Mathematics from Sudoku to Science with Fred Simons: Live with the Academic Outreach Team

Hello everyone. Thanks for joining us today. I'm Noah Chartoff here with John McNally and Fred Simons. And we are going to be discussing Fred Simon's life, work experience as a teacher. Quite a few things.

Yeah, I, I suppose that I have to tell a little bit about myself now.

A little more, A lot.

Yeah, absolutely, I will do that. Okay. Well, yes, well, somebody said that I studied in Amsterdam. That is of course quite correct, but that is a terrible long time ago. And now I have to say a little bit about my life in say one minute. And of course I have to leave out quite a lot. So what did I do? I studied mathematics in Amsterdam and I am a pure mathematician. At that time, that was around 1960s, say then you could study mathematics and that was pure mathematics. And applied mathematics was considered more or less as ugly or dirty. And you should not do that as a pure mathematician. Yes, that was the situation at that time.

Even today I've heard some people refer to studying applied mathematics as going over to the dark side. I, I know there is, there is still a disconnect between the two.

Yes. Well, I have to say, despite the fact that then I moved to a technological university, of course mathematic was something like what we call a surface topic in general, I still am a pure mathematician. That's a way of thinking, I think more than mathematics itself. So I moved to the technological university Eindhoven and I got my PhD there. And after some time people discovered that I could be a pretty good teacher rather than a moderate researcher. So I was offered a teaching position and my main task was to teach large classes of non mathematics students things like calculus and linear algebra. And the funny thing is that despite the fact that I was a pure mathematician, I was always interested in what would be the impact of using small computers contents of what you were teaching. And we are talking now around 1970. So there was not yet a personal computer, but some things like handheld calculators started to come out. So handheld calculators. I even have worked with a computer with 200 bytes a byte was to the unit later on a kilobyte and much later on a megabyte, but I still remember the byte. Only handheld calculators.

Wow. So like the arithmetic that you're doing is the entirety of that computer's capability. I'm guessing like normally these days we're used to that being just a fraction of its power, but back then it sounds like that was.

Yes, you are absolutely at the border of what was possible. Right. So all 200 bytes but it's amazing what you can do with 200 bytes when you really only have 200 bytes.

Okay, yeah, I've heard someone say that with what a TI83 calculator can do things like they manage the Apollo moon landings. That, that was, that was as powerful as the computers they had available then, things like that.

Exactly. Well, so what I studied at that time for myself privately is what can you do when you use a handheld calculator in more or less standard surface teaching of say, calculus?

You mentioned surface teaching surface topics a couple of times. What does that mean in this context?

That means that the teaching was not for mathematics students, the teaching was for future engineers. Well, of course, when an engineer studies mathematics, it has a different purpose than a mathematician mathematics. And that makes a difference in the way you teach and what you want to reach with your students. What should they do? How long, how much do they need to understand, really? What is understanding? What is mathematics? All these questions turn up. And so I discussed the surface teaching of mathematics to non mathematics students. People who may be in their professional life are going to use mathematics? More or less. Yeah. Okay, that's the idea. Well, and to mention, one of the strange things I ran into is that onto such a small computer you could add, say a program for numerical integration so you could compute your definite integrals. You could compute them on this small computer. And the students immediately asked, and I still, after well, that is now more than 50 years ago, I still do not have a good answer to that is why do I have the students, the engineering student mathematics student is different. Why do I as an engineering student have to compute this different, this difficult integral by hand? If my computer can do in tenths of a second.

This is a very engineering oriented question, but I think it's a very valid.

It is a mathematics orientated question. What should we teach in that day? By what mathematics do this engineering student really need? Does he really need the technique for finding definite integrals with pen and paper? If at that time, in that time the handheld calculator existed on which you could do it, and now, for example, automatica exists by which you can do it, the question still exists, what am.

I going to use this for? What am I going to need to know this for? Is a question that teachers sometimes dread in classrooms. But it's a crucial question. It shows that students are, are actually interested in how this is going to apply to their lives, that they're not just learning this by rote, they are trying to connect this to their current and future experience. So I love hearing that question whenever it comes up. But I see what you're saying. It's a difficult one to answer, especially to a population who is concerned with how the information that they are learning is going to help them do something concrete, do something specific. And you're saying that now when you say you don't have a great answer for that, are you saying that you think maybe people shouldn't have to learn how to do integration by hand?

Yes, I agree. That's indeed my opinion. I think quite a lot of people do not need at all to know how to compute such an integral by hand. I completely agree. And if I may make a more or less provocative remark, I would say I think that there is not a good reason why we should teach mathematics to everybody. My personal feeling is that we should teach mathematics and now discuss mathematics to people who are interested and in a sense, gifted to mathematics.

And your view on mathematics in that sense is the how to do things sort of by hand understanding each step along the way, rather than there's the sort of applied layer that has a much broader use where people can say, okay, I know that integration exists as a thing. I know the types of places I might want to do an integral. And now there's a tool that just does the integral because I recognize, oh, I want to do an integral now. So that that sort of mathematical layer that you were discussing is more specialized, you think? And the applied tool is the thing that we would want everyone to be able to say, yep, I know that that tool exists.

Again, I think this depends on what you call, well, what you think mathematics is.

Ah, okay. So when you said not everyone should be taught mathematics, there was a.

When I used the word mathematics, I. I would say, I mean mathematics in a MA mathematical way. You see, I think computing a definite integral is not mathematics. That's human computation. The mathematicians has found out what the definite integral is, and they found out techniques by which that can be computed. And finding these techniques is part of mathematics. The roots of these techniques is very different. It is, it is implemented in Mathematica. So why should the student know that? The student should know what a definite integral is. That's a difficult concept, I have to say. In my experience during the 50 years that I was working with that I met quite a lot of people who had the idea that when a student has computed by pen and paper, say, 250 definite integrals, technique one, technique two, etc. Oh, my goodness, what the definite integral is, I think that's absolutely wrong. By Computing definite integrals. You don't get an idea what a definite integral is.

Yeah.

Therefore, I think when you leave the computations to a computer, then you can spend more time in your course to understanding what the definite integral is. Yeah, but that means at the same time that this surface course becomes more mathematics orientated. And my experience is at the same time, it doesn't mean that the course becomes simpler for the students. On the contrary, it becomes more difficult for the students. For mastering a concept is more or less typical for mathematics. Not only, of course, not only for mathematics, quite a lot of other topics as well. But you have to master a concept instead of learning techniques.

This is beautiful. So, so many people are worried that when people get computational tools, they're going to approach the topic and learn how to do things without understanding what they're doing. But I'm hearing you say that the computational tools will let you skip the skill and drill repetition of things and focus on understanding. Focus on being able to really comprehend what is going on a deeper level.

Yes, exactly.

Do you have like some examples of materials that do this sort of thing on hand?

No, no, not more. At the moment, I did not yet finish by, what do you say? My life story. We are, we are only now at the moment about 1970. 1975.

Oh, I see. Not at this point in timeline.

Okay. And I was interested in these handheld computers and say, is this good for future engineers? And my answer was, yes, of course, things like that are good for future engineers. Again, there is some basic idea behind that. When you educate engineers or whatever sort of people, you should come, you should use the tools that I will use in practice, in their professional life. You should introduce these tools in the courses. So if you think that an engineer will use Mathematica in his professional life, for example, imf, then you should introduce Mathematica already in the first year of the student, you should use it. That's my idea. So now that it was 1975 when we did this, and I developed the courses, and I developed the courses and it moved on. And it moved on and I had quite a lot of this discussions on that with my colleagues. That's important. You cannot do things like this alone. You have to do with a lot of colleagues and you have to do with the faculty of the students, in this case the engineering faculty. And. Well, I had excellent context. You ask in what direction develop these courses? Well, as I said, they develop more or less in understanding in the concepts. And I have to say that indeed the good students like that very Much the students. The courses were very popular, but not many students made this course completely. It was a little bit too difficult for many of them, despite the fact that they liked it, they did not pass the examinations for the examinations also moved from reproduce a trick to more intrinsic questions.

Yeah, the question of how do you assess what people are learning and what should you be assessing? Is a very deep topic that we could probably talk about for hours. But yeah, this idea that, you know, students really do like to be challenged, but you know, what they don't necessarily like is one failed assessment turning into something that, you know, hangs around their neck is this is going to bring down my career now. Right. So yeah, that, that the whole side issue in education of, you know, the challenge is really what students want and like. But then you need a sort of structure that allows them to, you know, learn by failing a few times rather than just being, you know, left. Left with no, no recourse when they don't get something. The first time.

Maybe. Yes. Now there are many facets, faces of this dead English. Okay, yes. First of all, I'm discussing first year students.

Sure.

And for some other reason they do not understand themselves. They choose for I want to become an engineer because. But they are rarely deeply motivated to become an engineer. That's a very general problem at that time. I think that has not changed nowadays. So there are fakely motivated students. And if you want to help these students, you have to stimulate them. You have to make your courses interesting. And definitely discussing concepts was a very good tool for doing that. But after some time the student has to go to his faculty and say I have to pass an examination. Right. Then we run into the really very deep problem. We focus on understanding a little bit on what are we really doing. But how do you test understanding? And then many discussions with my colleagues and everywhere, not only on my university. And the amazing thing is that I saw that the contents of the nowadays calculus course. No, not nowadays, of the calculus courses. 50 years ago, but.

Right, right.

What was nowadays at the time?

Yeah, nowadays is 1970. 1980, but I think it's still the same. The contents of these courses were determined not really by what the student needed, by the things, but by the things that you can examine.

Yeah, that's been my experience.

You can examine how to solve a differential equation, but you cannot examine how to find a differential equation. You can make two or three exercises that please find this differential equation, but that's it. But you can make a large variety of differential equations that the students can solve on that examination compute an integral. The contents of the courses is to a large extent determined by the ways of what we can accept and not by what is needed. Very frustrating. It is really frustrating.

That is a crucial distinction.

But here we have another point only for which I do not have an answer. I have no idea how I test understanding my personal experiences. And some other people know that as well. I think is when you speak with or not in an oral examination, then from the way the student formulates it is clear in within a minute if he understands what he say what he is not understanding what he says. This interview basically I do not exactly know what that is. It is very funny. You could ask by for example, how much is 2 plus steel? And the way he formulates that it is 4 already has some contents in it that I can feel that if someone understands it or if he just learned it by heart.

Yes.

This interview based method of in a recent examination. I have no idea how to test it well. But here's another experience I have. I worked on large classes as I said and that means that we had hundreds of students that have to take an examination and well, and now and then I was happy to find a nice question for understanding and so exercise for understanding. And then next year all people who accompanied the exercises for the students said oh, that was asked yesterday last year on examination. So we should train that with you. And they started train that sort of questions with the students because of what could have asked on the examination. And here again indirect. The contents of the examinations determined what happened in the teaching. Not very good. But I think it is not. You cannot avoid it.

It's certainly fair to say that's not a solved problem.

Yeah, yeah, yeah. I. I think probably comes down to a number of factors. One of them being just, you know, if you have a class of 2, 300 people, some people would say that it. It would be prohibitive to have to talk to every one of them one on one. I don't entirely understand that explanation because we have to read their papers individually if we're not. If we're not doing that. But the probably that it pushes this issue one level higher that if you are talking to someone who is a student, the question has to always be how do you know that your assessment of whether or not they understand is accurate. Like I personally believe you're right. I believe that when you're talking to someone that you can tell pretty quickly whether they're just reciting something that they don't understand at all or they're really Thinking about it, they're engaged with it, they're interested in it, and they have understanding. I do believe you can tell that. But I also know an important part of assessment is the thinking that the same performance should get the same grade coming from two people. And when it's written, I think it's easier to confirm that that's happening. That, oh, well, this is exactly the same sort of argument. So it gets full points in a conversation. It can be hard to say exactly what it was about the interaction that makes you know that someone understands.

Right. I completely agree with you. But we. I had classes of even from 300 to 400, still a little bit larger. And well, I like doing that as a student, like doing that as well, as I said before, but it is completely impossible to speak with all. To do an examination with all the 300, 400 students orally. That's. That's impossible. And so you have to do it written. And I have not found a good way in finding how to test understanding. As I said before, on a written examination, in an oral examination, it is very simple.

I wanted to let you know, Fred, that there's a question that I think we will be answering later on because I'm enjoying hearing about these things in chronological order. But someone, one of our viewers, has asked what does the future of education look like? So that'll be something that if you wanted to answer now, I'd love to hear your thoughts on it. But if you wanted to wait until we get to closer to the future in the.

Of course, when I, when I was teaching to large classes or small classes, no matter what, when someone had a question I always wanted to ask immediately because of it lives in the audience. So I want to ask it now. Well, I did not think fairly deeply about it, but again, my personal opinion is that despite the fact that we have computers, a computer can never take over the role of a good teacher. A good teacher has something extra that he can give to the audience in the communication, all sorts of communication that a computer doesn't have. I am also deeply convinced that you can read a lot of tricks on, say, Internet or in a book, and you can see the tricks from the book or from the Internet. But that does not help you necessarily in understanding what you're doing. And I think for understanding personal lectures from school teachers, indispensable. That's my feeling about it. Well, what will happen in the future? Well, that strongly depends on what sort of people do you need in the future, how many mathematicians do you need and what would be the task of a mathematician. Well, there are, of course, tremendous large problems on which mathematicians work. But if I look to the engineering education, say, mathematics. How much mathematics does a mechanical engineer need?

Good question.

I do not know, but I think it is not much.

And the specifics of which ones are likely to change over time as well, based off of what the cutting edge of any given field looks like. Shorty, the specifics of which mathematics, say, a mechanical engineer needs to know is also going to change over time, surely, based off of the specifics of. Of what's cutting edge in that field at the time.

Oh, yes, of course, of course. There are some general ideas, you know, and in a sense, now you hit on the basic of mathematics. Mathematics is, in a sense, the generalization of everything. You make the things abstract, you formulate it in an abstract way, and you study the abstract description of what you were doing. I do not think that that was an idea 50 years ago as well. I do not think that when you make your mathematics applied, then you get a better understanding of the students. I do not think so. I do not even think that it's mathematics. Mathematics is abstract. It may be an abstract generalization of something very concrete, but at the end it is abstract. Yeah.

Now, for teaching, some of these ideas.

Even for surface teaching.

Yeah, well, for surface teaching as well as teaching, I know that you've created some tools that use computers that help people interact with some of these ideas. So. And I know that some of these were created more recently. But at what point in the story did you start to realize how useful computation and computer tools would be for the purposes of teaching or surface teaching?

Already? From the very beginning.

From the very beginning.

Fred has actually written a book back in 1985 that we were lucky enough to be able to get our hands on.

By the way, on the title of that book, Calculus with a PC. Now, everybody thinks that the PC is a personal computer, but at that time the PC was a pocket computer. Personal computer.

Oh, my goodness.

And there are nice basic programs in it. Very short basic programs that can do a lot of things. But of course, they are not necessarily nowadays at all. Mathematica can much more than my basic programs in that book. But the point then is you're looking all the time. No, what I found is that when you are starting using the computer tools of any kind, by the way, then it changes the way in which you want to teach the mathematics that you want to tell to the students. For example, a very simple thing, by the way, again, the integral and the definite integral. And the definite integral is a limit of a Riemann sum. And that's what you discuss with the students. So it is that and that and that. And here we have a Riemann sum. Now, with a pen and paper course, an old fashioned traditional course, you continue with the Riemannson and you do not compute a Riemann sum. But more or less theoretically, you find some results about definite integrals.

Right?

But when you define a definite integral as a Riemann solve that, you have a computer. With a computer you can compute Riemann.

Solves as as small, as small a step as you like, as many times.

As you like, with as small steps as you like. And you can see what happens. And et cetera, et cetera, et cetera. So I started doing that. I introduced computer Riemann job. And then you have to. You run into another problem that doesn't exist. In a pen and paper calculus course, you have a computer and the computer returns a sequence of numerical results. And then the question arises, what can you say if your computer returns a sequence of numerical results? About limitations, does the limit exist? Can you improve the. Can you improve the convergence? Things like the rate of convergence suddenly becomes important. And these are very basic numerical techniques. But as soon as a student is starting to use a computer for doing his computations, my feeling is that you should also know a little bit about the numerical background of things. And you must be aware of numerical problems that exist. And these numerical things became part of my course. It was a calculus course, okay, with a computer, but of course with concepts from numerical mathematics. Yeah, in that sense it changed the traditional course tremendously. It is by no means a pen and paper course.

Now I understand this is still material you are currently teaching, even though you're retired from the University of Eindhoven, that you have Mathematica materials for calculus lessons now.

Yes or no? I was discussing my life, wasn't it? So now we are around 1985.

Okay, so you do now have those things, but not in the story that you're telling. I understand, yes.

Okay, I come back to what you were saying. No, of course. What happens? 1985, there were a lot of people starting working in this direction. And there were many more people that did not want to use a computer in the teaching. Interesting. So we had a lot of discussion. What shall we do? What shall we do? I had some colleagues who are very enthusiastic. I had, unfortunately, I have to say, more colleagues who denied what I was doing. The development went on, I had international contacts, etc. A lot of people Were working on it. And well, I have to say I was quite far with what I reached. I was very happy. Well, then unfortunately, no again I was deeply confused that the contents of the courses had. At my university you have universities on all sorts of levels, but this is a pretty high level university. Was understanding. You had to focus on understanding. And now what happened then? Around 1986, 87 there was a reorganization of my university. There was short of money. They had to reduce the money with 25%. We had three research groups and I was in a teaching group. So of course there is. The teaching group had to disappear 25% back. 25 back of the university. And so I left the university. But of course I still had contacts and I used Mathematica in my courses. The personal computer, the pocket computer was replaced by Mathematica. And that was for me very. For the students a very inspiring business. But I was working with Mathematica. And the distributor of Mathematica asked me already a little bit before that time if I could also give courses on Mathematica to people from industry who were working with Mathematica. And there again I started to present courses on Mathematica. I worked with the same idea as in my calculus course. And that's the. That is understanding is more important than knowing how. And much of the documentation of Mathematica I think there is excellent documentation. So I do not want to say anything negative now, but I just want to remark that much of the documentation of Mathematica is that it only describes how to do it. This can be done in that way and this can be done in that way. And it is wonderful what all what can be done, but it doesn't give you a deep understanding of what Mathematica really is. How does it work? There is a front end and there is a kernel and the documentation hides all facts about the front end in the sense as much as possible is my feeling. Do say that there is a front end, but that's it. And that. Well, that's an interesting point.

Someone could know the definition and description of every function in Mathematica, but not no math. That is. That is a thing that could happen.

Yes, okay, that's true as well.

I mean, it's pretty unlikely to happen. But you know what I'm saying.

The course I was doing on Mathematica was with the internal structure of Mathematica. The claim is everything is an expression. But if you work with Mathematica, you never see expressions. 3 +4 is not an expression. An expression is head square bracket argument comma, argument, comma, etc. Square bracket. That is an expression.

Yes.

Three plus four is not an expression. Funny enough, right? It becomes only an expression when it goes from the front end where you type 3, 4 and you press Shift Enter. And then it goes to the kernel and the kernel makes an expression of it and the kernel does computations with expressions, and at the end there is an end expression, and that has to be typesetted in the front on your screen. And then it is typesetted, not in the form of an expression. So you never see expressions for some users.

Absolutely.

And quick note for something that is the basic idea about the courses that I presented on automatic. How does it work? What can you do if something goes wrong? But very often it goes wrong, and that is because of things like this. Suddenly the order of the arguments has changed. Very simple to understand, if you understand what you're doing.

Yeah. So I love this point about expressions being hidden. And a quick, quick note for people who might have been using Mathematica for a long time and not realized any of this. But. So if you're on a Windows machine, on any cell, if you press Control Shift and E, it will go from the sort of normal way of displaying something in a front end to more like an expression. Right? And on a Mac, I believe it's Command Shift E, it will do the same thing. So, yeah, so Control Shift E or Command Shift E, if you're someone who has, you know, for a very long time wondered, what exactly is the difference between an expression and what I'm seeing in the front end? There's. Yeah, so. And there's all sorts of things like that that are definitely not something that you would necessarily know when you're getting started. Because when you're getting started, you might not necessarily appreciate why that can be a helpful thing to know how to do during your first five minutes, let's say. But that's an excellent point that you're just making, Fred, about expressions compared to what you're seeing in the front end.

I completely agree with you. The discourse originated from something that was an introduction to Mathematica. But some people said, okay, this course is much more interesting for people who know Mathematica already, then they learn quite a lot from this basic principles course that I present. So it is not a course for the beginners, it is a course, four courses, actually, for the, say, more advanced users who want to know more about Mathematica. For the same. It is the same with mathematics and engineers. Say, how much do you have? Do you have to know about mathematics when you are an engineer. How much do you have to know about Mathematica when you use Mathematica? Not trivial, but yeah.

On that topic, I want to preface this by saying that I consider understanding to be a pure good thing, both in education and in general. I think it is a wonderful end in and of itself and something that everyone who is doing anything that they could describe as learning should be aiming for at all times. But playing devil's advocate here, I want to ask why do you think it is important for people to understand this sort of detail that you're describing? I mean, after all, well, someone can drive a car very well and in fact become a great driver without ever understanding what the engine is doing, you know, or, or how it's working literally under the hood. So what, what is the, the benefit that you saw? You mentioned being able to fix things when they broke, broke. And I wondered like what. What sorts of things does this understanding give us to student? Do you think that that will be beneficial to them?

That is a. In a sense, this is a terrible difficult question. First of all, we talk about understanding, but do we know ourselves what understanding is? Understanding is sort of a feeling, I feel if someone understands what he is doing. My first year students say we. I understand perfectly. I understand perfectly. Calculus for I can make all the. All examination exercises. I know all the tricks, I can apply all the tricks. And a lot of people think I understand because of I can use the trick. And I think that is more or less the situation with your car drive. Right? Yeah. So that is understanding on a very low level. But we also know that there are people who have a better understanding, a deeper understanding of what they are doing. I do not know what deep is, but yeah, well, how do I. There is some examples, but it doesn't quite appear so you can understand things on a terrible lot of levels. And that means too that we are talking now about teaching. And the level of your course, I think is determined better by the level of understanding that you require from the students. So my feeling is that 95% of the engineers, at least some years ago boss understanding of mathematics was enough when they could make the exercises. They do not need an understanding of mathematics. What I call an understanding of mathematics is a different sort of understanding. They need lower level, I think, in a sense. But if you want to be able to develop new ideas, you need a better understanding of what you are doing. You need a deeper understanding, otherwise you cannot develop new ideas. So you go deeper and deeper. Ah, now I think you go deeper and deeper and deeper and deeper. That's beautiful.

It's about not only being able to use the thing, you know, but to build on it and build with it, to use it to come up with new ideas.

That's yes, exactly. But you cannot teach that to everybody. Therefore, I think these ideas are not good for people who think that everybody should be able to become an engineer or a mathematician or etc. People have very different levels of capacities. I would say everybody agrees that for becoming a top pianist you have to be very musical. There are people who are less music musical. So the idea that everybody can be a top music musician is a crazy idea. The top is the same idea applies to men, most of at least the technical studies, the science studies. If you want to become a top physicist, you have to be very gifted for doing physics. And you? I, I cannot build top physicists or top mathematician. Well, I, I hope all these people have a much better level than I have, I hope, Yeah, I, I, I cannot do deep physical things. I do not understand physics well enough. I cannot eat mathematical things. I do not understand mathematics well enough. But me as a mathematician, anyway, I, as a mathematician, still have a mathematical way of thinking. But I do not want to teach to all students, although I thought that I, I think you have to differentiate between the classes.

So at this point in your story, you've gotten to the point where you're teaching courses for people who are wanting to know how to use computer tools, specifically Mathematica, better in their professional life, and you're still going at it with this attitude of understanding is what we're aiming for. Perhaps we differentiate that based off of where someone is in their career and where someone is in their prior training and these sorts of things. So where does it go from there?

Sorry, what is the question?

Go on with your story, is the question.

Yes, where does the story continue from there? Well, the story was a little bit, a sad story. I had to go back from my university in 1988 or something like that. After that, all the courses that I have developed were still given some time, time, but now, after five years, they were completely disappeared and we were completely back in the traditional situation we had when I started studying things like that.

Teaching to the test.

That, that's a pity, I think, but I, in a sense, I am also happy with it if it is not longer my responsibility.

Yeah, that's a nice part of it.

To take care of the level of education that we want to achieve. Now there are other people and they have their responsibilities. And I have to. To accept that. I have my ideas. My ideas did not change about the teaching, but in a sense, I regret that most of the things that I developed now simply have disappeared.

But, but like I said before, you're still using your own materials in the. The lessons you're teaching about Mathematica, is that that correct?

Yes.

Do you have any of those to show, to demonstrate the sort of thing that you're.

Oh, yes, of course I have. I always like to show things to other people.

That's the name of the game right here.

There is a thing we did not yet mention that for me personally is as important as understanding. And that is. Is the beauty of the things that you are understanding. And there is a beauty, but beauty is, I think, just as understanding. Very personal.

Yeah. Very subjective.

Yeah. I like. I can feel the depths of piano music of Hubert. And I know quite a lot of people who have no idea say, okay, they're playing the piano. And I hear much more when I said that is very personal. Of course, those things that I like very much that for me have a beauty can be more or less straightforward or simple or have no. No meaning at all for other people.

That's true.

What I can tell you is what I like very much about. Well, I propose that I do the following. You may say. Well, first of all, my first, first acquaintance with beauty of mathematics was already in the secondary school. And that made a very deep impression on me. It is very simple mathematics. So I. My first step is. I would like to show that to you, please. Then, you know, I developed a backlit for doing computations with Sudoku. And in that I also have some things that I consider as beauty and I can say show the beauty to. And then why do I like Mathematica? Because of. You can do beautiful things in Mathematica and you can make things that really have their own beauty and even are useful. So shall we go this way? And that you do. You'll interrupt me when there is something, etc. Right?

Absolutely, yes.

Okay. Yeah.

So I think we'll have to.

I have to go try to get back my screen here. Here it is. So I go back to my secondary school.

We can't see your screen yet.

You can't see the screen?

Not yet.

Not.

Not yet.

Do I have to share it somewhere?

Yep, it will. You'll need to press on the green button. There you go.

Yes. Okay, here we are again.

Now we got it.

Yeah. Now you can see my screen. Yes, yes. Well, secondary school. So I Was pretty young and we had geometry on the secondary school. And there was a very famous theorem that we were taught. And that is, this is typical, not English for me. We call this middle ordleinen, which is very more simple to pronounce for me the perpendicular bisectors. Okay, but you know, the three perpendicular bisectors run through one point. And when I was at the secondary school, you also had to prove that. We really had to learn proofs on the secondary school. And this is straightforward and not exactly beautiful, but nevertheless nice. This is the perpendicular bisector. What is a perpendicular bisector? That is the set of points that have an equal distance to this point, point and that point. Perpendicular bisector. So have a look at this perpendicular bisector. This one has the same distance to that point and to that point.

Right.

This point is on this bisect. Perpendicular bisector. So it has the same distance to this point and that point. From that it follows that this point has the same distance to that point and that point. So this point must be on the perpendicular bisector of this side of the triangle. So the three the things run through one point. Very straightforward, very straight.

The circle helps make that point very nicely. That it's the. The center of a circle and that these are radii.

No, not at all. From a method. Not at all. The circle doesn't play a role in the argument.

It does, it does to me because.

It follows from the seaworm that you can draw a circle through.

Well, it follows for me because it's saying that there's. That if the. If they met in something other than all one point, that would lead us to a circle with two centers that are at different points.

I think we can discuss that later on.

Yeah.

Probably better to move on from here.

Yeah. For me, this particular point.

Anyway, this is what I learned of the secondary school. But that was not very particular for me. It was a nice theorem. But it was nice and that was all. And then having seen this year, there was a similar theorem. Similar theorem. The altitudes of the triangle passed through one point. Okay. And you draw a picture like this.

That one I'd forgotten.

That's a similar. Similar as Europe. But it was shown in a way that I really think is extremely beautiful. But I interpret your faces correctly, as I normally do when I give a class that this should be pretty new to you. What you simply do is you try this line and this line and that line. And I made a button for that. What happens here? This triangle is exactly the same as this One as that one as that one.

Sure.

And these altitudes or altitudes are automatically because of this equals this, the perpendicular bisectors of the outer. And that is the argument. And that is really, for me an amazing argument. For whom? Who finds the idea that you have to draw these three lines? That is the beauty for me of this argument. It must be somewhat a Greek of two or three thousand years ago who already found that. But this is a marvelous idea to add to your triangle, this blue triangle. And then suddenly these two theorems are exactly the same. It's only one theorem.

This connects exactly to what you were saying before that.

Well, this is the thing that made a deep impression on me when I was at the secondary school. And I think this is really marvelous. This is typical mathematics, but this is beautiful, I think. Right. You may agree with me. You may see. Okay, not important at all. It is not important at all. The result is in a sense important. But. Okay, well, then I want to show you something that I also definitely.

I.

Want to throw in. I definitely consider that important. It connects exactly to what you were saying before. It's an example of real understanding showing by you being able to use your understanding of one fact to discover another one.

Yeah, but I am a little bit puzzled by the fact. If you know what is perpendicular bisector is, how do you arrive from understanding that concept in finding the blue line.

Around the triangle takes insight.

That's something I do not know how. That is the inspiration. Yeah, you need it. When you do mathematics, you need inspiration. The fellow who did that for the first time was highly inspired. I agree.

Okay, a quick point I wanted to make about the. So that the pictorial representation that had the circle, where the circle doesn't play a role in the actual proof, but nevertheless, to me, but for a lot of very important classroom applications, having a pictorial proof in scare quotes, where you haven't really shown the proof, but the picture sort of suggests everything it needs to do to the student.

Yeah, I completely agree with you. That is very important. I can give you some 20 other examples in which this plays a role. And the things that comes up in my mind now is that when you have to compute limits of functions or sequences or etc. Then you have the famous L' Hopital rule. And of course you know the rule of l'. Hopital. And that is a trick. And it is a very popular trick in some areas. It was never popular in the Netherlands. But it is a trick and it's useful. But the insight that you can tell to the people is why is this, why does this. Why, why do we find this limit? We have a numerator. We have a denominator, say. And then you can say, oh, but the numerator behaves like X to the power 4 and the denominator behaves like x to the power 7. This total behaves like 1 over x to the power 3. So it goes to 0 when x tends to infinity. Things like that you can do. And that is what I incorporate because I really wanted to have this sort of insight rather than. And giving an exact proof. That's, that's a different thing. I introduced, for example, the O notation, the big O and the small O notation in my calculus course. That is for improving insight. And so you mean it's so.

I've seen people take the quadratic equation, for example, that, that long, you know, square root of b squared -4a. See additives, all that. A very useful, handy way to solve any quadratic equation. And I've seen people use it for things like x squared minus 1 equals 0, something. Something like that. And I, I understand why they're doing it, but I would rather that they take a look at it and say, well, what does this actually say? You don't need to apply a trick every time.

Yeah, exactly. I completely agree. I have the same experience. But that is again the difference between knowing a trick and having insight when you are accustomed to read your formulas. The problem of mathematics, I think, is mathematicians work with formulas. But what is a formula? That is a mathematical way of writing down an idea.

Yes, it's a, It's a language.

What should a teacher of mathematics do? The teacher of mathematics should unravel this formula and present the ideas behind the formula. And his own notation, for example, is behind the rule of. Or, for example. This behaves like that. That behaves like that. That is, I think, how you should teach.

Thank you. Otherwise, it's like teaching a foreign language by just teaching. How do you respond to someone who says something to you in that language without ever teaching what the thing said to you means or the thing you're replying? We would consider that absolutely lunacy in any other field where we were teaching people how to communicate. And yet in mathematics, it's something people accept. And I'm glad to hear you didn't and don't.

It is not something that people accept. Maybe some people accept it, but mathematics is. If you do this, if you. But not.

I'm not talking about the field in general. I'm talking about way too many individual people.

There is not so much time left, I see on the clock on my computer.

Yes, we can go a little bit over our one hour that's scheduled, if you have time for it. But yeah, we should, we should definitely move forward.

Okay. Okay. Okay, then. Shall we then do the first? Because of we are all interested in Mathematica. Shall I first show you what I think a beautiful Mathematica applications. Right. So then I go to. To this furious problem. Yeah. Can you, can you read it?

Yes.

Looks good. If you want to make it a little larger, just because not everyone has a better screen.

Yeah, it's probably better for our audience.

Thank you. The problem that I'm going to consider is the following. I construct a list of pairs. So here you see my numbers. It starts with 31, 31, etc. I consider all these integers I consider as points in some or other space. And so we have some points will not play a role. But these are the points I have that are these numbers. And such a pair I consider as the left point, left number, the left point is connected with the right. So this is a trivial one for 31. So 38 is considered, is connected with 25. The problem is find all connected components. Given this sequence of pairs, find all connected components. Well, that is in a sense very simple. You can do it in this situation by hand in the following way. You can start with 31 and say, okay, what points are connected to 31? So that is 31. And is there another 31? Well, I cannot see it because of you are here on my screen. You do not see it. But okay, but you look is there another 31? And it is connected to that. And then you can, you can run and run over all these pairs and see what is connected with what, and you can write an algorithm for that. And.

Like I can see that on the lower line near the middle, 29 happens twice and is connected to 32 and 30, for example.

Yes, but a little more to the right. 29, 29, for example. So 32 is connected to 29 and 30 is connected to 29. So if you want to have the component of 29, you go also to 32 and you see what is connected with 32 and you see what is connected to 30 and so on. So that is tedious work, but you can do the work and you can program that. Yeah, so you can find disconnected components. Now, well, Mathematica, but you have to find the ID first. This was the straightforward approach. But you can see it here. If you can find a function that is constant on each component and has a different value on different components. Then you have a very nice function in Mathematica Function Gather by that gives you the components you're looking for. Thank you. Trust that you know the function Gather by Gatherby. You give it with a function and it looks at the domain and it separates the domain in subsets which have the same function value. So Getterby will solve my problem as soon as I have constructed a function that is constant on each component and different or different components. Right. That's the idea behind it. Now. So let us start with a function and with the property that for any two points we do not yet look at the pairs that for any two points have different values. When then we ask when we use Gather by, we can bet get back the single points. Of course, when all function values are different, then we can back the single points and that are the connected the classes when you have not yet looked at any pair. So we start with such a function now. Now, for example, let us look at say the fourth, the pair 421.

Okay.

For example, what does that mean? That means that because of now 4 and 21 have to belong to the same components. We have to join. The component of 4 is the component of 21. That's what we have to do. But how do we join them? Well, that is the idea. You compute the function value in the left point, you compute the function value in the right point. And the only thing you have to do is you have to find that these two function values have to be or as soon as you do that, you have joined for gather by the two components. That's the function gather by. Wonderful function, isn't it? You were a little bit overwhelmed, Tashi.

Okay, I think I need to see this.

In practice, you have the function value of the one component, you have the function value of another component. And when you make these two function cells equal, I.e. joining the components, that's the basic idea of this algorithm. So that is this command is the basic command we are looking for. But the only problem is that when you work with real function values with given function values, say 25 and 27, you cannot say, okay, the A was 27 and the B was 25. And now I put 27 is 25. Right? Right. So what is the trick for that? Simply do not define values for your function. Then the function value in A is different from the function value in B. So if you start with a function that doesn't have any value defined, this will work. So what is the solution Clear F, no values there. Do we said, I said before and then I call the function Getterby. That's all.

Oh my goodness.

Beautiful function. And now we can even make it a little bit more mathematical. For what do we do? We start with a pair P1, P2. Right. That's, that's what we consider. And we compute FP1 and FP2. Well, look what I do here. I start with a P and I map the function F on P. Then I arrive at the list FP1, FP2 list FP1, FP2 and that list expressions. Aha. Deep understanding of Mathematica has had set. Sorry, has had list. And now I want to make an assignment of that. How do you make an assignment of that? Simply replace the head of this expression list with the Mathematica name set. And now I have a half liner actually for solving this complicated problem.

And.

To emphasize, I think this is beautiful. This is really beautiful. This is one that you can do things like this in. Mathematica is one of the things I like of mathematics. Right, yeah.

And to emphasize for, you know, people who might not be as familiar with the syntax, a couple of things. So one, by leaving the function F abstract, in other words, not having it be, you know, some, some function. As you were saying, that's like X. Yeah. Where you're going to wind up setting, you know, 25 equal to 27 when you're, you know, going through the algorithm, which is not what you want to do. Right. So that's one of the things, is that you can leave F as a sort of abstract thing, which is concrete enough that you can type it and then the system can sort of work with it. But it's also abstract enough that it's not as specific as, oh, this is X squared or X cubed or something. So working at that abstract symbolic level lets you do a lot of really powerful things. And this is a cool example of that. And then the other thing that I love that you did here was that that double at. So that's applying a new head to an expression. So if you're a user who, you know, works mostly with front end stuff, that looks a little bit, excuse me, more like what you might be used to seeing with traditional mathematical notation like.

This guy right here, then.

Not me.

Right, right. So if you're more used to the traditional mathematical notation, taking a little bit of doing the exercise for yourself and saying, okay, here's something, I know what it means in mathematical notation. And then using that sort of control shift e Or command shift E, or there's other ways that you can read about this in documentation. It always sort of says what the alternate way of expressing the same concept would be. So you can play around with what's the way of expressing that in sort of the language of an expression so that it's got a head to it, which is that first part and then all the arguments that go into square brackets. And so, yeah, this is a really cool example that shows the power of knowing that.

Absolutely. It shows that it's the kind of thing that a human being could maybe understand but couldn't do practical work with based on these sorts of definitions. Like whenever you tell a person, oh, a function is something that outputs a value for an input, people are like, okay, fine, but with that vague a description, I can't do any actual, actual work, but a computer can. It can understand exactly the information that you give it and then do whatever you wanted to with that.

I have to say this is for me a little bit frustrating as well. For very often when I use Mathematica, something goes wrong. And then my first feeling is always that Mathematica is no, but it always turns out that Mathematica that I want wrong. And that, okay, that is for the user of Mathematica, painful. By the way, you like, you seem to like that and I like that. But I am very sure that quite a lot of Mathematica users do not like that because of it is highly unreadable. And you have to know quite a lot of Mathematica before you really understand. It is very, very, very, very difficult to. To see what is really happening. This is all sort of infix notations, etc. And that is very dangerous. So, for example, I want to show you this. When I type this, Some people like that. What did I type?

You typed apply x squared 2f as in.

But I can also read it as take F squared, take square of fx.

In other words, so all of these.

Symbols have a particular order.

Notations like that do not have a unique meaning. And I have really seen people making the mistake that this one was interpreted as F X squared. By the way, I do not, I do not even know what. Let me make sure it is FX squared. It is FX squared.

Wait, can you run that? Can you run that?

So you can see it by clicking on this edge and I pick double click. And it first computes F of X. And of that it takes the square. That is a double clicking technique for doing things like that. And you see it is F of X squared. So this is highly. Can be highly confusing notations like this. These notations are very dangerous.

Yeah, I got it wrong just now.

And similar to, similar to the way that when you were teaching people how to do like traditional mathematical notation, there's, you know, certain sort of notational things that you can do that do have a set meaning, but no one would actually write them that way because it's so confusing and ambiguous when you're seeing it. And so, you know, because there's different ways of notating things in, you know, Wolfram language program as well. Definitely be aware that if you're wanting things to be readable, you know, just, just in, in that particular case, you're just looking at with the, you know, input output, 11 square brackets is probably what you want there rather than using the notation. Right. Because if you just use the square brackets as someone who's writing something, then, you know, someone can just see. Oh, okay, I see where you meant the square brackets to be. But. But yeah, so understanding the order that these notational shortcuts get applied, it's like order of operations, but in a context which is specific to how this language is going to work.

Similar the two order of operations for arithmetic. If you were to put parentheses around the X squared part of that, it would read it as.

Of course, when Mathematica is fond of records. Everybody knows that Mathematica knows can you use a tremendous amount of brackets. And I think that's very good. There's nothing wrong with that. I use quite a lot of brackets because. But I always realized the, the difficulties when I use notations like this. Actually, I, I hate this at symbol and I prefer to write F square brackets at the end, closing square brackets for then I understand what I'm routing down and things like that can go wrong easily.

So the, the add symbol can be very useful. But it's ambiguous enough that you, you would prefer the clarity of those brackets.

By the way, this, therefore, in. In this solution is less dangerous than this.

Yep.

I, I was impressed at how little you had to write for the last version. But yes, the, the second to last version is much clearer to me in terms of what it's saying.

Yes, exactly. So that's also. For that reason, I gave courses on Automatica, you know. Yes, Automatica already. Yeah. Now I. There is still plenty I can say. Shall I say make? What shall we do?

Yes, I want to make. Before talking about Sudoku and some other details here, I want to emphasize for people who aren't familiar, that highlighting thing that you were doing so, for example, when you were clicking and then clicking again on width to highlight the entire thing, that's something that you can do if you're not familiar with this at home. If you have some big expression maybe that someone else has written and you're curious what it's doing and you're having trouble tracking the brackets and the symbols and the order that they're applied. When you continuously click on something like that, it will start where you are and then sort of expand outward to the next meaningful chunk of how things are going to.

It's a very powerful technique in Mathematica on my system anyway. It is also control dot, which is just a little bit easier to do instead of double clicking. I prefer the keyboard above the mouse. Control dot also works, but that is fine. Maybe we discussed understanding at the beginning and I think it is also nevertheless a good point to mention it here. One of the first things I do in my course is Mathematica course, is that when you work with Mathematica, you always have to realize that decimal numbers shown by Mathematica are not decimal numbers shown with pen and paper. Something for decimal numbers we spend on paper that are just rational numbers when you think about it. And decimal numbers shown by Mathematica are double flots or things like that, machine numbers. And that is very different. And it turns out that. That. Well, more than half of the audience I have when they go to a course of mine do not know this difference. And so they discover a lot of books in Mathematica that are not bugs in Mathematica, but are just consequences of the use. That you use double floats instead of rational numbers. Yep. And that's a very important thing to mention too. And it was also part of my course on my calculus course for engineering students, of course. And machine numbers are not decimal numbers. They are very different.

It goes both ways. There are concepts that are extremely hard for human beings to understand and easy for a computer to do. And other concepts very easy for a human, like 1.5. Hard for a machine to model that in the way that it does it.

1.5, I think still is easy. But I understand what you are meaning.

It is.

Shall we do one of the other exercises? One of the other beautiful things? Oh, please.

Yes, let's do it.

Yes. Then I close this.

Oh, yeah. Probably shouldn't have used an example that includes a power too, shouldn't I?

No, no, I will not do that. But I think this is 1.7.

1.7.

This is the Packlet Sudoku hint that I have Written and I show. And you can. With that, you can more or less solve Sudoku. So here I have an example.

But for people who aren't familiar, a Sudoku is a puzzle where you have to use all numbers in a row or column and can't repeat any.

Yeah, I will. I will explain a little bit. I enlarge it. So on here you have even not a standard Sudoku. For what do we have a grid? And we have to place the numbers 1 to 9 on every row, every column, every square block. I have here. But in this Sudoku, also on the two dice diagonal, there are a lot of variations of Shibuyaku, and this is one of them. And these are the given numbers, the big numbers. And what you see written down in the. With the small digits that are the candidates that still can be placed there. When you take in account that, for example, when you have here a nine, here cannot be a nine, There cannot be a nine, there cannot be nine. That there cannot be a nine because of a look at this role and so on. And that is computed by the program. Not really difficult, but it is shown here now. And what is solving a Sudoku actually seen from a mathematical point? Yeah, that is in every cell not yet filled cell, you have some candidates. And you want to skip candidates until there is only one candidate left in the cell. And then that is the situation. You can place that cell there and you can continue. So solving a Sudoku is actually skipping candidates. And there are a lot of logical rules for skipping candidates.

What do you mean by skipping in this context?

When I had a logical argument, for example, in that I could Skip the candidate 5 here, then only the candidate 6 was remaining. But if you have a cell in which there is only one candidate, that candidate must be placed. Placed in that cell.

Right.

Then I can. And in this way, I solved the Sudoku problem. Okay, now, so that is, in that sense, is solving a Sudoku puzzle is skipping candidates. Yeah. And as long as. Until there is only one candidate left in a cell, and then you can place that, and you can go to the other cells and see what can happen. Well, there are a lot of logical arguments used by that, and they have very deep names. There is a lot of theory about it. And some is dirty, I think ugly, and some is highly beautiful. And here I show you something that I think is beautiful. Have a look at this situation here. In this role, you have a candidate 2, a candidate 2, a candidate 2,. And a candidate 2.

Sure.

Right. So the two must be placed in some of These one of these four positions. Now, what can we say? There is a very remarkable argument here. Can be a 2. But let us assume that this is not a 2. When it is not a 2, there must be a 2 placed somewhere here. And the only possibility is here. So then we have it. When this is not a 2, this is a 2. But when this is a 2, this cannot be a 2. Right. Now we look at the diagonal. When the 2 is not here, it can only be here. So what is the funny situation? When the two is not here, it is here. So the two is in one of these two yellow cells. When the two is in one of these yellow cells, the two can be skipped from these two green cells and it can be skipped from this green cell. And that is what I consider as a beauty. Beautiful. This is a beautiful argument. I think as such. Yeah, you are maybe not not, But I think you have something for. For the arguments. This way of finding arguments for doing things right. And you have nice arguments and you have the ugly arguments that you have beautiful arguments. And in this situation, I think this is a beautiful argument. Not here, then two, not here, then here. So away with the two here and there. Simple, straightforward and interesting, I think. Right. Well, and my last example, I think very short. I can skip this again. That is Sudoku. What is really the Sudoku problem? We did this already. Now I showed it to you. This is a standard Sudoku. Okay, so what is the problem? All nines. The numbers none to nine here, one to nine here. In every row, every block and every column, you have to place the numbers 1, 1 to 9 and all different. So that's. That's a standard one. But there are many others. And I show you some of them.

Oh, I've never seen a jigsaw sticker before. That's interesting.

Look in the newspaper and you'll see plenty of them. This is a jigsaw figure for the. The blocks now are not longer three times three, but they have this sort of force. Right. Still more puzzles. We have seen something like that. You have the normal standard blocks, but you also have a diagonal and another diagonal in which you have to place the numbers 1 to 9 only once. Right? But there are multi Sudoku.

That's fun.

The Christmas show. You must have seen things like that at some places where they are popular when the holiday starts for people who want to spend a lot of time with this. Right. So this is a multi suit dog. And in our. We have some brokers like this. Here we have a jigsaw sudoku and here we have a diagonal Sudoku and they overlap here all these sorts of things exist, right? And now what is again for me a big discovery. I program finding solutions for this already since 2004 or 2005 already. So I'm many years busy with it. But this is what I discovered only I think one year ago. It is very simple. This is a simple Sudoku and I make a multi pseudo group of it. So it looks like that a multi pseudography is an extension of a simple pseudography. It looks like that, and everybody thinks it is. Here I have one Sudoku and the overlap is another Sudoku and you solve that one and then you go to that one and etc. But what I suddenly discovered is it is exactly the same, just like the theorem with the. All these Sudoku are the same. So what is really the situation? Let us try to explain it. Here we have in this multi suit document, we have a lot of positions. This is a position. That is a position. Every cell can be considered as a position. So we can make a long list of positions. And then we have what we call groups. A row is a group that are nine positions in which have the numbers one to nine. A column is a group. That is there are nine positions in which we have to place the numbers one to nine. This block is a group. There are nine positions. So what is really the Sudoku problem? We have a lot of positions. We have here. We have a lot of positions and we have a lot of groups. And groups consist of of Each group consists of nine positions. And mathematically spoken, the program is such that place the numbers 1 to 9 of the group of the positions such that in every group the numbers 1 to 9 turn up. Now I have a mathematical formulation for what a Sudoku personalist. But with this formulation, there is no difference at all between this multitudoku and between this dodiaconho Sudoku and between this and.

Now I know there's only like five minutes left in our time, but I have to ask here. Aren't there some situations where. Like with the multi Sudoku, with the. The five different.

Yes.

Aren't there ones where a single cell could be considered to be in two different columns at the same time?

So we're abstracting the notion of group or collection, right?

Exactly. Skip the idea of a row of a column and look at a group. When you look at this, send a Sudoku and you look at this cell. This cell is in this group, this cell is in this group, and this cell is in this Group, I see.

There's nothing wrong with it being in multiple groups. All cells in every Sudoku are in more than one group. At least they're in a row and a column, traditionally. And so. Yeah.

Huh.

I think that's a beautiful sort of.

This is, in a sense, mathematics, isn't it?

Yes.

Mathematical thinking. You have a problem and it is formulated by. In. In terms of matrices and rows and columns and blocks, etc. And that is a far too complicated formulation of what a Sudoku problem really is. And when you start thinking about it, what is it? As I said before, it is just, you have a list of positions has nothing to do with a matrix. You have a list of groups, and each group consists of nine positions. Has nothing to do with row or columns or blocks or whatever. You just have a list of positions and a list of groups, and you have to place the numbers on the positions such that in each group we arrive at 1 to 9. And that means that there exists much more pseudoquic that cannot represent it in such a way that have not the form of a matrix with blocks and row and columns, etc. But at the moment when I developed my packlets with my Sudoku functions in it, that turned out to be very fruitful. For since all Sudoku are the same I have for F, for both for multi Sudoku and for single Sudoku, all my functions work because of I abstracted the idea of a Sudoku to a list of positions and a list of groups, and that is behind my backlog.

That's a beautiful illustration of what we were discussing at the very beginning of the difference between mathematics, or what some might call abstract mathematics, and some application. Right. So like, each type of Sudoku you could think of as the sort of application of this abstract idea of the Sudoku problem.

Yeah, exactly, exactly. Each Sudoku you see printed here is just a special case of something more general. So this is a special case of something more general. And this. Where is it? Here is also a special case of the same general situation.

I love that this is something you've been working on for more than a decade, and you are finding insights to it as you're going that. Like that. The idea of you need to understand what you're doing in order to build on it is not just something for surface students, it's not just something for beginners. It's something we're doing throughout our entire lives. If we're doing this.

Well, yes, exactly. Exactly. And you even find an example of that in something simple, obviously.

Yeah, that's truly wonderful illustration of that idea and one thing that we often tell people, but I think this is actually probably now my new favorite specific example of this is that if you can abstract a problem to a certain level, then you just write the solution to that and then your, your solution to that will then trickle down to all of the specific instances of that problem that you've sort of abstracted to. Oh, this is my real problem that I need to solve.

But is that not the basis of scientific research, what you formulate now?

Yeah.

Is to discover something that's not just true now, but true in many different contexts.

Yes. But it's not, not exactly to be true. I think it is an ID that will work in all these situations. Right. I think an ID is better than being true. I can formulate, I can solve this and this, this general problem. And in that way I have so also solved that sort of problems and that sort of problems and that sort of problems and that sort of.

I think, I think we are at this point pretty much out of time, but I, I could go on for quite a while with this. So thank you so much for joining us. Anyone who's watching will be uploading. These documents are going to be made available. Yes, Fred. That people will be able to explore them themselves. So that will be linked to this video later on.

And thank you so much for joining us. This is, this has really been great both to get your perspective on the role of understanding compared to using a particular. One might call it trick, one might call it computational technique, but there's a difference between these sort of techniques and the deep understanding. And then also for sharing this really nice instance of how solving an abstract problem allows you to solve lots of specific problems and then also getting into some very beautiful ideas that visualization plays a really powerful role in.

Thank you everyone who joined us to watch.