endTime":1634.582,"body":"And there's, before we we're gonna move on from magic squares, but just to say"},{"startTime":1634.582,"endTime":1634.867,"body":"there's other kinds of ways you can explore, you can think about magic cubes, you"},{"startTime":1634.867,"endTime":1640.285,"body":"can think about different magic."},{"startTime":1640.425,"endTime":1645.07,"body":"So this one in the middle, it's not a magic square by our definition, but"},{"startTime":1645.07,"endTime":1645.38,"body":"in this square the product of ev of the numbers in every row column and"},{"startTime":1645.38,"endTime":1650.645,"body":"diagonal is constant."},{"startTime":1650.905,"endTime":1652.205,"body":"So that's a different kind of magic."},{"startTime":1652.865,"endTime":1656.165,"body":"And then on the right, this is called, this is an anti magic square."},{"startTime":1656.865,"endTime":1661.541,"body":"The requirement here is that the sums of the rose columns and diagonals must all"},{"startTime":1661.541,"endTime":1662.165,"body":"be different."},{"startTime":1663.425,"endTime":1666.725,"body":"So there's lots of scope for exploring, which is cool."},{"startTime":1667.915,"endTime":1672.125,"body":"Okay, let's think about another kind of number square Latin squares."},{"startTime":1672.865,"endTime":1678.121,"body":"So Latin squares, which were in the title of the lecture, they are well sudokus"},{"startTime":1678.121,"endTime":1680.925,"body":"are Latin squares, but not necessarily vice versa."},{"startTime":1681.565,"endTime":1685.015,"body":"'cause a Latin square requires you, this is an order nine one, it's a Sudoku"},{"startTime":1685.015,"endTime":1686.165,"body":"as well as it happens."},{"startTime":1686.665,"endTime":1691.202,"body":"But the numbers in this case, one to nine appear exactly once in each row"},{"startTime":1691.202,"endTime":1693.925,"body":"and in each column Sudoku have an extra requirement."},{"startTime":1693.995,"endTime":1697.045,"body":"They also need the numbers to appear once in each block."},{"startTime":1697.305,"endTime":1701.365,"body":"So every Sudoku is a Latin square, but not every Latin square is a Sudoku."},{"startTime":1702.065,"endTime":1705.085,"body":"Now Latin squares have loads of interesting applications."},{"startTime":1705.625,"endTime":1707.765,"body":"One of them is in experiment design."},{"startTime":1708.265,"endTime":1711.805,"body":"So here is a Latin square on a hillside in Wales."},{"startTime":1712.665,"endTime":1717.076,"body":"Um, what they were doing here was they were trying to look at different species"},{"startTime":1717.076,"endTime":1720.605,"body":"of tree and see how they did in in harsh weather conditions."},{"startTime":1721.225,"endTime":1725.251,"body":"Now when you're doing that, you want to kind of get rid of any extraneous"},{"startTime":1725.251,"endTime":1728.205,"body":"things that might make one tree do better than the other."},{"startTime":1728.225,"endTime":1732.704,"body":"But for kind of the wrong, not the thing you're looking for, you need a"},{"startTime":1732.704,"endTime":1736.885,"body":"very big patch of land to do an experiment involving five species of tree."},{"startTime":1737.265,"endTime":1741.779,"body":"So you can't guarantee it's impossible to guarantee that, you know, the soil conditions, the"},{"startTime":1741.779,"endTime":1742.08,"body":"humidity, the moisture content, the aspect of the slope, all of those things, you can't"},{"startTime":1742.08,"endTime":1749.605,"body":"make them completely constant across an area of land that big."},{"startTime":1750.025,"endTime":1754.55,"body":"So instead of having like, you know, plot one, plot two, plot three plot plot"},{"startTime":1754.55,"endTime":1754.852,"body":"four and plot five, what you do is to create a Latin square, a five"},{"startTime":1754.852,"endTime":1755.154,"body":"by five array, and you have a plot, um, or uh, you know, an area"},{"startTime":1755.154,"endTime":1767.525,"body":"of land for each species, one in each row and in each column."},{"startTime":1767.905,"endTime":1772.968,"body":"And that kind of, you know, separates them out and, and hopefully then has a"},{"startTime":1772.968,"endTime":1773.306,"body":"sort of sample of each part of the different bits of land so that it's"},{"startTime":1773.306,"endTime":1779.045,"body":"a fairer experiment."},{"startTime":1779.045,"endTime":1780.725,"body":"Somehow it gives you more robust data."},{"startTime":1782.345,"endTime":1784.325,"body":"Now we talked about order three Latin squares."},{"startTime":1784.325,"endTime":1787.085,"body":"You can think about order three, uh, order three magic squares."},{"startTime":1787.085,"endTime":1789.925,"body":"We can think about order three Latin squares here is one."},{"startTime":1790.425,"endTime":1793.672,"body":"So this would have, if it's numbers, you'd want the numbers one to three in"},{"startTime":1793.672,"endTime":1795.405,"body":"each row and in each column exactly once."},{"startTime":1795.945,"endTime":1797.685,"body":"Um, but it doesn't have to be numbers as we've seen."},{"startTime":1797.685,"endTime":1799.485,"body":"It could be trees or it could be colors."},{"startTime":1799.905,"endTime":1801.285,"body":"How many possible ways are there?"},{"startTime":1801.835,"endTime":1805.856,"body":"Well, if you've got your first row, and I haven't specified how I'm assigning these,"},{"startTime":1805.856,"endTime":1808.805,"body":"these, these colors to the numbers, but here's your first row."},{"startTime":1809.145,"endTime":1810.765,"body":"You can then think about the first column."},{"startTime":1810.875,"endTime":1812.485,"body":"There's only two ways to finish that."},{"startTime":1812.555,"endTime":1813.125,"body":"Here they are."},{"startTime":1813.745,"endTime":1818.805,"body":"And actually it turns out once you've got this far, the rest is determined, right?"},{"startTime":1818.805,"endTime":1822.205,"body":"There's only one Latin square that looks like each of these kinds."},{"startTime":1822.205,"endTime":1822.805,"body":"Here they are."},{"startTime":1823.985,"endTime":1827.316,"body":"So now there's two kind of basic Latin squares, but we need to know where"},{"startTime":1827.316,"endTime":1828.205,"body":"to put the numbers."},{"startTime":1828.865,"endTime":1830.845,"body":"So which number is gonna be red?"},{"startTime":1831.235,"endTime":1832.285,"body":"Well, there's three possibilities."},{"startTime":1832.305,"endTime":1832.925,"body":"One, two or three."},{"startTime":1833.315,"endTime":1837.125,"body":"Once you've chosen that, there's two possibilities for what's gonna be yellow."},{"startTime":1837.465,"endTime":1839.365,"body":"And then whatever's left must be green."},{"startTime":1839.705,"endTime":1845.437,"body":"So three times, two, six possibilities, meaning there are 12 order three Latin squares and"},{"startTime":1845.437,"endTime":1850.405,"body":"the numbers go up, but it's not as crazy as for magic squares."},{"startTime":1850.825,"endTime":1851.845,"body":"Here's some numbers."},{"startTime":1852.195,"endTime":1856.445,"body":"They're big numbers order six a mere 821 million."},{"startTime":1856.865,"endTime":1859.165,"body":"And we do know exactly how many there are of order nine."},{"startTime":1859.435,"endTime":1864.243,"body":"It's that rather large number here, 10 to the power 27, um, you know, five,"},{"startTime":1864.243,"endTime":1865.205,"body":"five times that."},{"startTime":1865.465,"endTime":1867.125,"body":"So we know how many there are of order nine."},{"startTime":1867.275,"endTime":1869.645,"body":"It's a smaller problem, not much smaller, but it's a bit smaller."},{"startTime":1870.905,"endTime":1877.734,"body":"Now why they called Latin squares to answer that, um, we need to see an"},{"startTime":1877.734,"endTime":1878.645,"body":"old puzzle."},{"startTime":1878.715,"endTime":1879.765,"body":"This is quite a cool puzzle."},{"startTime":1880.265,"endTime":1885.238,"body":"You take a standard pack of playing cards and what you do is you take"},{"startTime":1885.238,"endTime":1888.885,"body":"the top four value cards from each of the four suits."},{"startTime":1889.065,"endTime":1893.533,"body":"So the jack, queen, king, and ace from each of the suits, which are spades,"},{"startTime":1893.533,"endTime":1894.725,"body":"hearts, clubs, and diamonds."},{"startTime":1895.195,"endTime":1896.565,"body":"That gives you 16 cards."},{"startTime":1896.865,"endTime":1901.674,"body":"And your job is to arrange them in a square so that in every row"},{"startTime":1901.674,"endTime":1901.995,"body":"and in every column there is exactly one heart, um, one spade, one club, one"},{"startTime":1901.995,"endTime":1906.805,"body":"diamond."},{"startTime":1906.905,"endTime":1911.165,"body":"So one of each suit and exactly one of each card value."},{"startTime":1911.585,"endTime":1914.445,"body":"So there should be one king, for example, in each row, in each column."},{"startTime":1914.625,"endTime":1917.245,"body":"And there should be one diamond in each row, in each column."},{"startTime":1917.905,"endTime":1920.698,"body":"And implicit in this of course is, you know, you've got a standard pack of"},{"startTime":1920.698,"endTime":1920.885,"body":"cards."},{"startTime":1921.305,"endTime":1923.645,"body":"You, you should have each pair occurring exactly once."},{"startTime":1923.665,"endTime":1926.285,"body":"So like one king of hearts, one ace of spades in there."},{"startTime":1927.825,"endTime":1931.165,"body":"Now this has lots of solutions, it's a fun puzzle."},{"startTime":1931.265,"endTime":1935.787,"body":"But after a while, maybe you could do it a little bit after this, in"},{"startTime":1935.787,"endTime":1936.089,"body":"the 19th century in the court of, uh, Catherine, the great of Russia in Petersburg,"},{"startTime":1936.089,"endTime":1943.325,"body":"they were all trying to solve the 36 officers problem."},{"startTime":1943.675,"endTime":1945.925,"body":"That is exactly this, this card problem."},{"startTime":1946.065,"endTime":1953.171,"body":"But bigger in this problem, you have six army regiments and six ranks in each"},{"startTime":1953.171,"endTime":1953.645,"body":"regiment."},{"startTime":1953.645,"endTime":1957.323,"body":"So you have 36 soldiers in total, and you have to arrange them in a,"},{"startTime":1957.323,"endTime":1957.568,"body":"in a six by six square, such that in every row and in every column"},{"startTime":1957.568,"endTime":1957.813,"body":"there is one of each regimen and one of each rank, and no pair is"},{"startTime":1957.813,"endTime":1964.925,"body":"repeated."},{"startTime":1964.925,"endTime":1967.845,"body":"So you don't have like two generals from regimen one or something."},{"startTime":1968.905,"endTime":1971.925,"body":"So what we've got here, actually let's take with the cards."},{"startTime":1972.265,"endTime":1974.525,"body":"You've got two Latin squares happening at the same time."},{"startTime":1974.545,"endTime":1978.607,"body":"You've got a Latin square with the suits one, one of each suit in each"},{"startTime":1978.607,"endTime":1978.877,"body":"row in each column, and you've got a Latin square with the card values, one"},{"startTime":1978.877,"endTime":1984.565,"body":"in each row, one in each column."},{"startTime":1985.105,"endTime":1991.442,"body":"So you've got two Latin squares overlaid with this extra requirement that every pair of,"},{"startTime":1991.442,"endTime":1995.245,"body":"uh, objects occurs exactly once one ace of spades."},{"startTime":1996.145,"endTime":2000.273,"body":"So these, these, this kind of pairings, we have two Latin squares together that have"},{"startTime":2000.273,"endTime":2001.925,"body":"this special relationship with each other."},{"startTime":2002.345,"endTime":2004.765,"body":"We now call those orthogonal Latin squares."},{"startTime":2005.465,"endTime":2009.672,"body":"But when Catherine the Great couldn't do this problem and she asked a nearby mathematician"},{"startTime":2009.672,"endTime":2009.952,"body":"to help her, the guy she asked, um, who was visiting from Petersburg at the"},{"startTime":2009.952,"endTime":2017.245,"body":"time, was one of the best mathematicians of all time, Leonard Oiler."},{"startTime":2017.865,"endTime":2021.48,"body":"And when he started thinking about this problem, he realized that what you had was"},{"startTime":2021.48,"endTime":2022.685,"body":"two of these particular squares."},{"startTime":2023.105,"endTime":2026.285,"body":"And he would use, he used some notation in the first square."},{"startTime":2026.545,"endTime":2029.045,"body":"He called the things in that square, the set that was being used."},{"startTime":2029.185,"endTime":2031.645,"body":"He labeled them with Latin letters, A, B, C, D."},{"startTime":2031.865,"endTime":2035.999,"body":"And in the other squares, he labeled the things in that set, um, with Greek"},{"startTime":2035.999,"endTime":2038.205,"body":"letters, alpha, beta, gamma, delta and so on."},{"startTime":2038.465,"endTime":2041.005,"body":"So he called them Greco Latin squares."},{"startTime":2041.585,"endTime":2043.965,"body":"We now call them orthogonal Latin squares."},{"startTime":2044.145,"endTime":2047.584,"body":"And Latin squares is kind of a back formation from that because if you're just"},{"startTime":2047.584,"endTime":2050.565,"body":"dealing with one of them, we just call that a Latin square now."},{"startTime":2050.565,"endTime":2051.804,"body":"But that's where that came from."},{"startTime":2052.304,"endTime":2056.005,"body":"And he was asked, he asked himself, well actually, can we always make these things?"},{"startTime":2056.145,"endTime":2057.364,"body":"We know we can do it with the cards."},{"startTime":2057.364,"endTime":2058.284,"body":"That's order four."},{"startTime":2058.985,"endTime":2060.125,"body":"He could not do it either."},{"startTime":2060.465,"endTime":2065.067,"body":"For the 36 officers, he couldn't solve that problem and equals six version, he and"},{"startTime":2065.067,"endTime":2069.364,"body":"he realized he could do, whenever it's an old number, he could do it."},{"startTime":2070.225,"endTime":2072.364,"body":"Um, whenever it's a multiple of four, he could do it."},{"startTime":2073.315,"endTime":2076.889,"body":"It's impossible for n equals two, just like magic squares, he couldn't do it for"},{"startTime":2076.889,"endTime":2077.605,"body":"n equals six."},{"startTime":2078.344,"endTime":2084.005,"body":"So his conjecture is that you can always make these things, these orthogonal Latin squares,"},{"startTime":2084.005,"endTime":2084.383,"body":"unless your n the size of the square is even, but not a multiple of"},{"startTime":2084.383,"endTime":2090.045,"body":"four."},{"startTime":2090.065,"endTime":2091.645,"body":"So it's two more than a multiple of four."},{"startTime":2091.665,"endTime":2093.165,"body":"So you can't do it for n equals two."},{"startTime":2093.585,"endTime":2096.054,"body":"And he and he con injected, you can't do it for n equals six or"},{"startTime":2096.054,"endTime":2098.525,"body":"10 or 14 or any of these, but you can do it for everything else."},{"startTime":2099.305,"endTime":2102.736,"body":"Now oiler, as I said, one of the greatest mathematicians of ever, um, of all"},{"startTime":2102.736,"endTime":2102.965,"body":"time."},{"startTime":2103.425,"endTime":2107.165,"body":"And so when he conjectures something, you tend to believe it."},{"startTime":2107.545,"endTime":2113.361,"body":"So it was a great shock when in 19 59, 3 mathematicians found order, 10"},{"startTime":2113.361,"endTime":2114.525,"body":"oral Latin squares."},{"startTime":2114.705,"endTime":2116.725,"body":"It was on the front cover of Scientific American."},{"startTime":2116.785,"endTime":2119.445,"body":"It was all like world news exciting."},{"startTime":2120.265,"endTime":2124.778,"body":"Uh, and so this was kind of relatively recent mathematics when, um, I'm sure you"},{"startTime":2124.778,"endTime":2126.885,"body":"can already tell where this is going."},{"startTime":2126.885,"endTime":2129.245,"body":"This is the ideal way to structure your novel."},{"startTime":2129.705,"endTime":2129.925,"body":"Yes."},{"startTime":2130.585,"endTime":2137.611,"body":"So in 1978, George Pek structured his novel life, a user's manual around these order"},{"startTime":2137.611,"endTime":2139.485,"body":"10 orthogonal Latin squares."},{"startTime":2140.145,"endTime":2145.485,"body":"He set it in a Parisian apartment block, 10 by 10, a hundred rooms and"},{"startTime":2145.485,"endTime":2149.045,"body":"every chapter happened in one of the a different room."},{"startTime":2150.065,"endTime":2152.205,"body":"And he gave the chapters a unique flavor."},{"startTime":2152.345,"endTime":2156.781,"body":"And what he did was he tap particular characteristics or objects and he put them"},{"startTime":2156.781,"endTime":2157.965,"body":"in lists of 10."},{"startTime":2158.105,"endTime":2163.432,"body":"So it might be like fabrics and colors, and you use the orthogonal Latin square"},{"startTime":2163.432,"endTime":2168.405,"body":"structure to ensure that each pair from these sets occurred in exactly one chapter."},{"startTime":2168.585,"endTime":2172.725,"body":"So there's only one chapter, you know, that would have red silk in it."},{"startTime":2172.725,"endTime":2176.125,"body":"There's only one chapter that would have green velvet, something like this."},{"startTime":2176.665,"endTime":2180.005,"body":"So every chapter gets its unique flavor from this structure."},{"startTime":2181.145,"endTime":2184.965,"body":"Now the book actually that one of the main themes of the book is failure."},{"startTime":2185.195,"endTime":2187.645,"body":"Lots of the characters are trying to complete things, do things."},{"startTime":2187.645,"endTime":2190.999,"body":"One of 'em is trying to complete a lot of jigsaw puzzles of particular places,"},{"startTime":2190.999,"endTime":2192.565,"body":"and he isn't able to do it."},{"startTime":2193.025,"endTime":2198.215,"body":"Um, and so then what better structure to put under your book than this emblem"},{"startTime":2198.215,"endTime":2198.561,"body":"of the kind of, the only time, almost the oiler, this great mathematician actually failed"},{"startTime":2198.561,"endTime":2204.445,"body":"to predict something."},{"startTime":2204.785,"endTime":2207.045,"body":"So this is, you know, it's, it fits in with the book."},{"startTime":2207.675,"endTime":2209.525,"body":"There's another really cool bit of structure."},{"startTime":2210.185,"endTime":2212.925,"body":"How do you decide which room you're going to have in your next chapter?"},{"startTime":2213.635,"endTime":2216.965,"body":"Well, the stories follow a night's tour of a 10 by 10 chess board."},{"startTime":2217.545,"endTime":2219.005,"body":"But remember this is a book about failure."},{"startTime":2220.265,"endTime":2221.765,"body":"It only has 99 chapters."},{"startTime":2222.505,"endTime":2223.845,"body":"The tour is never completed."},{"startTime":2224.145,"endTime":2226.725,"body":"And I love what per he blames one of his own characters."},{"startTime":2227.145,"endTime":2232.826,"body":"He says the little girl on pages 295 and 394 is solely responsible for this"},{"startTime":2232.826,"endTime":2233.205,"body":"laps."},{"startTime":2233.205,"endTime":2234.165,"body":"So it's, it's very cool."},{"startTime":2234.745,"endTime":2238.85,"body":"Now, obviously this is the most important application of Latin squares, but occasionally there are"},{"startTime":2238.85,"endTime":2239.124,"body":"some, I mean, you know, the real world's over rated, but there there are some,"},{"startTime":2239.124,"endTime":2244.325,"body":"there are some other applications."},{"startTime":2244.325,"endTime":2247.125,"body":"I just wanna mention one and then we're gonna go back to Sudoku."},{"startTime":2248.025,"endTime":2252.514,"body":"Uh, so the one I want to mention involves a kind of extension of this"},{"startTime":2252.514,"endTime":2252.814,"body":"idea of these orthogonal Latin squares, two Latin squares that you put 'em on top"},{"startTime":2252.814,"endTime":2253.113,"body":"of each other, they're both Latin squares and every pair that you see from, from"},{"startTime":2253.113,"endTime":2265.685,"body":"the sets that are involved occurs exactly once they're one ace of spades rule."},{"startTime":2267.585,"endTime":2273.214,"body":"It turns out that if you are lucky, sometimes you can make more of these"},{"startTime":2273.214,"endTime":2273.965,"body":"orthogonal squares."},{"startTime":2274.105,"endTime":2278.956,"body":"In other words, you can have a whole set of, of Latin squares, of a"},{"startTime":2278.956,"endTime":2283.485,"body":"particular size where every pair of them has this, this property, they are orthogonal."},{"startTime":2283.865,"endTime":2287.84,"body":"And so you call a set like this a set of mutually orthogonal Latin squares,"},{"startTime":2287.84,"endTime":2289.165,"body":"which is a long phrase."},{"startTime":2290.025,"endTime":2294.809,"body":"Now, can we find these things we know, you know, sometimes you can't even find"},{"startTime":2294.809,"endTime":2296.085,"body":"two orthogonal Latin squares."},{"startTime":2296.085,"endTime":2299.085,"body":"In the case of the officer's problem, you couldn't do it, it's impossible."},{"startTime":2299.705,"endTime":2304.612,"body":"Uh, but sometimes you're, you are lucky and you are lucky when the order of"},{"startTime":2304.612,"endTime":2304.94,"body":"the square, so the size of the square, um, let's call it Q, when that"},{"startTime":2304.94,"endTime":2305.267,"body":"number Q is either a prime number or a power of a prime number, then"},{"startTime":2305.267,"endTime":2305.594,"body":"you can guarantee that you'll be able to find a set of Q minus one"},{"startTime":2305.594,"endTime":2320.645,"body":"mutually orthogonal Latin squares."},{"startTime":2321.385,"endTime":2324.125,"body":"So when qic equals three for example, it's a prime number."},{"startTime":2324.785,"endTime":2328.725,"body":"We can guarantee that you can find two orthogonal Latin squares."},{"startTime":2329.065,"endTime":2331.205,"body":"So that's one of the ones that actually oil managed to find."},{"startTime":2332.185,"endTime":2333.685,"body":"Now what use is that?"},{"startTime":2333.685,"endTime":2337.365,"body":"Well, it turns out there is an application in a rather unexpected place."},{"startTime":2337.785,"endTime":2342.433,"body":"Uh, it's when you want to send messages, they don't have to be secret, but"},{"startTime":2342.433,"endTime":2342.743,"body":"they're sent across what we call a noisy channel where, for example, maybe you are"},{"startTime":2342.743,"endTime":2343.052,"body":"like a space probe 150 million miles from Earth and you're sending a signal back"},{"startTime":2343.052,"endTime":2343.362,"body":"and you know that going across all that distance of space every so often, like"},{"startTime":2343.362,"endTime":2343.672,"body":"a passing neutrino will bump into you and maybe turn one of your ones into"},{"startTime":2343.672,"endTime":2362.885,"body":"a zero or something like this."},{"startTime":2363.145,"endTime":2366.735,"body":"So it's that kind of situation where sometimes a message can be corrupted a little"},{"startTime":2366.735,"endTime":2366.975,"body":"bit and you, and you want to still be able to understand it at the"},{"startTime":2366.975,"endTime":2370.805,"body":"other end."},{"startTime":2371.465,"endTime":2375.395,"body":"So what you don't want to do is just send, you know, say you are"},{"startTime":2375.395,"endTime":2379.325,"body":"wanting to send yes or no, and maybe yes is one and no is zero."},{"startTime":2379.585,"endTime":2384.601,"body":"If you just send that yes or no, if that's a small chance, but if"},{"startTime":2384.601,"endTime":2384.935,"body":"that message gets corrupted and it's only one letter that you're sending, when you receive"},{"startTime":2384.935,"endTime":2393.965,"body":"on the other end a message, you can't tell that something's gone wrong."},{"startTime":2394.385,"endTime":2396.525,"body":"You know, you're, you are expecting a one or a zero."},{"startTime":2396.865,"endTime":2400.405,"body":"If you receive a zero, who knows whether that was what was meant."},{"startTime":2400.785,"endTime":2405.244,"body":"So what you can do is you can add a bit of redundancy to this,"},{"startTime":2405.244,"endTime":2407.325,"body":"uh, a sort of a checking mechanism."},{"startTime":2407.665,"endTime":2411.407,"body":"So for instance, suppose you want to send yes or no, what you do is"},{"startTime":2411.407,"endTime":2412.405,"body":"to duplicate your message."},{"startTime":2412.985,"endTime":2417.335,"body":"So now you can either send one, one for yes or zero, zero for no,"},{"startTime":2417.335,"endTime":2417.625,"body":"and let's you know, we can assume that it's quite rare for there to be"},{"startTime":2417.625,"endTime":2422.845,"body":"any kind of problem."},{"startTime":2423.065,"endTime":2425.045,"body":"You know, not very many errors are introduced."},{"startTime":2425.585,"endTime":2430.367,"body":"We can assume, let's say that, that at worst there'll be one, one digit that"},{"startTime":2430.367,"endTime":2431.005,"body":"gets tweaked."},{"startTime":2432.225,"endTime":2435.637,"body":"So us on the receiving end, if we receive zero one, we know that that"},{"startTime":2435.637,"endTime":2437.685,"body":"is not a message that could have been sent."},{"startTime":2438.145,"endTime":2441.685,"body":"The, you know, we, we got a collection of messages that could be sent."},{"startTime":2441.775,"endTime":2445.682,"body":"We'll call that the code, the code words here would be one, one and n"},{"startTime":2445.682,"endTime":2446.725,"body":"naught, we receive naught."},{"startTime":2446.725,"endTime":2447.925,"body":"One, something's gone wrong."},{"startTime":2447.925,"endTime":2451.645,"body":"So we have detected the error, but we don't know what was originally said."},{"startTime":2452.275,"endTime":2457.16,"body":"However, if we send everything three times, which is bit inefficient, but if we do"},{"startTime":2457.16,"endTime":2457.485,"body":"that, so if I send either NN naught or 1 1 1 and and you"},{"startTime":2457.485,"endTime":2466.605,"body":"receive naught one, one, again, that's not a code word, so something's gone wrong."},{"startTime":2467.185,"endTime":2471.685,"body":"But if we're assuming at most one, uh, digit has been changed, then we, we"},{"startTime":2471.685,"endTime":2475.285,"body":"can say what the original message was, we can correct the error."},{"startTime":2475.745,"endTime":2478.645,"body":"So what's happening here is we've got a collection of words."},{"startTime":2478.705,"endTime":2482.376,"body":"We call it a code, and the words of the code words, so words, these"},{"startTime":2482.376,"endTime":2483.845,"body":"strings of, of digits or letters."},{"startTime":2484.665,"endTime":2488.943,"body":"In our case in this example, the code consists of the words naught, naught, naught,"},{"startTime":2488.943,"endTime":2490.085,"body":"and 1, 1 1."},{"startTime":2491.105,"endTime":2495.912,"body":"And then what we do is when we receive a message, if it's not a"},{"startTime":2495.912,"endTime":2496.233,"body":"correct code word, we know something's gone wrong and we've confined what was the original"},{"startTime":2496.233,"endTime":2503.925,"body":"message, the correct message by finding the closest code word."},{"startTime":2503.985,"endTime":2505.085,"body":"So what does that mean closest?"},{"startTime":2505.715,"endTime":2509.867,"body":"Well, there's this very simple idea of distance between words, which is the number of"},{"startTime":2509.867,"endTime":2511.805,"body":"places where they differ from each other."},{"startTime":2512.625,"endTime":2516.845,"body":"So, uh, for example, we've received 0 1 1."},{"startTime":2517.625,"endTime":2522.627,"body":"The distance between that and 0 0 0 is two because they differ in two"},{"startTime":2522.627,"endTime":2522.961,"body":"places, whereas the distance between 0, 1, 1 and 1 1 1 is one because"},{"startTime":2522.961,"endTime":2529.965,"body":"they only differ in that first position."},{"startTime":2531.105,"endTime":2534.245,"body":"So we do have a closest word and then that can be our answer."},{"startTime":2534.305,"endTime":2537.165,"body":"We think it's um, the original word was 1, 1, 1, right?"},{"startTime":2537.505,"endTime":2539.045,"body":"So that's how we correct the error."},{"startTime":2539.705,"endTime":2545.088,"body":"The reason that works, um, is because those code words that we've got, um, are"},{"startTime":2545.088,"endTime":2546.165,"body":"distanced three apart."},{"startTime":2546.165,"endTime":2548.325,"body":"They differ from each other in three places."},{"startTime":2549.225,"endTime":2552.845,"body":"So you can't be distance one from both of them."},{"startTime":2552.945,"endTime":2556.297,"body":"If you're distance one from this guy, you must be at least distance two from"},{"startTime":2556.297,"endTime":2558.085,"body":"that guy because they're three apart in total."},{"startTime":2559.145,"endTime":2562.97,"body":"So we can guarantee to be able to correct an error if our, our code"},{"startTime":2562.97,"endTime":2563.225,"body":"that we've got has this property that all of the words in it are at"},{"startTime":2563.225,"endTime":2568.325,"body":"least three apart from each other."},{"startTime":2568.515,"endTime":2569.845,"body":"They different, at least three places."},{"startTime":2571.235,"endTime":2571.525,"body":"Okay?"},{"startTime":2571.525,"endTime":2576.025,"body":"So remembering that what we'd really like, we'd like to be able to correct an"},{"startTime":2576.025,"endTime":2580.525,"body":"error but have as many messages as possible, as big a code as we can."},{"startTime":2580.725,"endTime":2583.685,"body":"'cause that gives the greatest versatility to, to our system."},{"startTime":2584.265,"endTime":2586.085,"body":"So how big can this be?"},{"startTime":2586.835,"endTime":2588.285,"body":"Well, let's have a think about this."},{"startTime":2588.665,"endTime":2592.245,"body":"Um, maybe we can make it bigger by allowing more letters, right?"},{"startTime":2592.305,"endTime":2595.765,"body":"So, um, we had three, we had zeros and ones."},{"startTime":2595.785,"endTime":2597.245,"body":"So our alphabet was rather small."},{"startTime":2597.245,"endTime":2598.565,"body":"Our alphabet just had two letters."},{"startTime":2599.205,"endTime":2600.845,"body":"I mean the English alphabet has 26 letters."},{"startTime":2600.845,"endTime":2601.685,"body":"Maybe we can use that."},{"startTime":2602.585,"endTime":2605.805,"body":"And we also, we had, we had words which only had three letters in."},{"startTime":2606.135,"endTime":2607.605,"body":"Maybe we can have slightly longer."},{"startTime":2607.825,"endTime":2612.347,"body":"So let's imagine that we've got code and it has got words in it that"},{"startTime":2612.347,"endTime":2615.965,"body":"have four letters and those letters can be drawn from an alphabet."},{"startTime":2616.345,"endTime":2618.125,"body":"And let's say the alphabet has Q letters."},{"startTime":2618.145,"endTime":2621.565,"body":"So we talked, we had two letters, zeros and ones, but we could have more"},{"startTime":2621.565,"endTime":2623.845,"body":"letters maybe that gives us more words to play with."},{"startTime":2625.185,"endTime":2628.815,"body":"So what's the best we could hope for if we want every, all of those"},{"startTime":2628.815,"endTime":2632.445,"body":"words that are in our code to be at least different, at least three places."},{"startTime":2633.315,"endTime":2637.965,"body":"Well, if you've got two words in the code, A, B, C, D and E,"},{"startTime":2637.965,"endTime":2638.275,"body":"F, G, H, it's A and B are the same two letters as ENF, that's"},{"startTime":2638.275,"endTime":2638.585,"body":"bad because then it would mean that those two words could only possibly differ in"},{"startTime":2638.585,"endTime":2649.125,"body":"two places the last two places."},{"startTime":2649.625,"endTime":2650.605,"body":"So that can't happen."},{"startTime":2651.145,"endTime":2655.005,"body":"So any two code words have got to have the first two letters can't match."},{"startTime":2655.825,"endTime":2657.965,"body":"So everyone has to have a different first two letters."},{"startTime":2658.465,"endTime":2660.405,"body":"How many different first two letters are there?"},{"startTime":2660.715,"endTime":2664.302,"body":"Well if your first two letters are A and B, there are Q choices for"},{"startTime":2664.302,"endTime":2664.542,"body":"a, it comes from an alphabet of Q letters and Q choices for B there"},{"startTime":2664.542,"endTime":2669.565,"body":"are q squared possible first two letters."},{"startTime":2670.505,"endTime":2674.817,"body":"And so that means the size of your code, you can't have more than that"},{"startTime":2674.817,"endTime":2677.405,"body":"'cause any two code words have a different beginning."},{"startTime":2678.505,"endTime":2682.645,"body":"So that's the absolute upper maximum that you can hope to achieve, right?"},{"startTime":2682.825,"endTime":2683.445,"body":"That's good."},{"startTime":2684.265,"endTime":2685.405,"body":"Um, if we can get there."},{"startTime":2685.985,"endTime":2687.525,"body":"Now here's Latin squares coming back."},{"startTime":2688.275,"endTime":2694.096,"body":"This is possible precisely when there is a pair of orthogonal Latin squares of order"},{"startTime":2694.096,"endTime":2694.485,"body":"Q."},{"startTime":2694.865,"endTime":2699.085,"body":"So squares of side Q, that is, that's super cool."},{"startTime":2699.115,"endTime":2700.485,"body":"What are Latin squares doing there?"},{"startTime":2700.845,"endTime":2701.885,"body":"\u003claugh\u003e, that seems really weird."},{"startTime":2702.115,"endTime":2703.445,"body":"I'll show you, I'll just do a little example."},{"startTime":2703.445,"endTime":2705.205,"body":"So that's the same theory I've written up at the top."},{"startTime":2705.745,"endTime":2709.405,"body":"I'm gonna give you just an example here, here I've got Q equals three."},{"startTime":2709.465,"endTime":2712.285,"body":"So I'm making three order three Latin squares."},{"startTime":2712.575,"endTime":2714.405,"body":"Those two are orthogonal to each other."},{"startTime":2715.665,"endTime":2719.765,"body":"Uh, and here's how we make the code for each row and column."},{"startTime":2720.545,"endTime":2722.485,"body":"Um, we get uh, a code word."},{"startTime":2722.705,"endTime":2726.826,"body":"So in the second row, first column, this is the code where we're gonna make"},{"startTime":2726.826,"endTime":2727.925,"body":"2 1, 2, 3."},{"startTime":2727.985,"endTime":2729.205,"body":"How do we get the last two bits?"},{"startTime":2729.205,"endTime":2733.262,"body":"So two for the second row, one for the first column, and then what's the"},{"startTime":2733.262,"endTime":2734.885,"body":"entry in the left hand square?"},{"startTime":2734.955,"endTime":2735.445,"body":"It's two."},{"startTime":2735.745,"endTime":2737.165,"body":"What's the entry in the right hand square?"},{"startTime":2737.165,"endTime":2737.565,"body":"It's three."},{"startTime":2738.225,"endTime":2740.445,"body":"So we can do that and we can make nine words in that way."},{"startTime":2741.225,"endTime":2743.765,"body":"And it turns out this is one of these codes."},{"startTime":2744.185,"endTime":2748.948,"body":"And the reason is that if you, any one of these words can be completely"},{"startTime":2748.948,"endTime":2749.265,"body":"determined just by knowing any two of its letters, that means no two of the"},{"startTime":2749.265,"endTime":2757.205,"body":"words can have the same two letters in the same places."},{"startTime":2757.865,"endTime":2762.175,"body":"For example, if you are given the last two letters, one of these words, what"},{"startTime":2762.175,"endTime":2763.325,"body":"does that tell you?"},{"startTime":2763.475,"endTime":2767.285,"body":"That tells you the entries in these squares, right?"},{"startTime":2767.285,"endTime":2768.565,"body":"But these are orthogonal."},{"startTime":2768.705,"endTime":2771.845,"body":"So every pair appears exactly once in that structure."},{"startTime":2772.265,"endTime":2775.715,"body":"And so we can say exactly which row and column it appears in by just"},{"startTime":2775.715,"endTime":2776.405,"body":"finding it that."},{"startTime":2776.405,"endTime":2778.405,"body":"So that's one you can check the other possibilities."},{"startTime":2779.025,"endTime":2783.98,"body":"So these, this code and all others you can produce like it do indeed differ"},{"startTime":2783.98,"endTime":2788.605,"body":"all the words differently, at least three letters so they can correct one error."},{"startTime":2788.905,"endTime":2791.885,"body":"So, and this is, we can get this from these, from these Latin squares."},{"startTime":2791.885,"endTime":2792.405,"body":"It's fantastic."},{"startTime":2792.895,"endTime":2794.405,"body":"There is a more general result."},{"startTime":2794.885,"endTime":2796.285,"body":"I will put it there for 10 seconds."},{"startTime":2796.865,"endTime":2801.965,"body":"You can actually do this if you have more mutually orthogonal squares, you can correct"},{"startTime":2801.965,"endTime":2802.645,"body":"more errors."},{"startTime":2803.305,"endTime":2804.485,"body":"So that's really, really cool."},{"startTime":2806.115,"endTime":2808.005,"body":"Okay, well let's get back to Sudoku."},{"startTime":2808.865,"endTime":2810.685,"body":"Um, in, in the last five minutes."},{"startTime":2811.065,"endTime":2816.213,"body":"So we've seen some Latin squares, not all of those vast number of Latin squares"},{"startTime":2816.213,"endTime":2819.645,"body":"of border nine of course are going to be Sudoku."},{"startTime":2820.185,"endTime":2824.565,"body":"Um, because we have an extra requirement for Sudoku, we need this block rule."},{"startTime":2825.385,"endTime":2830.885,"body":"So actually it is known how many or how many Sudoku grids there are."},{"startTime":2831.345,"endTime":2836.179,"body":"Um, I want to show you a really clever way of finding an estimate of"},{"startTime":2836.179,"endTime":2836.501,"body":"that number that I think is kind of nice and intuitive and it somehow, it"},{"startTime":2836.501,"endTime":2845.525,"body":"gives me more \u003claugh\u003e more information in a way than the exact number does."},{"startTime":2845.665,"endTime":2847.645,"body":"So I want to just show you what this idea is."},{"startTime":2848.065,"endTime":2849.925,"body":"So Sudoku have to have three properties."},{"startTime":2849.925,"endTime":2851.205,"body":"They have to be Latin squares."},{"startTime":2851.385,"endTime":2855.3,"body":"So the numbers appear once in each row and column, but they also are what"},{"startTime":2855.3,"endTime":2856.605,"body":"I'm gonna call block grids."},{"startTime":2856.995,"endTime":2859.565,"body":"That the numbers appear exactly once in each block."},{"startTime":2860.225,"endTime":2864.492,"body":"So what we're gonna try and do is first to establish how many block grids"},{"startTime":2864.492,"endTime":2864.777,"body":"there are, how many, so then we're not, they're not yet required to be Latin"},{"startTime":2864.777,"endTime":2869.045,"body":"squares."},{"startTime":2869.425,"endTime":2872.125,"body":"How many of these grids are there that satisfy this block rule?"},{"startTime":2872.625,"endTime":2874.845,"body":"So we'll do a mini Sudoku example first."},{"startTime":2875.865,"endTime":2881.383,"body":"Um, so in this this size of Sudoku grid, you have the numbers one to"},{"startTime":2881.383,"endTime":2881.751,"body":"four and we are just interested so far in, in making the ones that satisfy"},{"startTime":2881.751,"endTime":2888.005,"body":"the block requirement."},{"startTime":2888.265,"endTime":2891.327,"body":"So in other words, in each of these little blocks, you've gotta have the numbers"},{"startTime":2891.327,"endTime":2893.165,"body":"1, 2, 3, and four somewhere in some order."},{"startTime":2893.505,"endTime":2894.845,"body":"So how many ways there to do it?"},{"startTime":2895.155,"endTime":2898.463,"body":"Well, let's just focus on this top left hand block and we've gotta put the"},{"startTime":2898.463,"endTime":2899.125,"body":"number one somewhere."},{"startTime":2899.125,"endTime":2900.485,"body":"There's four places we could put it."},{"startTime":2900.785,"endTime":2904.425,"body":"And then there's three places left where the next number could go and then two"},{"startTime":2904.425,"endTime":2906.125,"body":"places where the next number could go."},{"startTime":2906.265,"endTime":2909.325,"body":"And then there's only one remaining place and the final number goes in there."},{"startTime":2909.985,"endTime":2913.205,"body":"So there's four choices times three times two times one."},{"startTime":2913.545,"endTime":2915.365,"body":"We call this number four factorial."},{"startTime":2915.515,"endTime":2916.045,"body":"It's the product."},{"startTime":2916.105,"endTime":2918.845,"body":"So n factorial is the product of the numbers from one up to N."},{"startTime":2919.025,"endTime":2923.983,"body":"So four factorial, that's the exclamation mark that's pronounced factorial, uh, which of course we"},{"startTime":2923.983,"endTime":2924.645,"body":"can calculate."},{"startTime":2924.645,"endTime":2925.325,"body":"It's 24."},{"startTime":2925.785,"endTime":2927.565,"body":"So there's 24 ways to make a block."},{"startTime":2928.015,"endTime":2929.005,"body":"We've got four blocks."},{"startTime":2929.365,"endTime":2931.285,"body":"Remember we don't care about the Rowan column rule yet."},{"startTime":2931.865,"endTime":2936.761,"body":"So is 24 times, 24 times 24 times 24, which I'm sure you've just worked"},{"startTime":2936.761,"endTime":2941.005,"body":"out in your head is about 331,000 ways to make a block grid."},{"startTime":2941.665,"endTime":2942.845,"body":"Now we need more than that."},{"startTime":2943.715,"endTime":2947.957,"body":"What we next want to work out is actually how many of these block grids"},{"startTime":2947.957,"endTime":2948.24,"body":"satisfy also the row rule that the numbers one to four appear exactly once in"},{"startTime":2948.24,"endTime":2952.765,"body":"each row."},{"startTime":2953.705,"endTime":2955.005,"body":"So let's think about that."},{"startTime":2955.775,"endTime":2957.165,"body":"Let's think about these top two rows."},{"startTime":2957.515,"endTime":2962.076,"body":"Once you filled in the left hand block, which of course there are 24 ways"},{"startTime":2962.076,"endTime":2965.725,"body":"to do, then there's only two ways to complete the top row."},{"startTime":2965.825,"endTime":2967.445,"body":"It could be three, four or four three."},{"startTime":2967.715,"endTime":2969.845,"body":"There's only two ways to complete the second row."},{"startTime":2970.345,"endTime":2971.365,"body":"One, two or two one."},{"startTime":2972.025,"endTime":2975.325,"body":"So four ways to complete these first two rows."},{"startTime":2975.945,"endTime":2979.765,"body":"Two times two, um, once you've chosen your block."},{"startTime":2979.765,"endTime":2983.69,"body":"So 24 times, four ways to complete the first two rows and then you can"},{"startTime":2983.69,"endTime":2986.045,"body":"do the second two rows in the same way."},{"startTime":2986.785,"endTime":2989.822,"body":"So if you do that, don't worry about the detail, but I just want you"},{"startTime":2989.822,"endTime":2991.645,"body":"to see the, the idea of this, this argument."},{"startTime":2992.105,"endTime":2997.845,"body":"You get some number 9,216 block grids that also satisfy the row rule."},{"startTime":2998.465,"endTime":3002.548,"body":"Now what we then do is we say actually that is a particular fraction of"},{"startTime":3002.548,"endTime":3003.365,"body":"the total number."},{"startTime":3004.315,"endTime":3007.845,"body":"It's oh, it's 1 36 of all of the block grids."},{"startTime":3008.145,"endTime":3009.725,"body":"Um, satisfy this row rule."},{"startTime":3010.025,"endTime":3014.44,"body":"And you can do the same exact calculation but just uh, tilted by 90 degrees"},{"startTime":3014.44,"endTime":3014.735,"body":"and say that by the same reasoning, 1 36 of the block grids satisfy the"},{"startTime":3014.735,"endTime":3019.445,"body":"column rule."},{"startTime":3019.795,"endTime":3023.912,"body":"What we want to know is what proportion of the grids satisfy both the row"},{"startTime":3023.912,"endTime":3025.285,"body":"rule and the column rule?"},{"startTime":3025.285,"endTime":3026.405,"body":"Those are our Sudoku."},{"startTime":3026.835,"endTime":3031.22,"body":"Okay, so let's move back up to the standard size with exactly the same kind"},{"startTime":3031.22,"endTime":3031.805,"body":"of ideas."},{"startTime":3032.665,"endTime":3034.885,"body":"The number of order nine block grids."},{"startTime":3034.885,"endTime":3038.626,"body":"So we don't care about rows and columns yet, but those standard Sudoku had these"},{"startTime":3038.626,"endTime":3039.125,"body":"nine blocks."},{"startTime":3039.475,"endTime":3041.645,"body":"Each block contains the numbers one to nine in some order."},{"startTime":3042.305,"endTime":3046.335,"body":"The number of those is, this looks quite harmless, but nine factorial raised to the"},{"startTime":3046.335,"endTime":3048.485,"body":"power nine is a very, very large number."},{"startTime":3049.005,"endTime":3052.675,"body":"\u003claugh\u003e, we don't need to calculate it, but that's how many order nine block grids"},{"startTime":3052.675,"endTime":3053.165,"body":"there are."},{"startTime":3053.825,"endTime":3056.165,"body":"And we can do a calculation, it's a bit more involved."},{"startTime":3056.395,"endTime":3061.455,"body":"That tells us how many of those block grids satisfy the row rule and it's"},{"startTime":3061.455,"endTime":3062.805,"body":"one 10th of them."},{"startTime":3063.545,"endTime":3065.445,"body":"You'll notice I haven't told you what n is."},{"startTime":3065.725,"endTime":3069.559,"body":"I I will put it in the transcript, but it's a very large number, which"},{"startTime":3069.559,"endTime":3071.605,"body":"is known exactly it's something like 128 trillion."},{"startTime":3072.465,"endTime":3075.948,"body":"We, we don't need to write it down explicitly right now, but we can work"},{"startTime":3075.948,"endTime":3076.645,"body":"out this fraction."},{"startTime":3077.105,"endTime":3078.805,"body":"One 10th of them satisfy the row rule."},{"startTime":3079.105,"endTime":3082.565,"body":"And so also one 10th of them satisfy the column rule."},{"startTime":3083.105,"endTime":3087.661,"body":"Now here's the assumption that we're gonna make, which is an incorrect assumption, but it's"},{"startTime":3087.661,"endTime":3087.965,"body":"useful."},{"startTime":3088.405,"endTime":3093.738,"body":"\u003claugh\u003e, uh, we are gonna assume that satisfying the row rule is independent of satisfying"},{"startTime":3093.738,"endTime":3094.805,"body":"the column rule."},{"startTime":3095.265,"endTime":3099.826,"body":"The reason that's a helpful assumption, even though it's not actually quite correct, it's nearly"},{"startTime":3099.826,"endTime":3100.13,"body":"correct, we're not quite, is that if you've got two independent events, then that you"},{"startTime":3100.13,"endTime":3108.645,"body":"can multiply their probabilities to find out the probability of both of them happening."},{"startTime":3108.675,"endTime":3109.205,"body":"Same time."},{"startTime":3109.905,"endTime":3114.123,"body":"For example, if you have a pack of cards, one 13th of those cards are"},{"startTime":3114.123,"endTime":3114.405,"body":"ACEs."},{"startTime":3114.905,"endTime":3117.365,"body":"One quarter of those cards are hearts."},{"startTime":3118.225,"endTime":3124.0,"body":"So because those independent, we can say that one 13th times one quarter, which is"},{"startTime":3124.0,"endTime":3125.925,"body":"one 52nd, 52, 1 50."},{"startTime":3125.925,"endTime":3129.958,"body":"Second of the cards are ACEs of hearts, which is, that's what we find the"},{"startTime":3129.958,"endTime":3130.765,"body":"pack of cards."},{"startTime":3131.665,"endTime":3137.345,"body":"So if being satisfying the row rule is independent from satisfying the column rule, then"},{"startTime":3137.345,"endTime":3137.724,"body":"we could say that one of, one of all of the block grids are pseudo"},{"startTime":3137.724,"endTime":3143.405,"body":"kus."},{"startTime":3144.385,"endTime":3148.695,"body":"And if you work out that number, it's about this big number, 6.65, seven times"},{"startTime":3148.695,"endTime":3149.845,"body":"10 to the 21."},{"startTime":3149.985,"endTime":3150.405,"body":"Big number."},{"startTime":3150.985,"endTime":3155.401,"body":"What's really brilliant is I've said it's, this isn't, they're not quite independent, but they're"},{"startTime":3155.401,"endTime":3156.285,"body":"really nearly independent."},{"startTime":3156.505,"endTime":3157.925,"body":"You can work out the exact answer."},{"startTime":3157.945,"endTime":3161.485,"body":"It requires a lot of fiddly calculation and lots of computer time."},{"startTime":3161.985,"endTime":3166.795,"body":"And I'm sure you were almost there in your heads, but it's this, but if"},{"startTime":3166.795,"endTime":3171.605,"body":"you look that's really very, very close to this estimate, it's a really great estimate."},{"startTime":3171.605,"endTime":3173.765,"body":"I think it's, it's a useful calculation to do."},{"startTime":3173.835,"endTime":3175.765,"body":"It's only not 0.2 of a percent out."},{"startTime":3176.305,"endTime":3180.045,"body":"So that's how many we did mention one other question."},{"startTime":3180.705,"endTime":3184.264,"body":"We know we've got all these grids now, but of course to make puzzles you"},{"startTime":3184.264,"endTime":3185.925,"body":"need to remove some of the numbers."},{"startTime":3186.305,"endTime":3189.525,"body":"How many clues is enough to guarantee that you're gonna have a unique answer?"},{"startTime":3190.065,"endTime":3194.045,"body":"So this is counterintuitive again, um, look at this grid."},{"startTime":3194.825,"endTime":3197.125,"body":"It has 77 of the numbers filled in."},{"startTime":3197.665,"endTime":3201.685,"body":"Surely, surely we can complete this, but we can't."},{"startTime":3202.545,"endTime":3204.805,"body":"The and the top row on the second row, sorry there."},{"startTime":3205.105,"endTime":3208.285,"body":"Um, the missing numbers are one and two and on the, what's that?"},{"startTime":3208.285,"endTime":3211.713,"body":"The seventh row, the missing numbers are also one and two, there is no way"},{"startTime":3211.713,"endTime":3213.085,"body":"to know how they should go."},{"startTime":3213.475,"endTime":3216.316,"body":"It's either one, two on the top and two one on the bottom or the"},{"startTime":3216.316,"endTime":3216.885,"body":"other way round."},{"startTime":3217.385,"endTime":3219.325,"body":"We do not have enough information to write this down."},{"startTime":3219.745,"endTime":3223.956,"body":"So unless you are careful, you, you know, you risk getting into these situations where"},{"startTime":3223.956,"endTime":3227.325,"body":"even if you have lots of numbers, you can't complete the grid."},{"startTime":3227.325,"endTime":3228.685,"body":"So this is not a trivial problem."},{"startTime":3228.745,"endTime":3232.645,"body":"You can't just remove 30 of the numbers and think it's all gonna be fine."},{"startTime":3232.935,"endTime":3236.26,"body":"There has to be some, you have to be really careful that you still have"},{"startTime":3236.26,"endTime":3236.925,"body":"a unique solution."},{"startTime":3238.345,"endTime":3243.871,"body":"Um, and so we talk about minimal grids, which is where, um, you've got some"},{"startTime":3243.871,"endTime":3247.925,"body":"of the entries filled in and it's got a unique solution."},{"startTime":3248.385,"endTime":3253.237,"body":"But if you remove any one of those entries, then what you have then has"},{"startTime":3253.237,"endTime":3253.885,"body":"multiple solutions."},{"startTime":3254.345,"endTime":3257.904,"body":"So those minimal grids are kind of the best case scenario in terms of you"},{"startTime":3257.904,"endTime":3259.565,"body":"can complete this with a unique solution."},{"startTime":3260.105,"endTime":3263.645,"body":"Um, but anything less does not give you uniqueness."},{"startTime":3263.905,"endTime":3266.188,"body":"And then, you know, if you want to make an easier puzzle, you can add"},{"startTime":3266.188,"endTime":3266.645,"body":"a few things."},{"startTime":3267.145,"endTime":3269.005,"body":"You know, you've got a unique solution by that time."},{"startTime":3269.465,"endTime":3271.205,"body":"Um, but, but you can't remove anything."},{"startTime":3272.065,"endTime":3273.765,"body":"Now how many do you need, do you think?"},{"startTime":3274.395,"endTime":3276.525,"body":"Well, it turns out there's a lot of activity on this."},{"startTime":3277.265,"endTime":3280.205,"body":"It took about 20 years to resolve, I guess."},{"startTime":3280.865,"endTime":3285.56,"body":"Um, in about 10 years ago it was found you have to have 17 entries,"},{"startTime":3285.56,"endTime":3287.125,"body":"at least not any 17."},{"startTime":3287.255,"endTime":3291.305,"body":"There are lots of seven ways to put 17 numbers in that don't give you"},{"startTime":3291.305,"endTime":3295.085,"body":"an unique solution, but 17, easy enough, as long as you choose them carefully."},{"startTime":3295.305,"endTime":3297.365,"body":"If you have 16, it's never enough."},{"startTime":3298.155,"endTime":3299.765,"body":"16 is not going to be enough."},{"startTime":3300.205,"endTime":3304.015,"body":"Wherever you put them, however cleverly you place them, you will, you will not get"},{"startTime":3304.015,"endTime":3305.285,"body":"a unique solution with 16."},{"startTime":3305.425,"endTime":3309.182,"body":"But with 17, yes, but what's not known, and if you're going to become mathematicians,"},{"startTime":3309.182,"endTime":3310.685,"body":"this is still an open problem."},{"startTime":3311.425,"endTime":3313.245,"body":"Uh, how many of these minimal grids are there?"},{"startTime":3313.245,"endTime":3314.485,"body":"What are these minimal grids?"},{"startTime":3314.985,"endTime":3318.685,"body":"And this is an ongoing, uh, area of active investigation."},{"startTime":3319.425,"endTime":3323.61,"body":"And so just to finish, I'm gonna mention in one minute some variations of Sudoku"},{"startTime":3323.61,"endTime":3325.285,"body":"like we saw with magic squares."},{"startTime":3325.865,"endTime":3332.393,"body":"Um, many variations for you to explore bigger doku, go as big as you like"},{"startTime":3332.393,"endTime":3335.005,"body":"cube doku or hyper cube doku."},{"startTime":3335.225,"endTime":3336.845,"body":"Go as high as dimensional as you like."},{"startTime":3337.365,"endTime":3337.965,"body":"I like this one."},{"startTime":3338.065,"endTime":3340.725,"body":"Ven doku, uh, these are all Sudoku grids."},{"startTime":3340.805,"endTime":3341.645,"body":"I haven't put any number in."},{"startTime":3341.645,"endTime":3345.802,"body":"You'll notice because I \u003claugh\u003e, but you can, you could imagine a puzzle where you"},{"startTime":3345.802,"endTime":3349.405,"body":"had three P Sudoku grids satisfying the Sudoku rules, but there's an overlap."},{"startTime":3349.405,"endTime":3353.071,"body":"So that gives you more information about the places that are in two or three"},{"startTime":3353.071,"endTime":3353.805,"body":"of these grids."},{"startTime":3353.955,"endTime":3356.285,"body":"That means you can get away with fewer clues at the beginning."},{"startTime":3356.345,"endTime":3357.605,"body":"So that's kind of an interesting question."},{"startTime":3357.945,"endTime":3359.205,"body":"How few clues do you need?"},{"startTime":3359.905,"endTime":3362.365,"body":"Um, magic doku, it's also a magic square."},{"startTime":3362.745,"endTime":3364.885,"body":"Got the low shoe, magic square in the middle as well."},{"startTime":3364.885,"endTime":3366.885,"body":"Just added, added little thing."},{"startTime":3367.065,"endTime":3371.205,"body":"And this last one, now I don't know if I've invented, well, I have invented"},{"startTime":3371.205,"endTime":3373.965,"body":"it, but I dunno if anyone else has invented it."},{"startTime":3374.205,"endTime":3377.005,"body":"I was trying to think what's like that anti magic square."},{"startTime":3377.465,"endTime":3380.965,"body":"So here I present the su don't coup \u003claugh\u003e."},{"startTime":3381.465,"endTime":3385.205,"body":"The rule here is every rule has to break exactly once."},{"startTime":3385.425,"endTime":3392.289,"body":"So every row, column and block must contain exactly three of the numbers, but not"},{"startTime":3392.289,"endTime":3393.205,"body":"all four."},{"startTime":3393.545,"endTime":3396.285,"body":"So the top row, it's got 1, 2, 3, it hasn't got a four."},{"startTime":3396.905,"endTime":3398.805,"body":"So this missing square, there's empty square."},{"startTime":3399.635,"endTime":3403.125,"body":"What could go in that square in order to maintain this rule?"},{"startTime":3403.475,"endTime":3406.413,"body":"Well, in that block that it's in, you've got one, two, and three, you haven't"},{"startTime":3406.413,"endTime":3406.805,"body":"got four."},{"startTime":3406.825,"endTime":3410.005,"body":"So if you put four in there, you'd have a correct block not allowed."},{"startTime":3410.185,"endTime":3411.045,"body":"So it can't be four."},{"startTime":3411.715,"endTime":3412.925,"body":"Look at the row it's in."},{"startTime":3413.345,"endTime":3415.165,"body":"Um, you can't put one in there."},{"startTime":3415.165,"endTime":3416.325,"body":"Otherwise that row would be right."},{"startTime":3417.145,"endTime":3419.365,"body":"And the column, you can't put three in there."},{"startTime":3419.685,"endTime":3421.245,"body":"Otherwise that column would be right."},{"startTime":3421.745,"endTime":3425.392,"body":"So for a genuine, so don't coup, the only number that can go in that"},{"startTime":3425.392,"endTime":3426.365,"body":"missing space is two."},{"startTime":3427.265,"endTime":3431.485,"body":"Now I have not tried to make an order nine, so don't coup."},{"startTime":3432.265,"endTime":3437.085,"body":"If you've got a, a free hour, uh, I invite you to do that."},{"startTime":3437.285,"endTime":3439.525,"body":"I don't even know if it's possible, but there we are."},{"startTime":3439.525,"endTime":3441.085,"body":"So that's plenty from me."},{"startTime":3441.455,"endTime":3442.445,"body":"We'll come back next time."},{"startTime":3442.505,"endTime":3447.321,"body":"I'm gonna be talking about, uh, mathematical puzzles and paradoxes, including the one that I've"},{"startTime":3447.321,"endTime":3448.285,"body":"hinted at there."},{"startTime":3448.625,"endTime":3449.965,"body":"But we'll stop there for today."},{"startTime":3449.965,"endTime":3450.525,"body":"Thank you very much,"},{"startTime":3458.215,"endTime":3458.565,"body":"Sarah."},{"startTime":3458.615,"endTime":3459.565,"body":"Thank you very much."},{"startTime":3459.785,"endTime":3463.342,"body":"And uh, just at the beginning before we were starting, we were talking about the"},{"startTime":3463.342,"endTime":3463.579,"body":"puzzles in the paper and Sarah saying, well, I work so much on this that"},{"startTime":3463.579,"endTime":3468.085,"body":"I actually do the crossword."},{"startTime":3468.745,"endTime":3472.098,"body":"And I was thinking, well, maybe after this I'll have a go at the pseudo"},{"startTime":3472.098,"endTime":3475.005,"body":"ku, but after this I think I'll be sticking to the word search."},{"startTime":3476.465,"endTime":3478.205,"body":"Um, that was, that was brilliant."},{"startTime":3478.205,"endTime":3479.045,"body":"Thank you so much."},{"startTime":3479.305,"endTime":3481.005,"body":"I'm sure you guys have some questions."},{"startTime":3481.265,"endTime":3484.245,"body":"I'm gonna start you off just with, um, a little one from here."},{"startTime":3484.625,"endTime":3486.445,"body":"Uh, so lifer uses manual."},{"startTime":3486.555,"endTime":3487.725,"body":"This is an interesting idea."},{"startTime":3487.865,"endTime":3492.497,"body":"So, um, taking that idea, are there ways that you can actually, I mean these"},{"startTime":3492.497,"endTime":3492.806,"body":"are, this is incredible the amount of mathematics that goes into preparing the game, let"},{"startTime":3492.806,"endTime":3498.365,"body":"alone doing the game."},{"startTime":3498.365,"endTime":3498.525,"body":"Yes."},{"startTime":3498.525,"endTime":3502.851,"body":"But are there sort of applications that you can use practically for these sorts of,"},{"startTime":3502.851,"endTime":3504.005,"body":"this sort of logic?"},{"startTime":3504.145,"endTime":3507.845,"body":"And are there any sort of careers that come that could come out of it?"},{"startTime":3508.305,"endTime":3512.936,"body":"So I mean, yes, being able to be logical, \u003claugh\u003e is a good skill to"},{"startTime":3512.936,"endTime":3513.245,"body":"have."},{"startTime":3513.365,"endTime":3517.278,"body":"I mean, so we talked about there are some applications directly of, of these Latin"},{"startTime":3517.278,"endTime":3519.365,"body":"squares and other kinds of structures like this."},{"startTime":3519.825,"endTime":3524.213,"body":"Um, but the process of thinking things through and being able to say we reason"},{"startTime":3524.213,"endTime":3527.725,"body":"our way through a problem, that is really great skill to have."},{"startTime":3527.725,"endTime":3531.845,"body":"It can, you know, I mean, any job that's a useful skill to have."},{"startTime":3532.145,"endTime":3536.256,"body":"But for example, I mean of course you can become a mathematician, but you could"},{"startTime":3536.256,"endTime":3536.805,"body":"computer programmers."},{"startTime":3537.145,"endTime":3539.245,"body":"Mm-hmm, \u003caffirmative\u003e, this is exactly what you do."},{"startTime":3539.245,"endTime":3542.47,"body":"You have to cover every, if you're writing a program, you have to cover every"},{"startTime":3542.47,"endTime":3542.685,"body":"eventuality."},{"startTime":3542.745,"endTime":3544.765,"body":"You have to make sure you haven't missed out any possibility."},{"startTime":3544.865,"endTime":3548.625,"body":"You've got to follow your way through a kind of branching narrative of, of the"},{"startTime":3548.625,"endTime":3551.885,"body":"algorithm and make sure that it does what you want it to do."},{"startTime":3552.185,"endTime":3555.94,"body":"So this is for me, I mean, writing a program is very, like constructing a"},{"startTime":3555.94,"endTime":3559.445,"body":"mathematical proof and these same kinds of ideas of how is this fitting together?"},{"startTime":3559.685,"endTime":3563.405,"body":"I have to know how it all works are just absolutely relevant there."},{"startTime":3563.465,"endTime":3564.005,"body":"So yeah."},{"startTime":3565.025,"endTime":3567.805,"body":"Um, when, when you were talking about Greek Latin squares Yeah."},{"startTime":3567.805,"endTime":3571.245,"body":"You, you mentioned the, um, was it four m plus two?"},{"startTime":3571.265,"endTime":3572.085,"body":"Um, the Yes."},{"startTime":3572.345,"endTime":3574.365,"body":"Um, 6, 10, 14 sequence."},{"startTime":3574.365,"endTime":3574.605,"body":"Yeah."},{"startTime":3574.825,"endTime":3576.565,"body":"And that the ten one had been solved."},{"startTime":3576.785,"endTime":3577.005,"body":"Yes."},{"startTime":3577.365,"endTime":3580.445,"body":"I think Ronald Fisher proved that the six one couldn't be."},{"startTime":3580.825,"endTime":3581.045,"body":"Yes."},{"startTime":3581.305,"endTime":3584.605,"body":"Um, has any progress been made on 14 18, 2, 2,"},{"startTime":3584.745,"endTime":3585.085,"body":"Et cetera?"},{"startTime":3585.085,"endTime":3585.245,"body":"Yes."},{"startTime":3585.245,"endTime":3588.485,"body":"So in fact, it's kind of the worst possible outcome for oiler because, so n"},{"startTime":3588.485,"endTime":3589.565,"body":"equals two can't be done."},{"startTime":3589.865,"endTime":3593.061,"body":"Uh, n equals six, actually it was a, a guy called Gaston Terry who proved"},{"startTime":3593.061,"endTime":3595.405,"body":"that in the right at the beginning of the 20th century."},{"startTime":3595.425,"endTime":3600.111,"body":"He proved it was impossible and it was aptly named a proof by exhaustion because"},{"startTime":3600.111,"endTime":3604.485,"body":"he had to check not absolutely every possibility, but he managed to reduce it."},{"startTime":3604.485,"endTime":3607.125,"body":"He said everything that we could try is equivalent to one of this."},{"startTime":3607.255,"endTime":3609.205,"body":"Still very, very long list of things to check."},{"startTime":3609.265,"endTime":3612.565,"body":"And he did do that checking by hand in like 1905."},{"startTime":3613.465,"endTime":3617.125,"body":"Um, yes, 1959, the order 10 ones were found."},{"startTime":3617.705,"endTime":3621.73,"body":"And that kind of opened the flood gates because then suddenly there was this, oh,"},{"startTime":3621.73,"endTime":3621.998,"body":"maybe people had sort of assumed there weren't gonna be these others, but actually it's"},{"startTime":3621.998,"endTime":3629.245,"body":"been shown since then that you can do it for every single order."},{"startTime":3629.265,"endTime":3631.205,"body":"So 14, 18, all of those, they exist."},{"startTime":3631.205,"endTime":3633.005,"body":"These, these orthogonal Latin squares."},{"startTime":3633.025,"endTime":3637.21,"body":"The only two numbers where it fails are the ones the oiler \u003claugh\u003e had had"},{"startTime":3637.21,"endTime":3638.885,"body":"direct experience of two and six."},{"startTime":3639.425,"endTime":3643.437,"body":"And so, you know, unfortunately yes, his dataset was not big enough, uh, to, to"},{"startTime":3643.437,"endTime":3645.845,"body":"see that what, that the pattern did not continue."},{"startTime":3645.905,"endTime":3651.421,"body":"So you can make order, um, n orthogonal Latin squares for every N except for"},{"startTime":3651.421,"endTime":3652.525,"body":"two and six."},{"startTime":3653.385,"endTime":3653.925,"body":"If you"},{"startTime":3654.745,"endTime":3660.634,"body":"Set your terms and you set your constraints, is it theoretically possible to solve any"},{"startTime":3660.634,"endTime":3662.205,"body":"puzzle in the world?"},{"startTime":3663.165,"endTime":3663.925,"body":"Hmm, \u003claugh\u003e."},{"startTime":3664.275,"endTime":3668.939,"body":"Okay, well if the theoretically, but the puzzle has to be set up in order"},{"startTime":3668.939,"endTime":3670.805,"body":"to be able to be solved."},{"startTime":3670.865,"endTime":3674.034,"body":"So this is one of the things about Sudoku that I say, you know, you"},{"startTime":3674.034,"endTime":3675.725,"body":"have, there has to be a unique solution."},{"startTime":3675.725,"endTime":3678.485,"body":"There has to be a solution and only one."},{"startTime":3679.025,"endTime":3684.307,"body":"So if your puzzle has been designed kind of correctly then such that that's the"},{"startTime":3684.307,"endTime":3687.125,"body":"case, then yeah, theoretically it can be solved."},{"startTime":3687.185,"endTime":3689.405,"body":"But it's of course actually doing it."},{"startTime":3689.505,"endTime":3693.521,"body":"As we all know, \u003claugh\u003e, sometimes those killer Sudoku, uh, you know, you may not"},{"startTime":3693.521,"endTime":3693.788,"body":"necessarily do it in the most elegant way, but it is theoretically possible to do"},{"startTime":3693.788,"endTime":3697.805,"body":"it."},{"startTime":3697.825,"endTime":3698.045,"body":"Yes,"},{"startTime":3698.295,"endTime":3698.925,"body":"There we go."},{"startTime":3699.235,"endTime":3700.405,"body":"Okay, thank you so much."},{"startTime":3700.405,"endTime":3702.405,"body":"Please join me in thanking Professor Sarah."}]}