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Well, Simon's eight was really quite difficult, I think.

Â On the other hand, looking at the description on this forum, I think many

Â people found it difficult because it hadn't yet made this transition to doing

Â mathematical thinking. They were trying to, to do something that,

Â that wasn't really required. This was very definitely the, the case

Â with this one. Remember the, that the things I keep

Â repeating are, one of the essences of mathematical thinking is that you, you,

Â you, first of all, you ask yourself, what is this asking me?

Â What is, what do I know? What does it tells me?

Â What kind of objects is it talking about? And what do I need to do in, in, in terms

Â of, what's my, my target. So you have to stop >> You pause, you

Â reflect, you think about it. What you don't do, at least in the

Â beginning, is say, does this remind me of a problem that I've solved before that I

Â can just instantly apply that previous technique, because that can lead you in,

Â in, in completely the wrong direction. For example, the various words here, you

Â know? And in high-school, a good strategy was

Â look for key words, and try to map them into techniques.

Â But it really isn't a good strategy it, in, in terms of advanced mathematics,

Â okay? There are many questions, many problems we

Â have about perfect squares. The two are about the natural numbers.

Â Okay, so what? This isn't about the natural numbers, this

Â is about the integers. Completely different set of numbers.

Â So, well not a completely different set, it's a larger set of numbers.

Â So this is about the integers and the fact that some of the words are often used in

Â discussions about the natural numbers, well, so be it.

Â I mean words get used for all sorts of contexts.

Â Okay, so this is a question about the integers.

Â And if it's about the integers, then why don't we just take m equals n equals zero.

Â Then m squared plus m n, plus m squared equals zero, which is zero squared, and we

Â are done. Folks, this is not a trick question.

Â This is a question that gets at the heart of mathematical thinking, and the heart of

Â mathematical thinking isn't knowing a ton of techniques from mathematics.

Â The heart of mathematical thinking is, what do I know, and what do I need to do

Â with what I know? And what am I talking about.

Â Stop, slow down, reflect, think, mathematical thinking at this level is not

Â a sprint. All of the techniques you're taught in

Â high school, were taught in high school were very good for succeeding in high

Â school. And there's certainly leave you with a lot

Â of valuable skills. But we've been there, done that.

Â You know, if we've been out of high school, if we've mastered that.

Â Then we try to do something else. High school sort of teaches you how to lay

Â bricks. What we're doing now is seeing how can I

Â take all of those bricks, which are valuable things to have, and use them to

Â build a house. We're now going from being bricklayers to

Â architects. We certainly need all of those bricks, you

Â know, I'm definitely not knocking high school, right.

Â We're use this stuff all the time. But that was providing us with the basic

Â tools. Now we're learning how to make use of

Â these tools. Okay.

Â That's what mathematical thinking is about.

Â And sometimes we need a lot of tools and sometimes we can get by with something

Â very simple because we just ask ourselves, what am I trying to do now.

Â And we're not getting seduced into thinking that this is just another variant

Â of something we've met before. Think of every problem is a new problem.

Â Then you'll find that things get much more doable because you'll be focusing on

Â thinking not applying techniques. Okay that's the end of my sermon.

Â Well, it's not really the end of my sermon because that's what the whole course is

Â about. But let me just move on now to question

Â two. And in this case you probably haven't seen

Â anything quite like this before. So you have to start by asking yourself,

Â how could it happen that the answer being was a perfect square?

Â I mean, just how could that come about? What, what kind of thing must happen?

Â Well, let's just write it this way. How can we have mn+1 equal to p-squared

Â for sum p. Well at this point, one of those bricks

Â that I was taught in high school becomes really useful.

Â Because if mn plus 1 equals p squared, that means mn equals p squared minus 1.

Â And one of the bricks I had drilled into me in high school, was that p squared

Â minus 1 is P minus 1, P plus 1. So, what we're about to do is take

Â something that was drilled to us in high school and make clever use of it.

Â Okay? Because, if A minus P minus 1P plus 1,

Â then we could have M equals P minus 1 and N equals P plus 1.

Â In that case, p would be m plus 1. Now, m is the number we're given.

Â We're trying to find an n. And I'm just using this to say, what could

Â that n be? What must that n look like?

Â Go given an m, we'll be able to take p equals m plus 1.

Â And then, from this one, n equals p plus 2.

Â And then we found the n. That is m plus 2.

Â Oops! M plus 2.

Â Okay? P plus 1.

Â P is m plus 1 and n is p plus 1. So n is m plus 1.

Â So given the m, we've already found the n. So given n, let's just summarize it now.

Â Given m, take n equals m plus 2. Then mn plus 1 Equals m, m plus 2 plus 1

Â which is m squared plus 2 m plus 1 which is m plus 1 squared.

Â So we've found, we've, we've, we've, we've answered the question we've said that

Â given an m take n equals m plus 2 Then MN plus 1 is M plus N squared which is a

Â perfect square. Now if I hadn't gone through this.

Â It would have appeared that this was a rabbit out of a hat trick.

Â And unfortunately, it's a consequence of the way that mathematicians often write

Â their papers, that they don't include all of the reasoning, they just give you the

Â conclusions. This will be an easy simpler relatively

Â simple example. But if I had simply said, given m, why did

Â we take m plus 2, you would have said, how on Earth did he come up with that?

Â What made him think of that? You know?

Â Then, and, and I'll look as though I've got some kind of magical ability.

Â No I don't. I just went in and said, well, how could I

Â possibly get to that answer? And then it was just something I learned

Â in high school. Okay?

Â So, you, you know? One of the techniques is, just say, how

Â could I possibly get the answer that I'm asked to find?

Â Now, I'm couching this in terms of, of classroom questions.

Â But the same, the same issue arises when you're dealing with mathematical problems

Â in the real world. Look at the problem, what does it tell

Â you? What you have to do, how can I get the

Â answer that I'm, that I'm going for? This might be a simple, classroom type

Â example but it has many of the elements, a good mathematical thinking.

Â And in this case, it was how come that I'm supposedly arrive as soon as Alex, so the.

Â Start looking at that, it just drops out, okay?

Â Recognizing that was the entire key to the thing.

Â Once you're doing that. Straightforward.

Â Okay let's move on and look at the next one in question three.

Â Well, question 2 is another 1 of these things, that.

Â When you first read it, you think. Wow, this is asking for something really

Â deep. How could I possibly come up with a, with

Â a quadratic, all of whose values are composite?

Â How could I guarantee that all of the values are composite?

Â You know? Sounds like it's going to be really deep,

Â but not if you, if you sort of just take a breath.

Â Sit back and say, you know, how could it happen?

Â That numbers are composite. Well if they're composite, they have to be

Â the product of two, two other numbers. Okay, wait a minute.

Â Why don't we just take? If that's going to always be the product

Â of two whole numbers, let's make it the product of two whole numbers.

Â And lo and behold that is indeed of the form n squared plus bn plus c, where b and

Â c are positive. That's all there is to it.

Â We just wrote one down instantly. We didn't have to prove that one exists,

Â we just, we just ran that we did prove it, we did it by just writing one down.

Â And all we have to say to ourselves is, oh this sounds complicated but no, all it

Â means is that the values are products. And quadratics are products of things.

Â So we just write it. We explicitly make it a product of two

Â things. The value will always be the product of

Â two numbers, n plus 1 and n plus 2. N plus 2.

Â And these are all positive integers. So, we'll have maybe 2 and 3, or whatever.

Â So these are always greater than 1. So this thing is composite.

Â N squared plus 3n plus 2 is always composite.

Â That's it. Didn't involve any advanced machinery.

Â Just thinking about what the problem asked us to do.

Â Very similar with this 1. You know it sounds deep, well it's about

Â the Goldbach Conjecture. A problem that's been around for hundreds

Â of years, hasn't been solved, how on earth can we do anything with the Goldbach

Â Conjecture. Well, the answer is, just look at what it

Â tells us and ask us what we can do with it.

Â And for 1st of all, we observe that this is talking about numbers n Bigger than 5

Â is odd. Well, if n is an odd number bigger than 5,

Â then n is of the form 2k plus 3, where k is Bigger than 1.

Â I mean, normally, we think about numbers as being of the form, 2k plus 1.

Â But because I'm, I'm looking for prime numbers here.

Â I'm going to write it as 2k plus 3. And I can do that, because n is bigger

Â than 5. So for numbers bigger than 5, any odd

Â number bigger than 5 is of the form, 2k plus 3 where k is bigger than one.

Â Well, in that case, since 2k is bigger than 2, because k is bigger than 1, 2k is

Â an even number bigger than 2, so by Goldbach's Conjecture 2k equals p plus q

Â where p and q are primes. In which case, n, which is 2k plus 3, is p

Â plus q plus 3. The sum of three primes.

Â We're done. Actually, it wasn't difficult at all.

Â Once we sort of thought about what it says.

Â All of the complexity is in this unsolved problem, Goldbach's Conjecture.

Â If we assume Golbach's Conjecture, which we're allowed to, 'because it says, if

Â that's true, then every even number bigger than 5 is the sum of three primes, one of

Â which, in the case of our proof, is 3. That was it!

Â Okay? Actually it wasn't difficult at all, it

Â just looked as though it might be when we first met it.

Â So, another lesson we can learn is, don't get put off because something looks

Â complicated. Until you think about it, you don't really

Â know whether it's complicated or not. Okay.

Â Couple more on this on this assignment sheet, and then we're done with it.

Â The question five, the question 6, the last question, they're both induction

Â proofs, and so what I'm going to do is I'm going to do example 5, and then I'll leave

Â you to do number 6 if you've already tried it and failed and you came here looking

Â for an answer. You're not going to find it explicitly.

Â But hopefully, by, by watching me go through another example, namely, number 5,

Â you'll be able to go back and do number 6. Because these things are all very similar.

Â At least the ones I'm giving you are all very similar.

Â Not all induction proofs are similar. But these, the ones I'm giving you from,

Â from number theory are all very similar. Okay.

Â Now what the first thing you have to do is, is express this is for some kind of an

Â equation you know, with a formula in it. Because otherwise we're going to deal with

Â this expression, the sum of the first n odd numbers.

Â So we're going to have to write down a formula for the sum of the first n odd

Â numbers. So, what it asks us to do is show that 1

Â plus 3 plus 5 plus and then the nth one is 2n minus 1.

Â Now we have to show that equals n squared. Okay.

Â The hardest part of this whole thing is figuring out what the last term is.

Â Okay. So, that's what we have to prove and we're

Â going to do it by induction. And we taught to use by induction so, we

Â have to begin by looking at the first case.

Â For n equals 1, this becomes just 1 equals 1 squared, which is true.

Â So it's true for n equal 1. Okay, so now let's assume the result.

Â Let me call that star. So let me now assume star.

Â And now I need to prove the same, corresponding result with an n plus one in

Â place of n. So why don't I just add the next term to

Â both sides. The next term would be 2n plus 1.

Â So both sides of star, in which case, I get 1 plus 3 plus 5 plus, plus 2n minus 1,

Â plus 2 n plus 1. This is the first n plus 1.

Â Okay that's the first, n plus 1, odd numbers.

Â This is the next case in the induction. And now, now when I add, 2 n plus 1 to the

Â right hand side it becomes that. Wait a minute, I can now use another of

Â those bricks that I learned in high school.

Â Thank you to my high school math teachers. Because now, I can just get this one

Â straight out. This was drilled into me.

Â Boy, it's good to have these things at my fingertips.

Â That's n plus 1 squared. We're done.

Â It's the same formula with n plus 1 in place of n.

Â Wasn't difficult at all. In fact, all of the machinery I needed to

Â solve this one, I learned in high school. The one thing I didn't learn in high

Â school, not, at least, not very well, was how to strategize about a novel problem.

Â And that's what this course is about. This is, how do you take all of that great

Â stuff you learned in high school, and strategize, and use it, and get new

Â results, and think about new problems. But before you do that, you have to ask

Â yourself what is the problem I'm given, what do I know about it, and what am I

Â having to prove. And if you misread a sentence, you going

Â to end up doing the wrong problem. And there were a lot of, a lot of people

Â writing on the forums where what they were really seeing was, you know, I

Â demonstrated all of my high school skills in great virtuosity, and solved a problem

Â that I wasn't asked to solve. Well you know in this course there's very

Â little penalty for doing that, except you feel bad.

Â But you know, if this wasn't a course, if you were working for a large corporation

Â you might be out of a job or demoted or moved to a less interesting job.

Â So this is a low penalty experience for how to think mathematically.

Â Which is why these MOOC's are great. Okay?

Â Well, as I say, I'm going to leave you do question 6 and, and, and having seen

Â another example, I hope you're not put off by the formulas and the complexity in

Â question 6. It's really just logical thinking, okay?

Â Mathematical thinking. Okay.

Â So much for assignment 8. Seemed hard at the time, actually was

Â hard. But, hopefully it doesn't seem quite so

Â hard now. It's not hard, if you become a good

Â mathematical thinker, and that's achieving, that is what the whole thing's

Â about. Okay.

Â