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Pascal's Triangle Patterns · Acedra
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Pascal's Triangle Patterns
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So today we're going to talk about Pascal's triangle. It's one of my favorite things in math - particularly because whenever I talk to other mathematicians they always have some interesting patterns they like talking about. It's almost never the same one. So Pascal's triangle is sort of like an addition triangle. So we start with the number one and every time I draw one assume that everything around it is a bunch of zeros so what we do is we have one and then we add whatever is to the left
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of it - so it's neighbor here is 0 and we just drop that down in between - then I do the same thing on the right. Alright cool now I have two ones in this row and I'll do the same thing so there's a 0 we can imagine here so we drop that down. There's one in this neighbor now so we can have a two and then there's one and the zero to its right we make another one. And we just keep doing this for a while and you can see that the triangle just keeps getting bigger this is nothing
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that's gonna stop. So we can just keep doing this for as long as we want and for me as long as I want is when the first double-digit number shows up because that's when adding gets hard. So Pascal was also a combinatorist so in particular like problems where "If I have N things and I want to choose K of them, how many different ways can I do that?" And it turns out that Pascal's triangle is really like "childlike adding game" as seems to be actually encodes N choose K. So if I have N things -
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in particular let's say i have four things and I want to choose let's say one of them then I go to the fourth row I move over one and that's how many ways I can do it four ways. I if I have four things let's say I have like four circles and I want to only choose one of them - there are four ways i can do that right. I can choose this one, I can choose that one I can choose that one, I can choose that one And it turns out Pascal's triangle says if I go down to the
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fourth row and - where it like this is the 0th row - and I choose one thing so I move one over then that's exactly how many ways I can do it. (Brady Haran) So if i told you to choose two circles- Then there are six ways to do it. You notice that the sides are always ones Like, this isn't going to change because we just keep doing 1+0 and then oh, look 1 + 0, 1 plus 0. And this is just gonna keep happening so we can define this entire diagonal as "+0" because every time I
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go down 1 i'm not changing anything But then we get to this row right and we have one plus one plus one plus one because I have 0 1 2 3 4 5 so i can define this row I felt like it as "+1" then this pattern looks less fun right. I have One, a Three But the difference between three and one is 2. The difference between six and three is three the difference between 10 and 6 is 4 so it turns out this pattern is + 1 + 2 + 3 keeps going. So around here One may be tempted to look for more
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patterns like right here is like plus what three plus six but that's when you start to notice what we've been doing all along which was just adding the number to the top-left like Pascal wanted us to anyway. Let's just look at something else one that might be tempted to do when they see a bunch of numbers and that's add them in a different way. So this is 1. One plus 1 is 2. One plus 2 plus 1 is 4. One plus 3 is 4 plus 3 is 7 plus 1 is 8. You'll notice that some point I just stopped adding
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because we can actually rewrite everything over here as the powers of 2. So the sum of the rows of Pascal's triangle is two - like if I'm on the Nth row and I call the first one row zero then the sum of that row is 2 to the N. Let's just read these as numbers let's not add them let's not treat them as anything else So the first row is 1 the next row is 11 The next row is a hundred and twenty one. The next row is thirteen thrity-one. Then we get Then we get fourteen six-four-one
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And then we get to this *laughs* So ,right now these are actually the powers of 11 too So this next row looks pretty frightening because if I wrote it out just as it is, I get a pretty big number right like once I put in these commas, this is around 15 million (brady) That's not 11 to the 5 It's not. But it turns out Pascal you know his triangle doesn't quit that early so what's happening here is that when I was writing these as digits before, I was actually considering that
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place value right - like the ones place the tens place the hundreds place So I'm just going to do the same thing where I rewrite these numbers as ones and tens and hundreds place So the first one will be one times 10 to the 0 because this is the 0th row Then we have 1 times 10 to the 1 plus 1 times 10 to the 0 (1x10^1+1x10^0) so this one's 1 this one's 11. Uh, let's do 1 times 10 squared plus 1 times te- oh 2 times 10 to the one my bad - plus 1 times 10 to 0. That"s a hundred twenty-one
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So let's let's skip on down to our friend here We do 1 times 10 to the 5 Plus 5 10 to the 4 plus 10 10 to the 3 plus 10(10) squared? - yeah 5 to the 10 to the 1 and 1 times 10 to the 0 and this is 11 to the 5 (Brady) So it is 11^5 It is 11^5, it's just hiding So now I have this really cool thing that I found extremely recently called um, the hockey sticks method i think some people call it. It's just a really cool trick that you can do on Pascal's triangle another result of the fact that
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it's the coefficients of like a binomial expansion which is all of - it will make more sense once I draw hockey stick What you do is you pick a one somewhere on the edge and you just go down diagonally so i'm going to pick this one and then i'm going to go down diagonally and then I'm just gonna keep going and then at some point I'm just gonna veer off in this case I have to veer off because I didn't draw any more rows. So the number that I veer off to
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is the sum of all the numbers I traced down 1 plus 2 Plus 3 plus 4 equals 10 So this row - or this diagonal rather - equals 10 So I'm gonna start with this one and it's gonna be awful real quickly actually I'm just gonna all right to work on small ones. Uh, let's use this one Because... why not. We can do the other direction, right and we get 1 plus two plus three equals 6 I'm gonna do Pascal's triangle mod 2 So, brief aside - mod 2 in this case is just going to mean that every time a number
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is odd I write it as a one and every time a number is even I write it as a zero (Brady) Cool cool right (Brady) That's still a Pascal Triangle, is that? Yeah, just mod two. It doesn't like preserve the actual value of the numbers as much, but it totally tells you whether or not something is even or odd and that's all we care about right now. So then I'm going to outline all of the diagonals that are all ones and all the rows are all ones
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So i'm going to do this - which isn't the world's prettiest thing but it's pretty nice
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and this shape should maybe start to look a little familiar if you played - I dunno - legend of zelda before. It's th-also the Triforce but more importantly it's sierpinski's triangle so for those who don't recognize sierpinski's triangle I start with an equilateral triangle I inscribe another one upside down and then i can just keep doing this on all of the upward facing triangles as many times as I want so yeah that's can just go on for a long time (Brady) Cool, yeah?
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And it turns out that the way that the evens and odds show up in Pascal's triangle when I convert them to mod 2, I can make Pascal triangle as big as I want and I'm just going to keep getting iterations of sierpinski's triangle so that's really cool (Brady) That's awesome! That's really cool by itself but it gets cooler using the mod 2 construction There's something else we can do in mod two though Um, in particular we can write down the same
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things but let's write them now as - we've got ones and zeroes, right? And for most people ones and zeros kinda remind them of binary So, let's just see what happens when we transcribe these from binary You know one binary - 1 and then 11 in binary is 3 101 is 5 111 is 15 and then the next one, 10001 is 17 Now these may not immediately appear pretty cool but if I kept going the next one is 51 The next one is 85 After that is 255 and here I'm gonna stop and I'm
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going to look at what these numbers are three is prime Five is prime 15 is 5 times 3 17 is prime 51 is 17 times 3 85 is 17 x 5 And 255 is 17 x 15 So you'll notice that every time I get a prime i go back and just multiply by all the things that came before it and I can generate the next you know n minus 1 entries So the primes that we have here actually at least the ones that I'm - that we're certain of - are actually Fermat (pronounced fer-mat) primes Or Fermat (pronounced fer-mot) primes rather
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Which means that they can be written of the form 2 to the 2 to the n plus 1 Somehow every time I talk to a mathematician- even 10 minutes ago when i asked a different mathematician for an idea he told me something I've never heard before So even if maybe somewhere out in the world all of the things you could do with this triangle are done I don't know if there's any one person who knows them all and maybe there's only one person who knows one of them and we just have to keep exploring the
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triangle and seeing if we can collaboratively explain everything inside of it but it's an infinite triangle right so maybe there are things on the hundredth line down that we haven't explored yet I feel like it wouldn't be a numberphile video if somehow the Fibonacci sequence didn't show up So i'm going to show you where the Fibonacci sequence is in the Pascal's triangle because... this is a numberphile video yeah so what we're going to do is we're gonna add up the shallow diagonals
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So what that means here is the triangle kind of inscribed diagonals already right like I could look at this one and just kind of follow this but a shallow diagonal means that i'm going to kind of go off at a slightly higher angle that I would have if I was just following the way I drew the triangle so for example the first Fibonacci number is one so that's my shallow diagonal And then this one doesn't have any other shallow neighbors so it's just by itself also So now we have one,one.
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So this shallow diagonal is a one and then this one So we do this and you get 2 Then we do this one with this two - we get three We do this one with this 3 with this one we get five And you know how this game goes So we have 1,4 and three here so we're going to get eight We get- I'm goina do one more in case you don't believe me We're going to do five plus six plus one and this one down here that I didn't add yet
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And we get 13 (Brady) Cool So there you have it - the two biggest recurrence relations in math that involve addition probably um tied together in the Triangle They're not unique things you know you can have one and it encodes the other and it's all just related in really beautiful ways it's so beautiful you know, this triangle has all of these really cool properties about it and again like you said earlier it feels like it's really childlike thing right like I just you know add the things next
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to me and I just plop them down under but I've been playing with this triangle since... I want to say 8th or 9th grade and I learned half of these things in the past month and there are still other things that I could ramble on about about this triangle and this isn't even my field like I'm still an undergraduate I didn't even study this triangle professionally so just imagine what people know about the triangle that I don't even know that they don't even know amongst themselves like it's
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amazing
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