 Thank you very much. It's a real pleasure to be here. I've had a lot of fun hanging out with some of you today Especially talking with some of the students. I want to tell you a bit about Some of my favorite math fun facts and to do that. I first have to tell you what a math fun fact is So what are math fun facts? I? started When I started teaching several years ago my lower division course courses such as a calculus course I would often start off with a five-minute math fun fact which could be Just an interesting idea for many different areas of mathematics just to show my students that math is actually pretty cool So what are they? Well, they're their ideas that often Involve a change of perspective so it'll change the way you think about something. It might generate some questions The idea is to arouse your curiosity and fascination with math and they should be fun often There's some element of surprise Or that you get from from hearing the fun fact and as I mentioned I've used them not not just in classes to wet wet people's appetite for mathematics, but also for project ideas So why do I do fun facts? Well, I do them just to show that math is actually more than just Dealing with numbers or dealing with just learning calculus I mean after all calculus is is a very useful subject and if you're a scientist, it's important to know but There isn't you don't get to see in calculus some of the very cool and neat things that have been developed more recently than calculus Calculus is 400 years old, right? So you sort of get a sense that it's all cut and dried and it's just about formulas if you just end your Mathematical career with calculus So I like to do that and I also Want to show my students that math is both a science and an art It's the science of patterns and the art of meaningfully engaging with those patterns And so hopefully you'll get some surprise insight or joy out of seeing some of these and So if you're interested I actually collected all my the fun facts many fun facts on a website you can find if you just Google math fun facts and I started this as a hobby back when the internet was still young If I were to do it today, I probably would be writing a blog But it's it's in the form of a website and it gets about a million hits a year If you don't have internet connection and you have an iPhone well, there's there's an app for that Okay So what I'm going to do is I'm going to show you some of my favorite fun facts I of course I love them all but I'll show you some of the ones that I think you might enjoy and along the way I'm also going to try to trace some of the The the the people we have to thank for some of these fun facts some of the contributions by different groups of people Now why is it hard to tell to give a talk like this? Well, one reason it's hard is that some of you know some of these fun facts already. Okay, and so if that's True for you then don't spoil the fun by shouting out the answer if you know it Let the people who haven't seen it before Enjoy the fun fact, but you can you can obviously play along and think along with me So I hope you won't spoil the fun for others. Okay, so let's let's just get started So I'm going to start with a very simple trick. I'd like you guys all to do So I'd like you to each of you To pick a number. Let's say pick a number between one and ten Okay in your head everybody done that. Yes. Okay good now what I'd like you to do is To square the number Which means multiplied by itself. Yes, have you done that great Okay, and now I'd like you to from this new Quantity in your head. I'd like you to subtract your original number from this quantity Yes, good Okay, great now take this quantity that's in your head and divide by your original number Everybody done that yes Okay, great and Let's see You tell me your name Holly what's your favorite number? three I'd like everybody to add three to your To the number that's in your head right now. Yes okay, great and Let's see. What about you with the cap J. What's your favorite number bigger than ten? Well, it's just favorite number whatever Fifty oh, that's actually pretty big. Yeah, go ahead and add fifty to your the number in your head and blame it on him if you Can't do it Okay Okay, as everybody have that everybody should have some number in your head. Yes Okay, now what I want you to do is I want you to subtract your original number Got that Yes, okay great So now I'm gonna make a prediction that the number you're thinking of right now and if I get it right don't hold back your applause Is 52? Okay So this is I'm sure you've all how many people played a game like this before maybe back when you were kids Okay, now to be honest. I actually didn't prepare this trick I just made it up in my head as we went along and I'm gonna show you a trick I'm gonna show you how I did it. It's actually pretty simple It's based on a very simple idea. It's basically based on algebra, which all of you learned in in middle school So you guys start with a number right like Holly. What did you start with? Three okay, because it's your favorite number great. Well, I started with another number. I called it x Yes Oh, that's great. I don't want to run makes is that way I write like this I started with x and then basically what I did is a series of operations that By the end eliminated the x Just doing algebra in my head now, of course I'm not doing something so simple that it's obvious what I'm doing It would have been a stupid trick if I said think of a number add three Subtract your original number. You're all thinking of three You wouldn't have been impressed right But this is what we did and you can follow along here x what did I say? Square it then what did I say? Subtract your original number then what did I say divide by your original number? I mean if you take x squared minus x and you divide by x you get x minus one then let's add Holly's number Which is oh it was three right x plus two and then let's do J's Trick what did I what do you say to add 50? Then get x plus 52 and Then we said subtract your original number and what you ended up with was 52. Yes Nifty right, so this is actually a points out something really useful And that is even if you are learning something that you think is boring in some class You can turn that around into an opportunity to be creative right come up with a magic trick that uses this idea Right this is how a lot of these kinds of mental tricks work now. What did I write down here? I said thanks here. Well What did we just use we just used? Algebra like the beginnings of algebra and one of the people that we have to think of course There's there's been a long history of people Many cultures developing the basic ideas in algebra, but al-hwad is me who is a Persian mathematician From long ago Probably wrote what we would call the foundational book in algebra where where people start taking Objects are taking equations and subtracting the same thing from both sides and manipulating them as we might do today In fact, if you look at the name of the book, what do you see in the name of the book? Algebra algebra Algebra that's that's where actually where we get algebra So this is let's we can all thank al-hwad is me for algebra All right Great, that's the first fun fact. Let's let's give al-hwad is me a big. Thank you actually another cool fact is that This name when when transliterated into Latin actually is is pronounced algorithms me So it's where we get our name algorithm the word algorithm in in computer science Okay, great Here's a second fun fact What I'd like you to do is to look at these four quantities and try to decide which one you think is largest Okay, so a is the number of stars in the universe B is a number of seconds since the Big Bang C is the number of orderings of a deck of 52 cards and D is the number of ways to fill out an nc double a bracket okay, and Don't calculate just give your intuition. Okay, don't calculate. I know some of you can figure this some of these things out Just use your intuition. Okay Which one do you think is largest? So let's see by a show of hands. How many people think the largest quantity here is a B C D Okay, wow pretty evenly split although many there were actually probably fewer D's All right, well, let's see what the right answer is I'm gonna go from smallest to largest the smallest here in place number four is oh Yeah, I didn't even ask you what you think is smallest I saw what you thought was largest. Let's see. Which one do you think is smallest show of hands? How many people think the smallest number on this list is stars? seconds orderings and Number of ways to fill out an nc double a bracket. Okay, that's even more evenly divided. Okay, the smallest quantity on this list is B The number of seconds since the Big Bang Yes, believe it or not number of seconds and so as we I think we estimate basically the the universe is about four three point seven billion years and in seconds. That's about ten to the 17 a little less than ten to the 18 Okay, a number that's 18 digits long It turns out the next largest is the number of ways to fill out a bracket and that's about the same size But a little larger. That's about ten to the 18 just over. Okay, that's two to the 63 it turns out because there's 64 teams in the tournament ah Okay, and the the the second largest is the number of stars in the universe. That's about ten to the 23 Really? Okay, that's a number. That's 23 digits long. This is what scientists estimate and the number of orderings of a deck of 52 cards is of course That's number one But guess what it is a number. That's basically about ten to the 68 a number that's 68 digits long That's so large that if for every star in the universe there was a whole nother universe of stars That would be the square of ten to the 23. That's still smaller Then the number of ways to order a deck of 52 cards What this also means is that if you take a deck of cards and you shuffle it you produce another configuration Right a deck of cards. I suppose somebody was sitting around shuffling a deck once per second since the beginning of time You wouldn't even run through a tiny infinitesimal fraction of the number of orderings of a deck of 52 cards You know what this means This means that every time you shuffle a deck of cards You're making history That's the first time most likely that that configuration's ever been seen in the whole history of the universe Isn't that amazing? You are making history by card shuffling Making history by Card shuffling. Hey, that's that's awesome. And of course, who do we have to thank? Now what did I say this? What did I say this? Thank the Chinese What are we? Who are we thanking for what? Yes, well, I think I just put that because I'm sure lots of cultures have developed playing cards, but Probably the first instance that we we know of our Chinese playing cards, okay and Playing cards have actually motivated a lot of interesting mathematics because after all there's a lot of patterns to study Right and so it mixes in combinatorics probability And just a lot of basic questions you might have fun with understanding patterns Okay, great. That's the second fun fact. You like that fun fact. Was that cool? awesome All right good Great, how many people here play football? And people watch football Yeah, okay, great. That's good enough Good enough. How long is a football field? It's undefined. Did somebody who's undefined? Okay, so let's see so a football field is a hundred yards. Yes, that's true It's a hundred yards and but let's suppose I include the the end zone. So there's two goalposts here. Yes So how long is it from goalpost to goalpost? 120 yards. Yes. Okay now suppose I run a a rope Across the field so that it's tight between the the the two poles. Yes, okay And now let's suppose that I add just one foot of slack That means a little an extra foot. Okay, so now it's a little loose. Yes Suppose then I pull The rope up at the center at the 50-yard line and ask what's going on there So now what's going to happen the the rope gets pulled up a little bit. Yes And I want to know how high does a rope get pulled up at the center of the field? Okay? Don't calculate just use your intuition. Okay, and you'll have four choices How many people think Here your four choices that when you pull up at the center that it'll it'll still be pretty tight And you can barely put your fingers under that's a the next one is you can crawl under that's B Next one is you can walk under and the the last one is you can drive a truck under. Okay, so vote for one of those Everybody vote. Okay, even if you don't just use your gut. Okay How many people think when you pull it up at the center by adding a foot to 120-yard long rope that it's Just enough to put your fingers under How people think you can crawl under Walk under Drive a truck under Okay, that's that that's probably The one that got the least number of votes or maybe the first one, but it turns out in fact Indeed you can drive a truck under What? it's actually gonna be about 13 feet high and How would you calculate it? Well, it's just a it's just a consequence of of what? It's just a consequence of the Pythagorean theorem and you can you can just calculate it at home and see See what happens. So we can thank Pythagoras for this Hmm. Yeah, and of course if you want a little bit of intuition, here's a little bit of intuition Some of you still have incredulous looks on your face. Here's a little bit of intuition Here is my here's my rope at the center. Here's the center of the field. Here's the rope and I'm gonna lift up this end. Yes Suppose I add just an inch to my arm How far will I have to raise this? rope to get an inch Several inches. Yes Right, and of course if you use the Pythagorean theorem, which says that the sum of the squares of the two Short sides of a triangle Equals a set the square of the long side. You can you can check this out. Ah Pretty nifty. How many did you like that trick? How do people like that trick? Yes, awesome. That's that's that's a fun fact Okay, while we're on the top of Pythagorean theorem, this is a very of course very very old theorem dates back to antiquity Pythagoras was perhaps the first Known proof although that's another thing about some of these really old facts is that Many of them were discovered independently by many groups of people and of course we are living here in In the West and we're often influenced I mean we tend to know the things that were the the cultures that we're part of But it's it's quite likely this is discovered in many other contexts But the first thing we know about is Pythagoras Pythagorean proof proof let me show you but there are many proofs and let me show you just a couple of proofs of Pythagorean theorem because they're cool One of them was actually proved by somebody who's not professionally a mathematician And this was one of our former presidents James Garfield came up with this proof Okay Which goes to show you that people can do math for fun, right? And it's a very simple proof and it basically what what what Garfield did is he drew if you have a right triangle and The two side lengths are C. So these are the green triangles. They're the same right triangle They're just oriented so that a the if one side is called a you can't see that but I'll write it here And one side is called B Well, we'll put them down so that one of them has a on the bottom and one on the has B on the bottom. Yes Okay, and then you form the the the yellow triangle Okay, and that's basically got two side lengths C Okay All right, so this is how this is what Garfield's argument was this is kind of nifty Garfield said hey, let's just use the formula for the area of a trapezoid Now of course who knows the formula for the area of a trapezoid Well, our president did right Okay, and it's not hard to actually to to guess what it is Right you how many people here know the formula for the area of a rectangle Yes, hype times width, right? Well, would you agree that if I just Lengthen one of the ends and shorten one of the ends that the thing would have the same area So basically if I take the left height and the right height and average them a Rectangle with that height and the same width would have the same area So that's what we'll do. What's the average of A and B? Well, this whole thing is area of the trapezoid is One half a plus B. That's the average of the two heights times the width and what's the width here? a plus B Right. Oh, okay. Great. So the area of the trapezoid if I calculate this is just one half Oh some more algebra a squared plus two a b plus b squared Correct me if I do something stupid Okay, great. That's the area of the trapezoid. Okay. Now. What's the area of the triangles? Well, the area of the triangles this is one half a B and this is one half a B. Yes Right. Okay. Great. So here's a question then. What is the area of the? Triang of the yellow triangle. Well, it's just the whole area minus the area minus the green area Yes So if I take this expression and I subtract the green area, would you agree? I'm taking this expression and subtracting a B because a half plus a half a B is a B. Yes. Oh Okay, great. So the area of this piece this final piece is simply This is one half a oops a squared plus B squared Because I'm subtracted off the this portion here with This portion here Yes, oh But what is another way to calculate the area of the triangle with two side length C? That's right Well, that's just also equivalently one half C squared so a squared plus B squared equals C squared if you just multiply everything by 2 Okay. Oh Pretty nifty. How many people like that proof like Garfield's proof? Yeah. Yeah, good. Okay. Some of you liked it Okay. Now this is not my favorite proof, but it's it's kind of nifty because it shows that that Even politicians can do math which you might not know today, but Okay, no, sorry. I'll just stop with the ragging on Okay, great Let me show you let me show you what what I consider to be my favorite proof my favorite Pythagorean proof And this one's going to go by really quickly So here's the idea Let's so of course we're looking at a triangle like this and I think we agree that that What I'm trying to verify is that the area of this which is a squared plus the area of this which is B squared is The same as the area of this Which is C squared? Right, would everybody agree with that if the side lengths are a b and c Okay, now What did you agree if I'm trying to show that a squared plus B squared equals C squared? That it's equivalent to Multiplying both sides by some constants, right? I mean you can't you can't stop me for multiplying both sides by the same thing Would you agree that if that if one is true the other is true? So it's enough for me to show that a multiple of a squared plus B squared equals a multiple of C squared Yes, you with me on this. Oh, but the multiple K is just a scaling Right, so I could change. I don't have to be looking at squares. I could be I could change the thing. I'm looking at So for instance instead of showing the sum of the areas of the two smaller squares is the area of the bigger square I could instead take some other object like. Oh, I don't know a I could take a smiley face right and I could show that the area of that smiley face is the sum of the areas of These two smiley faces Right if I wanted to right Any I could put any figure there I want Okay, good. So if you buy that well, I'm not gonna put that figure then here's the figure. I'm actually going to use I Will take my original triangle and Instead of using a smiley face or a square I will show you that if I take a copy of the original triangle a similar triangle It's actually the same triangle there and put a copy of the triangle on every side a similar triangle on every side This is not exactly the same Thing but there's a copy of the triangle oops Try again. This has got to be similar. Okay, and then here's another one Okay, those are basically similar copies of the triangle around all three sides. Yes All I have to verify is the sum of the areas of the two small triangles is the same as the area of the Big triangle. Yes, so here's how I'll do it. It's very simple all I'll do is Drop an altitude So that I have Cut the the green triangle to two pieces and I'm just going to fold over This triangle and fold over this triangle and you see they actually completely match when they fold over and they match To the same size as the triangle on the other side because that is that's just the big thing if you fold this over oops If you fold this over it'll cover the whole thing So the sum of those two triangles actually is the bigger triangle area Okay, cool how people thought that was cool Okay, you don't have to say that just to make me happy, but I think it's a little I think it's pretty nifty. Okay, great Let's move on. Oh Did you know that there is in fact a spherical Pythagorean theorem Spherical Pythagorean theorem really what does it mean to what does it mean to be a spherical Pythagorean theorem anybody have any ideas? We should be looking at what? Triangles yes, where on a sphere and one of the angles in the triangle better be a Right triangle so let's draw a picture Now you pick three points on a sphere now this the sides can't be straight straight So they're gonna be a little curved But you're just drawing the shortest the shortest path between two points on the sphere and they're gonna be slightly curved yes and To talk about an angle being right you mean that basically if you look really close at a corner It's gonna have a right angle. Yes everybody with me Now the weird thing about spherical triangles is That the sum of the angles don't necessarily add up to 180 in fact you could have a triangle where all the angles are right Right I go to the take the north pole go down to the equator move over 90 degrees and then come back up to the north pole All the angles there will be right But when we talk about right just pick one of them that happens to be 90 degrees and I will call the two sides next to the right triangle a and b and the third side C and Now the question is what's the relationship between a b and c and so here's a here's a relationship it turns out Turns out that the cosine of C equals the cosine of a Times the cosine of B Oh, I have to be really careful here. This is for a unit sphere. That is a sphere of radius one Or if you don't like that if you don't want to deal with a sphere of unit radius one This is for the case where if I have a sphere of radius R. I divide all these things by R Okay, so that this is radius R Interesting this is the spherical path I think it doesn't look anything like the regulars Spirit but that green theorem does it does it Well, in fact, it actually is so this is kind of nifty this is quite nifty and The Let me show you why this is really cool. So how many people have here had a little calculus just a little bit of calculus awesome Okay, well, so check out what happens here the first thing to notice is Would you agree that if my sphere got really big Radius gets big this thing gets flatter and flatter and flatter. Yes And so that triangle for this formula should change into the regular Pythagorean theorem if R goes to infinity. Yes, oh Okay, but if R goes to infinity, what's cosine? What's C over R doing if R goes to infinity C over R is going to something Really really tiny. Yes, okay, and if you take the cosine of something really really tiny Then what do you get? Very good. You get one but let's let's that's well So if that's true then one equals one times one and we knew that already So let's do a little take a little finer approximation Cosine of something really really tiny equals one plus another term minus another term Taylor series Cosine of X is one minus. What's the next term of the Taylor series? Well, if you calculate it's x squared over two it turns out, okay? Oh really? So if I take the next term in the Taylor series this thing just becomes one minus Approximately C squared over two R squared and we're saying that's equivalent to approximately one minus a squared over two R squared times one minus b squared over two R squared and When all is said and done if you start Canceling things and ignoring things which is what we often do when you work with approximations You'll find that C squared is approximately a squared plus B squared and viola the regular Pythagorean theorem 50 Who do we have to thank for this? Well, it turns out that we have a lot of astronomers who love mathematics a lot of astronomers Both in the Greek and Islamic world Astronomers to thank for for the origins of the spherical Pythagorean theorem you can find evidence of this in their writings Cool, how many people like that? Is that cool like that? Awesome. Great Here's a fun fact. I really love I love this fun fact. This is so awesome. So here's a question for you If I take a number like the square root of two and I square it, what do I get? Two Okay, no interesting So it's possible to take an irrational number and raise it to a rational power and get something rational Yes rational means written as a fraction Okay, so is here's a question is it possible to take something irrational and Raise it to an irrational and get something rational Is that possible? How many people think? No How many people think yes? Okay, hmm well Let me prove a theorem for you Which just means let me show you why it's true theorem. Yes, it can be done That's a great theorem just say yes Okay Here's an argument. This is one thing that's really clever. This is what's fun about this fact is not the result necessarily But how you prove this result? Because what's gonna happen is We're gonna show that this is true without showing you a specific example of why it's true What it's completely non-constructive and this is this is how it goes. This is really great. This is awesome So here we go. So here's a question. Let's look at How about let's just do the square root of two the square root of two Is it rational? Well, I don't know But if it is if yes We're done We found an irrational to an irrational that's rational correct. Yes So what's the other possibility? If no, then the square root of two to the square root of two is irrational. Would you agree with that? Okay, so you can't stop me From raising it to an irrational power and what I get is a square root of two to the Square root of two times the square root of two otherwise known as two and that's two and we're done How many people like that? That's awesome Nifty so either way we know it's true But we don't we we don't know whether root to the root to actually is rational Actually, some people do and what do you think the answer is? It turns out to be irrational Turns out to be irrational who do we have to thank for this? I have no idea This is one of the things that's passed down folklore and people just go ooh and ah and they share it It's kind of like sharing a joke on Twitter that you didn't write Okay. Oh, I love this one. This one's great This one's awesome if you have a piece of paper you can do this take a Take draw a triangle any triangle just make sure it's not special like make sure it's not a right triangle for instance Oh, that looks right. Yeah It's this actually really hard to draw something very general, but I'm gonna try okay. Here we go Here's a triangle and I'm gonna ask you to mark nine special points on on this triangle related to this triangle, okay? If you want to do this on your own sheet of paper you can I What I'll do is I'll ask you to mark the midpoints of each of each side. Yes That's one set of points Midpoints of every side. Okay, great Now the next thing I'm gonna ask you to do is to drop an altitude From each corner to the opposite side But what that means is pretend that that one side is the floor Drop something from the corner and see where it lands. Are you with me? So that's one that's that it's gonna be a right angle in this case Now if I imagine this side being the floor and I drop something from it It'll actually go to there approximately. Yes And if I drop something to that side from the other side, it'll actually Go here and it's actually a fact that all these altitudes across somewhere. They will cross Across somewhere which is not a point. I'm interested in right now But they happen to cross the three points. I want you to mark though are these points the points where they They land on the other side. So here's three more points Are you with me? That's six points Okay, now look at where the altitudes all cross That's called the orthocenter by the way We're at all crop where those cross and look at the point That's halfway between the orthocenter and the three corners. So in this case This point the point where they all cross halfway to each of the three sides is Is this point here's where there's one here There's one here and There's one here Those are nine points And if you've done this right if your things to scale by the time you're done those nine points all will lie on a perfect circle. Oh really, this is called the nine-point circle and For that we have to thank a German mathematician fairbac fairbac nine-point circle and This is early 1800s 1822 or so Nifty now, I know some of you drew an acute triangle or a obtuse triangle. It still works You just have to extend the edges the altitudes may not fall inside the triangle to fall outside the triangle Okay, but it still works. It's nifty Alright, how many people like this one like this one awesome great Great, we're on our way. Oh, this one's a great one. I love this one because it's all about friendship This is awesome. This is called the friendship theorem Friendship theorem says this suppose you go to a party and At the party suppose it's the case that every two people Have exactly one friend in common Okay, every two people have exactly one friend in common then something amazing is true if this is true then You read that. Oh, yeah, then it turns out that there will be There'll be one person who is friends with Everybody. Oh, yeah, really Yeah, so if you go to party where where any two people have exactly one friend in common There's one person who actually is friends with everybody and in fact what what what we know a little bit more It turns out that the graph of connections between people basically looks like a bunch of triangles that are connected at a Look that looks like this. This is what this is what the the the connections must look like It must look like this a bunch of triangles connected at a At a point that's basically the way it must look and And that person that person the center's friends with everybody Now does this have any practical application? Well, I don't know I mean go to have go to a party and start asking people who they know it might be kind of creepy Okay, but who do we have to thank for this These are several famous Combinatorialists Paul Airdish some of you may have heard of one of the most prolific Mathematicians and people like to count how many paper connections they have with Paul Airdish How many people have heard of an Airdish number? Okay, that's Paul Airdish An Alfred Renye is Hungarian and Vera Sosh is Hungarian And if any of you have gone to Budapest you have an opportunity I had a student once studied with Vera Sosh and They've got a great Combinatorialist there who proved lots of really cool theorems the friendship theorem Awesome, great. Okay Let's do a few more here So here's a here's a fun fact if you take a bunch of cubes The cube of one plus two plus three plus all the cubes one cube plus two cubes Let's let's count that actually can we do that? What is four cubed? 64 what's three cubed and what's 64 plus 27? mmm 91 right plus eight 99 plus one 100 really? Oh, that's interesting if I just add one cube plus two cubed. What do I get? nine oh 100 nine what if I add one two cube cube three cubed 27 plus nine is 36 Something funny going on nine thirty six a hundred those seem really special In fact, what are they? They're they're all squares and in fact the sum of cubes in this case turns out to be the sum of the the numbers squared Ten squared is a hundred. Yeah, right Six squared is thirty six and two three squared is nine So this is a this is a cool fact. It's it's true in general okay, and If you want you can prove it by induction those of you who know what induction is and Of course, we have a couple of people to thank here and the reason I said we can thank them is big is because Well first of all first of all our Aria Bada is one of the first mentions of this of noting this but al-karaji is probably the first person to prove it and What al-karaji is also known for is for Developing the method of induction This is another Persian mathematician Okay, but I'm going to show you a completely different proof which is an induction It's just a it's a proof by picture and this is this is cool enough for me to do okay, so here we go I am going to Would you agree that if I? want to To do this let's see Would you agree that this one cubed is just the volume of a little cube of side length one? Okay, and two cubed is just the volume of a cube of side length two. Yes Yeah, and three cubes three cubes. Okay, you get the picture All right now. Oh just wait. This is so awesome. Look Okay, I feel compelled to do one more. Is that okay? Okay Perfect Okay now check this out I'm gonna show you why the if you if you stack all these cubes in a different way You'll see the right-hand side is the same why I'll just put everything in one layer And I'll just show you the top so there's the first cube, but I've shown you just the top Well, if I want I can break up the blue two by two into a Two by one a two by one. That's the first layer of four squares Yes, and each layer and the second layer goes right here and Would you agree this is one plus two squared? And if I want to do one plus two cubed I can just well I got to show you why one plus two plus three squared is the same. Yes So you can't stop me from adding Three more squares on each side and then filling out the rest with a two by one A three by three by two and a two by three and a three by three This my friends is layer one, which I'll label a This is layer two, which I'll label b. I'll have to I'll have to to use one of these Together with one of these and then c will be the third layer. Yeah That's one plus two plus three squared on that side, right? And now you can see what's going to happen in the last case. I'll just I'll just draw this out Um Just one more thing and if you like it you can you can clap at the end There we go two by four three by four four by four four by three four by two four by one And if you want to match these up a b c d However, how am I going to do it? I'll call that a I'll call that b and I'll match b up with this one I'll match c up with this one And I'll match d up with This one one plus two plus three plus four Squared same number of squares how many people like that awesome Yeah, because you get insight from uh from doing this uh doing a picture proof all right I think we just have a couple more a couple more here. So here this how do you know what a magic square is? Yeah magic square. It's a array a square. Will you fill in numbers? Such that every row column and two diagonals add up to what? The same number whatever it is. Okay, so here's an example of a magic square Here's one uh one two three four five six seven eight nine Oh, cool. How did you do that? That's so awesome. Well, I used a special trick I won't tell you what that trick is though But this is a magic square size three in every row column diagonal adds up to 15 okay now Magic squares have been around a long time celebrated studied in many cultures china persia some of the the in the year in europe and even in africa and the reason we know this is that We have the the writings of muhammad ibn muhammad from an area that's now nigeria It wasn't at the time and he came up with a construction a very interesting construction For doing a magic square and i'll show you what his construction does in the case of a five by five There are other methods. I just want to show you this one because it's kind of nifty So here's what he said Basically The rule is the following You start in it actually doesn't matter. We'll start in the upper right square And then we're going to take we'll do night's moves All in the same direction. So if I go one and then I go down to two That's a night's moves on it on a board chest board, right? And I'll then I'll go down to three. Yes And now what I have to imagine is this board wraps around So if I leave the bottom, I enter at the top and leave the side under the other side. Yes And so I'll keep going as far as I can and let's see if I go down Make sure I'm doing this right. This is four. I believe yes Down and over down over left one down two. Yes left one down two Now what happens here with five? Uh-oh I can't go down to the next one because it one's already filled. Yes So what what he says is then you do a special rule you move over two units And put six there Yes And then you just continue six goes down to seven, which is here oops wrong Seven seven goes down to eight, etc. And just keep going And every time you just remember the rule if you can't keep going move over to The nice thing about this rule is it produces a magic square Not only with all the rows and columns and both diagonals That add up to the same number, but all the diagonals with wrap around add up to the same number Kind of nifty Really neat contribution from an african mathematician And if that wasn't enough magic squares are so awesome Here's another thing that was only recently noted Uh about magic squares Three by three magic squares. So if I look at three by three magic squares check this out Something that's really nifty Is if you take this magic square and you just pretend that the first row is a number Like 816 Uh and square it. Why would you do that? Well because something amazing happens if you read the rows as numbers and square them Guess what when you what happens when you sum up these squares You get a million 35,369. Yay Is that fun how people think that's fun Okay, maybe that's not fun, but if you now take the rows and read them backwards As a number 618 squared plus 753 squared plus 294 squared And you sum them guess what you get Whoa say whoa if you didn't expect that whoa, okay great, but that's not all Guess what if you read the columns as numbers and square them and sum them one way And read them backwards up the columns sum them square them one way You actually get the same number in but not the same number as this one In other words the same property holds uh reading the columns it holds for the diagonals It's kind of amazing fact probably uh the first place that I know if it was noted was in a martin gardener book But uh an undergraduate actually asked the question is this true for all three by threes And uh in this uh undergraduate uh at harvey mud actually proved that it actually holds for all all three by three magic squares You just have to interpret if you have more than one digit you have to carry So if that middle number were an 11 instead of a one the first row would be read as 916 And in reverse it would be read as uh 6 718 Okay, um before I do my final fun fact I just want to point out another fun fact Which is this year what's sort of the noble prize of mathematics uh is uh went to um four mathematicians uh and It was the first time we actually awarded a uh noble prize to a woman First time to a mathematician from latin america and the first time to an indian mathematician And the reason this is significant is if you think a little bit about uh the history of of uh scholarship not just mathematics but scholarship in general There have been many groups for which it's been hard Uh for those groups to be able to study or to have an education right some of the early women mathematicians actually had to do Take extraordinary lengths to actually get educated right so women weren't admitted into into universities at one point And some of the famous women mathematicians actually Started off by using pseudonyms in their correspondence. So sophie germane did that with um gauss Just because because of the fear that they would not be received And so of course we've come a long way But it's taken a long time for us to actually start recognizing the the work of Of diverse groups of mathematicians and of course we we hope this trend continues All right last trick Last fun fact is a trick. It's a card trick And to do that i'm going to Switch to the document camera This is this is really cool. It's one of my favorite card tricks So here's a deck. Yes Looks like an ordinary deck Mixed but of course you don't believe me. So i'm going to have somebody shuffle the deck Who here can shuffle the deck a riffle shuffle? So i'm gonna do a riffle shuffle. Okay, you're a little hard to reach but Um, can one of you riffle shuffle? No, okay You can riffle shuffle. Okay. Good. So i'm going to give him the deck. Just give it one shuffle Yeah, you don't need to give it a flourish. Just shuffle with it. Yeah, very good. Thank you very much. All right. It's a new combination Yeah, that's right. He just made history Okay so Is there any possible way after you shuffle the deck that I could know the the ordering of this deck? No, I wasn't in cahoots with him. I'll pay you later So would you believe then that I was born with an amazing ability I was no Amazing ability to feel the colors of the cards with my hands Now it's actually not a very refined ability. I can't tell you what color this card is but I can pull them off the deck in In pairs Red black opposite colors. Okay, and so just to uh, so you can see that red black. Yes Red black red black Red black now, of course if I'm just doing this by random Randomly then the chances that I'm doing this is actually about one and two for each pair And as soon as I do more than eight of them, it's one in 500 512 to be exact Yes One in the 1024 chance that I got all those to be pairs of red black. Yes And if you like I could keep going Or you could just believe me. I have an amazing ability Yeah, and we could all go home No No Okay, well a magician never reveals his secrets But I'm not a magician so I will This is so awesome because this is a trick it's a mathematical card trick So you don't actually need you don't have to worry about it's not sleight of hand You're not going to mess it up unless you if you just remember the rules it'll work This is so awesome. Let me show you Okay, so clearly I stacked the deck beforehand. Yes Yes Okay, but I had to do so in such a way that no matter how he shuffled I would still know something about the deck and this brings up another lesson from mathematics because math is the science of patterns after all And what we're trying to do is understand patterns that remain after a riffle shuffle. Yes It turns out we know that a single shuffle a riffle shuffle is not enough to mix the deck In fact, it was proved in the early 90s that it takes seven shuffles to mix the deck well of 52 cards That was proved by persidae conus and uh But so yeah, so there is something that happens after one shuffle There's still order and so here's what I did before the trick started. I actually stacked the deck in a special way I stacked them Okay, so I stacked them so that they're alternating red black red black red black And this one actually still is quite Ordered like that. Okay. Now it's not but I stacked them red black. I'll just do a few here. Yes Okay, so that's what I did Now of course when you show people at the beginning They won't notice it because the the the cards clump, right? You should show them that it looks mixed then you invite somebody to to shuffle. Yes Now the this is the only part you have to remember if you forget this the trick won't work or may not work. Okay When you hand the deck to your friend To shuffle you have to cut the deck You have to remember to cut the deck so the colors on the bottom are different colors. Are you with me? If you remember that then the trick will work no matter how we riffle shuffles And I advise you when you ask him to riffle shuffle it To make sure you tell him to shuffle it once because if they start shuffling more than once you're in trouble. Okay Or you should have another deck prepared as I did Okay, so Remind me your name again John so John is going to drop cards. Yes, but how is he going to drop them? I don't know If they'll fall off either one side or the other right? Yes So let's watch underneath what's happening when he drops cards from either side of the deck So which of these two sides is going to fall first? I don't know. I'm not sure. Maybe it falls from this side, but if it falls from this side What color must the next card be no matter which side it falls from? Red So the next one will be red even if it falls from the same side Now what happens when that falls? What's the two colors? Black red and we're back to where we started which sides are going to fall from now I don't know. Maybe red But what's the next card have to be? Black And so no matter how he shuffles even if he starts falling a lot from the same side Or if it alternates it will be It'll come off now won't be red black anymore. Do you see here? It's not red black anymore But every pair Starting from the top is going to be a red black pair. Yes And then the only thing you have to do to make sure people are confused about it is you don't show Good save You don't show them what you're doing You you you fish around to make it sound like you're you're picking cards randomly, but you're just picking them off the top Two by two by two Anyways, I hope you enjoyed that and uh, have a good night