I'm very excited to be here at Northumbria University today, and it's great to see lots of you.
So what do we think about when we think about maths? I think maths is about truth and certainty and unchanging truth, that sort of certainty.
Over time, we get through proof. I think the mathematics is about precision and logic and rigour, and I think the maths is about all of those things.
But I also think the mathematics is about play and creativity and curiosity.
It's about inspiration and frustration. It's about enjoyment.
And I think sometimes when we look at a maths textbook that's not necessarily apparent, we read about Pythagoras theorem,
about the sides of a right angle triangle and we learn Pythagoras Theorem,
which is a fantastic theorem which has been known to be true for more than two thousand years.
That unchanging certainty.
But we don't see the journey that Pythagoras and his colleagues took to finding Pythagoras is there the roots of exploration,
the adventure that they went on exploring right tangled triangles to reach that theorem?
For me, it's really important that mathematics is done by humans. I'm interested in that process by which we do mathematics.
How do we find new truths? How do we explore and come to understand pieces of mathematics?
And I think that as people, we can all enjoy mathematics so we can all have those mathematical adventures.
I think that as humans, we have an instinct for pattern infrastructure.
We are curious about patterns. I think that's something that we have as people, and that's very mathematical.
So for example, I've got here some scarves that I knitted.
I like knitting in my spare time, I made some scarves. I'll tell you a bit more about them later on.
But even without my telling you about the underlying mathematics,
I think we can maybe already just see some pattern straight away and maybe wondering what's different between the two.
So it's sort of blue, the green one at the top of the purple one. And there are some aspects that are kind of similar.
And yet you sort of look slightly different. If you slightly unfocused your eyes, you can sort of see some patterns there maybe be what's going on.
I'm kind of I'm kind of intrigued by be curious about that or also I like crochet recently,
so I'm very excited in the mathematical potential of crochet.
So I made some mathematical crochet blankets and again, without my telling you what's going on.
I think our instincts as humans is to start looking and see, well, what is that structure?
What are the patterns, what's going on there?
So somebody who captured this really nicely, in my opinion, is Frances Sue, who's a mathematician in the US.
Frances says mathematics makes the mind its playground. We play with patterns and within the structure of certain axioms.
We exercise freedom in exploring their consequences.
Joyful as any truths we find on this mathematics for Human Flourishing essay, it's available online.
I strongly recommend. I think it's a beautiful description of what it is to do.
Mathematics and mathematics for human flourishing itself is a phrase I think is really interesting.
So I thought before I tell you any more about mathematics, maybe I'll tell you about my summer holiday because in September,
I spent a week on the Isle of Skye of the northwest coast of Scotland.
And I like going for walks. And for me, the experience of walking up a hill is a lot like going for a mathematical adventure.
For me, there are a lot of parallels between setting out on a journey, and you're not quite sure what you're going to take.
You're not quite sure what you'll find when you reach the top or indeed whether you'll reach the top.
So this is called Ben Chiana vague in the sky.
It's not a very big hill. For those of you who know the Isle of Skye, there were some very serious mountains, the killings.
They're very beautiful. I'm not that serious a mountain walker.
Also, the weather was not fantastic, so I didn't really want to set off into the clouds.
So I picked Ben. She had a bag. So I've got a couple of pictures of this hill from different angles,
and I decided to set myself the challenge of climbing this hill and see what I find along the way.
And for me, this was just the right level of challenge for the day.
It was large enough that it was feeling like a challenge, but I felt like I had a chance of reaching the top.
This is a hill that lots of people have climbs before. There are guides online, so before I set out on my walk, I looked online.
I sort of checked out the path. I got my trusty Ordnance Survey map, so I'd look to the routes and all of those kinds of things.
So I guess some mathematicians are like mountaineering pioneers.
They are the first people to climb Everest or to find a new route up some particular mountain.
That's where mathematical research is about. But I think that we can all have mathematical adventures like me climbing Ben Big.
It doesn't matter that other people have been up there before. For me, this was my adventure, so I had a map and an online suggestion of a route.
What I'd like to do today is to invite you to come on a mathematical adventure with me, and I will be your guide.
So what is this mathematical adventure? Well, I have a particular problem in mind, and to do this,
I'd really like a volunteer to give me five whole numbers between between one and 30, let's say.
So is there somebody who'd like to volunteer to give me five whole numbers, please? Yes. Twenty six.
Keep going, I will five of them. 15. Three.
Seven. Eight perfect, thank you so much.
Bring somebody else, please give me five whole numbers between one and 30.
Yes. Thirty. One.
Three. Eighteen. Twenty seven, thank you very much.
Somebody else, please give me five whole numbers between one and 30. Yes.
13, 13, three. Well, 12.
Twenty three. And 30, thank you very much.
Somebody else, please give me five phone numbers, yes. Two, three, five.
Seven, 11. I feel like I might have come across this number somewhere before.
Thank you very much. Let's have a couple more examples.
Yes. Nine. Fourteen.
Twenty one. Twenty seven. And 11, thank you very much.
And finally, let's have one more set of phone numbers. Yes, right to the back.
Two, four, six, eight, 10. You see mathematicians love patterns.
Thank you very much. So. What do I do?
You give me five numbers.
And then a colour in three of them, and I add them up, hopefully correctly arithmetic in public is no my favourite activity in the world.
And I put the title on the right hand side on those titles have something in common.
Those titles are all multiples of three.
So maybe some of you know this sort of quick way to check whether the numbers are multiple of three, you can add the digits.
So fifty seven, five plus seven, that's 12.
Although I happen to know that 12 is a multiple of three already or I do the same thing, one plus two gets me three, that's a multiple of three.
So you can check that these are multiple three. So each time you've given me five whole numbers, I've picked three that add up to a multiple of three.
I call it the main. So our question, our pension of egg for today, if you like,
is can we pick five whole numbers so that there aren't three that add up to a multiple of three?
And I put it a constraint that that the whole numbers had to be between one and 30, that's because I didn't want you to give me seven digit numbers.
I was going to have to add up in real time in front of you. The problem?
I'm interested in just any whole numbers. So they might be positive numbers. They might be negative numbers, they might be zero.
Maybe I should just address one thing right away because sometimes when I ask this question, somebody says, Well, what about zero zero zero zero zero?
And that's a kind of interesting case to explore.
But if you pick your five phone number three zero zero zero zero zero, I'm going to choose three of them to be zero zero zero, which adds up to zero.
And for the avoidance of doubt, I think the zero is a multiple of three because it's three times a whole number.
So I just want to get that one cleared up right away.
So the question can we pick five whole numbers so that there aren't three that add up to a multiple of three?
And I like this question because it is not obvious what the answer is on the best questions in my experience, that did not have an obvious answer.
So maybe I'll tell you a bit more about my walk while you mull over this problem in maybe your subconscious mind,
my subconscious mind is sometimes a better mathematician than my conscious mind.
I can go to sleep and wake up in the morning with an idea or a problem, or doing the washing up and looking at the garden.
And I have an idea on a maths problem, so sometimes subconscious thoughts is good.
So here's my roots up until a big well, so you can't really see it. This is where I parked my car.
I always think Beach on sleep on the beach is a good sign at the start of a walk.
So I park my car and the route goes behind the houses.
You can just see there and then out along the headland and then you sort of turn up the hill and you can't see the hill very well from this view.
So I'm setting out across the headland, and at this point I see I started to wonder whether this was such a good idea at all.
In fact, I started to think this might not be a very good idea, although you can see some blue sky that was very thick clouds.
What you can't tell from the photos that there was a very strong wind blowing.
And where I parked my car was nice and themselves, and I thought, Well, this is lovely.
And then I walked out along the headland and this very strong wind arrived.
And what you can't see, but I read in the online guide is that when you start going up the hill here,
you end up sort of walking along the edge of a cliff. The ground just drops away to the side.
The wind's blowing this way and the ground dropping way this way.
And I'm starting to wonder whether this is such a good idea and thinking, Can I really do this?
And I tell by you, but I definitely have that experience, but I look at a massive problem.
I definitely have that experience of looking at this thinking. I'm not sure I can do this or even I'm pretty sure I can't do this.
So I'm interested in that. How do you overcome that reaction? My.
I think you need some some sort of belief that you might be able to solve a massive problem in order to engage with it.
And sometimes that comes from experience. I've seen a problem like this. I've got some ideas of what I could do or my friends.
My teacher gave me this problem. They know what maths skills I have.
I can tackle this. But actually, it's an interesting question to think when a mathematician solves a world famous unsolved problem.
How did they have the bravery to embark on that?
Did they set out on that problem thinking, Well, maybe I can do this, maybe I can be the one to reach the top of Everest first?
Or maybe I've talked to some. Some mathematicians told me, well, what I was doing was exploring the foothills.
I had some ideas. I just wanted to sort of play around and see what I could do with those hills.
And then I happened to find this path up to the top and it got me there. But I'm kind of interested in where is that role of belief?
Just believing I can solve a problem won't solve the problem, certainly.
But if I if I don't feel like I believe it, if I don't believe I might be able to solve the problem, how do I engage with the problem?
One source of thinking I've done on this is around the role of mindset, so I'm sure some of you have come across the work of Carol Dweck,
who is a psychologist in Stanford,
and she and her research team did the initial research and distinguish between what they called a fixed mindset and a growth mindset.
So she says here a growth mindset is about believing people can develop their abilities.
So do I believe that I have a fixed capacity for mathematics and whatever I do, that's my level in mathematics.
Or do I believe that that can change somebody else's work?
So this is Joe Bhola, who's a professor of mathematics education at Stanford and has looked at this a lot in the context of mathematics.
And Joe Paula says people with a growth mindset are those who believe that smartness increases with hard work,
whereas those with a fixed mindset believe that you can learn things, but you can't change your basic level of intelligence.
So my understanding of this is not that if we all work hard enough, we can all win Olympic gold medals,
we can all when fields, medals kind of top prises in mathematics that we can all climb Everest.
I don't think that's what this is saying, but I think this is saying that if we work hard, we put in the right kind of work.
We take on board feedback or we can develop, we can all get better. And the research that people like Harold Récord Joe Paula have done suggests
that people with a growth mindset do better than people with a fixed mindset.
And I think that makes sense in a mathematical context.
If you look at the problem and you think, I'm not sure about this trying it, having a go feels like taking a bit of a risk.
And if you try a maths problem and you don't solve it and you have a fixed mindset,
you might take that as evidence that your level of mathematical ability wasn't enough for that problem.
Whereas if you have the growth mindset and you try a problem and you don't manage to solve it,
you can use that as a learning opportunity to say, Well, what can I take away from this?
And I think that's a really interesting kind of distinction. So I don't think that people are fixed mindset people or growth mindset people.
It's not like everybody has one of these labels.
I don't think it's even as simple as in the context of mathematics as a place to play the piano or playing football or being an artist or whatever.
I don't think it's as simple as I have a growth mindset about maths or fixed mindsets about maths.
I think lots of us have a bit of both. But sometimes when I'm working on something and I'm finding it difficult,
I just kind of just take a moment to think, OK, I want to have this sort of growth attitude towards this.
So let's go back to this problem with our five hole numbers, so I've got a few more examples here,
but one of the things I really like about this problem is there's something concrete to try.
Sometimes when I look at a maths problem, I think I don't really have any ideas. Where do I start?
One of the things I love about this problem is that we can start by just trying some examples.
And the question was, can we find five whole numbers so that there aren't three that add up to a multiple of three?
What we could just test out some more examples. You gave me some numbers.
It seemed like we could always find three numbers add up to a multiple of three.
We could just keep testing some numbers and try to get a bit of a feel for what's going on.
And one of the things is the way I phrased the problem doesn't tell us what the answer is.
So there are sort of two possible ways out of this problem.
One way out of this problem is, yes, there are five numbers so that you can't have three months for three.
Here is an example of such a set of five numbers. Another way out of the problem would be No, it's impossible.
Here's a proof that whatever five numbers I pick, I can always choose three that add up to a multiple of three.
And the question is, though I phrased it doesn't tell us which it is,
which is much more representative of the world of research mathematics than if I just sort of said,
Well, let's show that the answer is this or show that the answer is this when we're doing research, maths and mathematics, we don't know the answer.
But those questions where you don't know, am I trying to come up with an example? Say, You can't do this?
Am I trying to find a proof? It can be hard to know what to do. And I was reading a book by the mathematician a bit little.
Littlewood was one of the great British mathematicians of the first part of the 20th century,
and Littlewoods expressed exactly this question in the context of research mathematics,
he says complete indecision between yes and no in an exciting new problem is agonising when you go all out one way, either yes or no.
The thought keeps nagging that it's an even chance that it ought to be the other way.
The difference when you do know when, for example, we are looking for a new proof is enormous.
And I think mathematical history bears this out. Sometimes you have this really difficult question a lot of people have thought about.
Finally, somebody sold said, and we know this thing is true because we've got a proof.
And then there's a flurry of more proofs over the coming years or decades.
Somehow, you know, for sure, it's true because there's one proof and that kind of liberates you to find other ways of thinking about the problem.
Whereas while I'm not quite sure it can be really hard to fully commit to, I'm going for it.
I'm going to try to find the proof. Whatever it may be. So I think that in maths, sometimes we are very fixed on getting the answer to the problem.
Maybe especially in kind of educational context, there's a lot of pressure. I've got to get the answer to the problem and that makes a lot of sense.
But I think that we could lose sight of the process by which we get to the answer.
And I think that the process there is a really important part of doing mathematics.
Maybe, maybe I don't get the right answer, but maybe I don't quite reach an answer at all.
But maybe I've still led to a whole lot. Maybe I've enjoyed being curious about mathematics.
Maybe I've being creative in my approach, taking making the most of the process regardless of the outcome.
So I guess that feels to me a bit like going for a walk up a hill.
The goal is not to be at the top of the hill. If the goal was to be at the top of the hill and I had a whole lot of money,
maybe I'd just get somebody with a helicopter to just fly me in and drop me at the top of the hill.
But that wouldn't do the job right.
It's not the same as the struggle of trying to climb up the hill and the winds trying to blow me off the cliff and try to fall off,
not fall off the cliff. And I try not to get lost in the fall and all of those struggles.
That's part of the process, and I enjoy the walking up. I think those of us who like [INAUDIBLE] walking kind of enjoy that.
So I thought I'd take a couple of pictures of things I enjoyed as I was walking up the hill.
There's an extinct volcano and this one on the islands next door. I thought that was an exciting moment.
I found a big furry caterpillar who doesn't love a big furry caterpillar.
I'm enjoying the moment. I don't have a mathematical analogy for Big Furry Caterpillar.
I'm not quite sure how to translate that. So just embracing the process for a moment.
You gave me your five numbers. I didn't have prior knowledge of what those were.
I typed them into my computer and fairly quickly I could find three that added up to a multiple of three.
That is not because I am amazing at mental arithmetic, I am not amazing with magic, I am not terrible at it, but I'm not amazing at it.
The slow bit of the process was me acting up the three numbers,
but I have since added up all of the other combinations of three numbers to find once the work.
So how was I checking those examples quickly?
So I want to illustrate this with pictures of bars of chocolate.
I like this way of thinking about it, so I've got three pictures here representing three types of numbers.
So the picture on the left. Without counting the squares, I can tell the remainder when I divide the number of squares by three.
Right, because their columns are three, I've got three rows I can see the thing on the left is a multiple of three at least remained two zero.
When I divide by three, I don't have to count the squares and the one in the middle.
These are made to one when I divide by three and the one on the right.
These are the two when I divide by three, and every type of number in the world is one of these three types.
For the purposes of this problem, I don't care how many columns in this bar of chocolate are only care about the remainder.
So what I'm doing is I'm taking them notes you're giving me and I'm replacing them by zero or one or two.
And I'm looking for patents within those zeros and ones and twos. And that makes the numbers much more manageable.
Small, I can cope with zeros and ones and twos.
So for example, if I had these five numbers, I've just replaced by zeros, ones and twos that I've picked out the first three.
And I've got one of each type here and I see looking at the pictures of bars of chocolate.
We can see that if we've got one of each type,
we're going to be able to use the remainder one square and sort of fill in this gap and make it multiple of three.
That's a really useful strategy that gives us a way of tackling this problem that feels much more manageable.
How many possible combinations of five whole numbers are that in the world?
Well, like loads. I mean, officially infinitely many. There are a lot of whole numbers in the world.
I can't check them all with this way of thinking.
This problem is a manageable kind of small problem now because I'm only I only need to think about zeros and ones and twos.
There's no point in my writing a million. I'll replace it by one. So let's just focus on zeros, ones and twos.
This is like a really good moment. Whichever roots of the problem we're trying to go for.
So if we're trying to find an example of five whole numbers, this gives us a way of kind of focussing in and looking at zeros, ones and twos.
Just sort of think about that if we try to prove there's no combination of five numbers.
This might give me a use where thinking about it. So I feel like this is a good moment.
I feel like we should feel happy with ourselves at this point. Just a cautionary tale, though.
So this is the mathematician Julia Robinson, very distinguished mathematician in the US and 20th century and somebody I think an administrator who had
sort of file a report or something asked her one day to describe her typical week as a mathematician.
So Julia Robinson described her typical week, like this Monday try to prove theorem.
Tuesday tried to prove theorem. Wed try to prove their.
Thursday. Try to prove theorem. Friday.
Theorem false. Somebody told me once I shouldn't use this quote because it was too depressing.
I don't find this quite depressing, and the reason I don't find it depressing is that Julia Robinson was a really very distinguished mathematician.
And if somebody is distinguished as Delia Robinson could spend a week trying to prove something that turned out not to be correct,
that makes me feel a whole lot better about all the times I've spent in my life trying to prove things that have turned out not to be correct.
That is not time wasted, Littlewood said. There's this problem of, well, if I try to prove it, the answer is yes or no and.
Julie Robinson reminds us that we might spend a week trying to predict the answers, yes, and it turns out the answer is no.
Or vice versa. Another mathematician I was reading about recently.
Turns out, also sees parallels between mathematics and mountain climbing.
And I only discovered this quite after I planned this talk. I was really happy to find this.
So this mathematician says there's a strong parallel between mountain climbing and mathematics research.
Well, first attempts on a summit are made. The struggle is to find any route.
Once on the top.
Other possible routes up may be discerned, and sometimes a safer or shorter route could be chosen for the descent or for subsequent ascents.
This is all my walk up and down arc. I thought I'd look back. Bottom here somewhere.
The coolants, these famous mountain range,
that's the dark shadow in the corner you can't see because they're in the clouds, just to put this in perspective.
So this quote continues in mathematics. The challenge is finding a proof in the first place.
Once found, almost any competent mathematician could easily find an alternative, often much better or shorter proof, at least in mountaineering.
We know that the mountain is there and that if we can find a way up or reach the summit, we should triumph in mathematics.
We do not always know that there is a result or if the proposition is only a figment of the imagination, let alone whether a proof can be found.
So again, I don't want to be depressed by this, Kathleen all and so really interesting mathematician.
Amazing person. I recommend going and reading about how if you haven't come across how previously
she was also a very serious mountaineer in the way that I am not safe.
But this idea that you could spend time trying to find a route up a mountain only to discover the mountain is not there.
How do you have the bravery to just go for it? I think that's a really interesting question.
I don't really have any good answers. I just think it's an interesting question.
Let me go back to these bars of chocolate for a moment, because these bars of chocolate are a way of visualising something called modular arithmetic,
which is one of my favourite ideas in mathematics, is immensely powerful.
So you might have come across it in mathematics or computer science, coding, different kind of contexts.
So here we're working March three and we're saying every number is a multiple of
three or one more than a multiple of three or two more than a multiple of three. And this turns out to be an extraordinarily powerful idea.
It's really useful tool for thinking about all sorts of things in number theory and cryptography in other contexts.
So one nice thing that you can do with modular arithmetic is you can multiply. So I did when I was at primary school,
maybe you had to do this to see these times tables you go one two three four five six one
two three four five six and you fill in all of the kind of times tables of the Great Times,
table practise or something. So it's kind of interesting to do that and the world of modular arithmetic.
So in the world of Model three, there are only three different types of numbers.
So that's why I've got a three by three square. So I got two two zero one two zero one two just fill in the Times table, so zero times zero is zero.
If I take multiple three multiplied by multiple three, I get a multiple of three zero times.
One is one zero two, I get zero times, two is zero one time zero is zero.
One times one is one one times to is. This still feels quite familiar to times zero zero g times.
One is two. Two times two. Sort of is full, but we're thinking in a more free world, that means it's the same as what if I do it three times to,
I guess, a number of the middle type with these three pictures here. So I'm going with a one here.
So this is like a multiplication table and the Model three world and I got up one day decided what I need to do is make a crochet version of this.
So here is my crochet multiplication table. What three?
It seemed like a good idea at the time. What is less clear to me, in hindsight,
is why it seemed like a good idea at the time to make multiplication tables in crochet mode two three four five six seven eight nine 10.
So each square is about this big. You know, each of the nine here is about this big.
It takes me between 10 or 15 minutes to make a square.
But I did make quite lots of them, and those pictures I showed you earlier on are crochet multiplication tables.
So I think the one on the left is more seven and the one on the right is more eight.
And what I love about this visualisation is that we can start to see differences fundamental structural differences between multiplication,
WhatsApp and multiplication mode age. So it's slightly tricky to say.
I'm afraid the Greens have sort of slightly merged in colour has a slight deficiency,
but there are differences in pattern like in the mode seven table.
But there's this sort of Sudoku property.
So apart from all the zeros around the top and the left hand column, each row in each column has all of the numbers in it.
This is a bit like a sea stock that doesn't happen on the right, on the right.
There's a row, for example, that just alternate 040 for that sort of green green, green green.
It's something fundamentally different that turns out to boil down to the fact the seven is prime and eight is not.
So there's lots to explore there. And I'm not going to go into all the details now.
But I love the visualisation and I say modular arithmetic helps us make sense of the scarves I showed you earlier on.
These are my prime distribution scarves. So the deal is if we look at the top one, the top one, there are six rows on this stuff.
We're looking at the numbers, what six?
And you can be a white number with a green frame or green number with a white frame and the white numbers with green frames and not prime numbers.
And the green ones with white frames are prime numbers.
So the prime numbers, the individual numbers, the numbers that can be divided by themselves and by one, but not by anything else.
So if you start from this end and count down the rows,
you go one two three four five six and then next column seven, eight, nine, 10, 11, 12 and so on.
And you can see two three five seven 11, which somebody helpfully picks as an example earlier on our prime numbers.
The bottom one same story, but I've got seven rows instead of six.
So the purple scarf is working more seven. The green one, which is blue in real life, is working.
What? Six? And what I love is that I can start to see differences again.
What six? Apart from two and three, all the primes are lining up in the top row in the fifth row.
What this scarf illustrates is the gorgeous fact that apart from two and three,
every prime in the world is one more than a multiple of six or one less than a multiple six.
Well, the bottom scarf illustrates, is this a whole lot more complicated note?
Seven. So if the bottom row, apart from seven, has no primes because there are no prime multiples, the seven apart from seven,
but the primes the kind of scattered throughout you might see sort of diagonals in the bottom one.
I don't know whether you can sort of see these diagonal stripes, the problems.
Those are, in fact, the same stripes from the Watch six one just kind of staggered because we're working about seven.
So that was a little kind of diversion into the, well, too much of the arithmetic,
which I love for its mathematical, mathematically intriguing properties is mathematically powerful.
Properties has lots of applications, too, so let's try going back to our problem about our five hole numbers and see,
can we use this sort of zero one two way of thinking this mode? Three way of thinking?
So here's the numbers that once you gave me, these are the examples that I did afterwards.
So what I can do is take each of these and replace each number by zero or one or two to reflect is
it a multiple of three or one more than a multiple of three or two more than a multiple of three? And then?
Well, I don't know about you, I find that very difficult to read because it's got a whole load of numbers crossed out,
so there's a tidied up version with just zeros and ones and twos.
And. Maybe I start to find some patterns, so things like in the not the bottom rope, but the one above, I've got three zeros.
I've got three numbers that are multiple of three.
I love it when people give me an example with three numbers in about four three because I know if I add three months,
wheels see three, I get a multiple of three. So you give me three multiple three.
I'm very happy. The bottom one, we've got one of each type.
And we said before, if we put one of each type that going to add to give a multiple of three.
What else have we got? So the fourth wrote down, We've got one one on one.
I've got three numbers the rule one more than the multiple of three I can use. All of those one leftover squares make another column.
I get a multiple of three. I think the top row I've got three numbers.
There are two more than a multiple of three. So what I can see is that there are some particular combinations of zeros, ones and twos.
And if I find those amongst my five numbers there, zero, one or two.
I'm in good shape. So if I have three multiples of three, we're done.
If we've got three numbers there, one more than a multiple of three. We're done.
We got three numbers that are two more than a multiple of three. We're done. And if we got one of each type, we're done.
This feels like a whole lot of progress I think I did about you, I'm feeling like we're pretty near the top of the hill at this point.
So let me show you some pictures of being near the top of the hill. This blue sky, there's sunshine.
The wind is blowing a gale off chasing the path, not right next to the cliff,
but the path a bit farther in so that I don't fall over the edge, but I'm getting near the top.
This is a good feeling. It's just there. So what's going on with us?
Zeros, ones and twos? We got to pick five numbers that it will zero, one or two.
And if we find one of these four bad combinations that we know that we're done, we've got three numbers to add up to multiple three.
So I can just check, right? So here's a first bunch of checking.
So I try to be really systematic about listing my five types of numbers.
So starting with zero zero zero zero zero. So I've got five numbers through a multiple of three and I'm working through and
I couldn't fit them all on the slide and still have slightly readable font size. But here's a first chunk and then I look at these, and for each one,
I have like one of these bad combinations and I'm a very patient person and I did this for you.
You can say thank you later. And each row I've been able to colour in.
So each row here I've been able to find three numbers that add up to a multiple of three.
And that's great. So how many combinations are that?
Well, there are three possibilities for the first column and three for the second of three for the third and three.
So that's three times three times, three times three times three. That's two hundred and forty three possibilities.
I've managed to fit seventy five on here, at least another hundred and sixty eight to go.
I do it by you. I'm not very excited about the idea of checking another hundred and sixty eight cases.
It wasn't that exciting checking those seventy five.
So I feel like I can solve the problem because I feel like if I can contain my just be patient for long enough,
I check my next hundred and sixty eight cases.
And I bet based on those seventy five, I will be able to find three numbers add up to multiple three and then I'll go,
Oh no, it's not possible to find five fold numbers so that there aren't three that add it to multiple three.
But I feel a bit flat about the whole thing because I mean, yuck.
It's horrible, isn't it? Just checking all those cases? I don't feel like I've understood what's going on behind the scenes.
I feel like there is some underlying mathematics here, and I just have kind of tapped into that, so I'm not so keen on that.
And actually, I go up to the top of the hill and it turns out it wasn't the top of the hill at all, and that's really kind of frustrating as well.
So, yeah, we've still got a little way to go. And I don't really have a better way of summing up this up apart from whatever this emerges.
So now I'm fed up because I'm not at the top of pantheon of egg and I've got a maybe way of doing the problem, but it's not very nice.
And let's just take a pause from this and let's have a digression.
This is really important to maths. Sometimes you have to know I just need to stop thinking about this problem for bit.
I just need to think about maybe another massive problem.
Maybe I just need to go for a walk, or maybe I need to go and have a cup of tea with my friend or whatever it may be.
And there's a sort of fine balance between I have to spend long enough persevering on the problem.
I have to kind of persuade enough for my subconscious to kind of absorb everything it can about the problem.
But also, actually, I've just hit the point of frustration. I'm not making any progress.
It's going to walk away. So let's have a digression. OK.
This is maybe not the most exciting facts in the world, but one is equal to one squared.
But if I do one plus three plus two squared and if I do one plus three plus five, that's three squared.
And if I do one plus three plus five plus seven, that's four squared, we're seeing a pattern.
So I just keep going and it seems like if I add up the first half of many odd numbers,
I got a square number about a kind of nice patterns that I mean, you know, why not?
There's a nice pattern. So I think I could go all the way up to five where I ran out of space on the side.
So one plus three plus five plus seven plus nine plus 11+, 13 plus 15 plus 17 is nine squared.
And at this point, I kind of I'm willing to bet that if I do one plus three plus five plus all the way up to 19, I get 10 squared,
but I run out of room on the side and I didn't really feel like doing the calculation, so I thought I'd arrange my crochet blankets instead.
Because having gone to the trouble of making them, why not?
So one is this green slash blue square in the bottom right hand corner because I didn't make it one by one square.
And look, here's three the remaining squares in the two by two square and then his five, which are the squares.
We can still see them three by three square and his seven, a nine, 11 and 13, 15 and 17 and 19 and hollow adds up to 100.
Isn't that lovely? That's not remotely my idea. I'm not sure whether anybody has been silly enough to make it out of crochet blankets before,
but I just love this idea for the beauty, for the elegance.
I can see why the result is true and that for me, it's a sign of a really lovely mathematical argument.
There are lots of ways to saying that this is true. And some of them are nicer than other ways.
For me, beauty is an important part of my work as a mathematician. So one person who famously wrote about this was a mathematician, James Hardie,
who like Littlewoods, was one of the leading mathematicians of the first part of the 20th century,
in fact, and which worked together a lot as a very famous collaboration and hardy in his book, Right?
A mathematician like a painter or a poet is a maker of patterns.
The mathematicians patterns, like the painters or the poets must be beautiful.
The ideas like the colours or the words must fit together in a harmonious way.
Beauty is the first test. There is no permanent place in the world for ugly mathematics.
That's a pretty controversial statement. I think there's a whole bunch of ugly mathematics and probably does have a prominent place in the world.
I think possibly this is a slightly extreme view. But for me, parts of the role of beauty is in helping me guide me towards a particular approach.
If I'm looking at the [INAUDIBLE] I'm, I can see the way.
If it involves climbing across a lovely green, smooth, grassy bank and the way that involves fighting through the bushes,
it's the green smoothies bank all the way I want to have.
I want to choose the beautiful route if I can, if I'm working on a problem and I can see different lines of attack.
My instinct is to go for the one that I find aesthetically pleasing first.
And maybe it doesn't work. Maybe I have to come back and try another tag.
Or maybe I find an ugly solution, and then I try to find a nicer one later on.
Like Kathleen, other and show said somebody finds a proof. Maybe you can then find another way.
But but beauty can be powerful. I think beauty is a subjective in mathematics as it is in art or music or landscapes or cityscapes or whatever else.
We might find beauty.
So I don't think it's the case that because mathematicians don't all agree about what's beautiful, the mathematics doesn't count the speed to.
For me, that sort of feeling of what is beautiful mathematics feels a little bit like,
what do I think of when I think of a beautiful piece of music or a beautiful landscape?
I sometimes worry that we might use beauty to exclude people from mathematics, and it's really important to me that we don't do that.
If you look at a piece of mathematics and somebody is telling you it's beautiful and you don't think it's beautiful,
that doesn't mean you're not a mathematician. Different people have different tastes.
Also, I can't find a piece of mathematics beautiful until I've understood it.
And I mean, spent quite a long time understanding it before I can appreciate the beauty.
But I think the beauty is important in mathematics. One of the things about maths,
so maybe you have this experience is that you work on a problem for a long time and you don't really feel like
you're getting anywhere and you've tried writing out seventy five combinations of zeros and ones and twos.
So this is not really kind of doing it for you. Oh, I don't know what to do sometimes.
Sometimes the answer comes at just one of those moments of, Oh OK, I see what to do now.
It's not always like that. Sometimes it's really slow and incremental. Sometimes you have that moment.
So the question you're working on, let me just remind you,
was can we pick five phone numbers so that there aren't three that up to a multiple of three?
We haven't yet found an example, and we tried a bunch of examples and we've systematically checked 75 cases and it seems like we can't.
And the way we were doing this was we think about zeros and ones and twos, so I can think about having five numbers that will zero, one or two.
I'm picking out these bad combinations. If I've got three that are multiple of three, I have three that add up to three and so on.
So if I take five numbers that it will zero or one or two.
What might happen? Well, I might have a zero and one and a two, I might have all three types,
and if I have all three types, it's game over because that was one of the bad combinations.
So if I go two zero and a one and a two, those are going to adds up to give me a multiple of three.
We can stop right away. What happens if we don't have a zero to one and a two, so I don't have all three types represented amongst my five numbers.
So that means we've got almost two types in them, but maybe they're all the same type.
Maybe that's split between two types, but we've got a most two types of number. So we got five numbers split between these two types.
And that means we must have three that are the same.
If I've got five numbers that fall into two categories, I must have at least three in the same category.
And that's the other type of bad combination. So whatever five whole numbers we pick that are always three adults, a multiple of three,
and I don't know about you, but I think that's a really lovely argument. And I think it's a much more lovely argument than let's check 243 cases.
So that's one of those moments you think, Oh yes, that satisfaction when you reach the top of the hill?
That's a good feeling. There's a trigger point. It was properly the top of the hill and how many coins to the top of the hill.
I wanted to say for the moment, I wanted to take some photos. I was taking these photos on my phone one handed with my other hand,
hanging on to the trigger point because the wind's trying to blow me off the hill. I'm not really into selfies,
but I thought I should take a selfie to prove to you that it was me who climbed to top of the hill and also
to demonstrate that this walk the winds meant this walk really was more hair raising than I had expected.
So at this point, I'm thinking, OK, where next? What I'm thinking is I got to climb down this hill very carefully, not being blown off the cliff,
and I'm going to get my car and I'm going to find a nice, warm cafe where next with our maths problem?
Well, we could think about varying and ways.
Some mathematicians do this all the time.
We solve a problem and then we're like, Oh, OK, now I can ask all these other questions so we could think of.
The problem we just talked about is take three from five. I pick five numbers. Are there three that add up to a multiple of three?
So we could think, well, what if we do take two from three, if I pick three numbers, are there ways to the add up to a multiple of two?
That's a really nice question. Or this is me planning my more adventurous clients in the crew islands when the weather is not covering them in cloud.
What could I do? Take four from something?
Are there some number of numbers so that if I pick four, I can pick four of them that add up to a multiple of four, or I should keep going.
I could just kind of keep thinking, Well, there's lots to explore here. I have many mountains to climb.
We have many mathematical problems to climb.
So I don't think that having a mathematical adventure needs to be restricted to people whose job title is research and mathematics.
I don't think that mathematical adventuring has to mean being the first person
to climb Everest or being the first person to reach a particular summit.
I think that everybody can enjoy the experience a mathematical adventure can enjoy the play and the creativity and the curiosity along the way.
So I hope that perhaps you might be inspired to tackle your own mathematical adventures.
I just want to say thank you to my friends and reds for introducing me to this problem.
Take three, for five and for letting me share it with you today. I love this problem.
It's such a beautiful problem. And I also wanted to show you my complete sets of crochet blankets just to give you some more patterns to think about.
Thank you very much.