This is one of my favourite Maths questions. I encountered it among a list of past interview questions for Oxbridge admissions: how can we get from point A to point B using the edges of the cube, without going over the same edge twice?

I encourage you to try it yourself first, as getting to the solution gave me another glimpse at the beauty of Maths problem solving. And who knows? Maybe you’ll get to a different solution than the one below…

Now, let’s take a look at my process for coming to a solution. I often look at beautiful solutions and think to myself ‘how on Earth did they come up with that?’ So, I’ll show you my exact steps.

First, we need a way of representing the road we take from A to B. Since we can represent an edge by stating its two endpoints, let’s try to label all the vertices of the cube. The natural way to do that is with the standard 3D Cartesian coordinates.

As you can see, A is chosen to be the origin of the coordinate system. Also, for ease of notation B was chosen to be the point (1, 1, 1), which can be done without changing the problem (all we are doing is re-scaling the cube to have side length 1 which doesn’t affect the number of paths).

It was at this point my friend remarked that the cube was too hard to visualise on paper, which made me realise that we don’t actually have to visualise it in 3D. The only information we need is how the points are connected together – ie. where we can move to from a given point.

So, picking a random point say (1, 0, 1), where can we move? Looking at the diagram you can see that there are three points connected to this point: (1, 1, 1), (1, 0, 0) and (0, 0, 1). From here it is not too hard to deduce that any two points are connected by an edge if their coordinates are different in exactly one place (only in x, y or z from (x, y, z)). And we can use this to show the lines in a 2D way as follows:

Next, we need some logic for the way we arrange these points in a diagram. In other words, we need more symmetry in the picture. A cube is very symmetric (48 symmetries in total – can you work them all out?) and so our 2D representation should ideally be more symmetric. This was my second attempt:

…we can do better than that. The central part in the next one is less jumbled.

Now we’re getting somewhere. Do you notice something in the middle? The path in blue looks like a crumpled hexagon – what would happen if we unravelled it?

Much better. I’d say we’re almost at the point where we can start counting the ways to get from A to B, we just have a couple of little problems to iron out…

First, B is not symmetrical in the picture. Sure, you could say this is only an aesthetic thing, but symmetry in Maths often helps us to solve things more easily so let’s try to fix it. One idea would be to add 3 ‘B’s into the picture – one for each of the nodes connected to it. But, thinking about this a little deeper, we see that because the path has to end at B – and once we are there we cannot go back – then in fact B does not need to be connected to the rest of the picture. Instead, we can just draw some EXIT signs at the nodes connected to B. Once you reach one of these points, you have two choices: either “exit” the picture through a B (only one way to do that: move to B from that point), or continue along the hexagon. This observation will be very helpful when it comes to counting paths.

Second, we are not quite sure where to go from A. There is no obvious choice for a starting move. From A to… which of the three? Thankfully, due to the symmetry of the picture (and of the problem, of course), we can simply count the ways to get from A to B with the first move being ‘A to (0, 0, 1)’ and then multiply by 3 to get the result. So, let’s do exactly that!

Given our new-found knowledge, we can make very useful simplifications to the picture. Firstly, the line from A to (0, 0, 1) can be replaced simply by an ENTRY sign at (0, 0, 1), since that is our starting move (which we do in exactly one way). Secondly, since we cannot use that first line anymore, we can simply remove it.

Now things are really starting to take shape. In this form we see that the line through the point A is equivalent to just going from (1, 0, 0) to (0, 1, 0) or the reverse (once we are in A, we cannot go back to where we came from), and so we can simply delete the point A. Also, for neatness, let’s rotate the whole diagram 60 degrees clockwise.

Perfect! I’d say we are now in a position to start counting paths – remembering that we have to multiply by 3 at the end because of the symmetry of the first move. In fact, looking at our much simplified – and very symmetric – diagram, it becomes apparent that we can also count all of the paths that start by going from the ENTRY POINT to (1, 0, 1) and then multiply by 2 at the end. I told you symmetry was going to be helpful!

The following picture should help with the counting:

It perhaps looks a little complicated so let’s break it down. Following all the arrows, or doing it on your own, you can hopefully see that by going through an edge once and starting as decided above (ENTRY to (1,0,1)), we have the following:

• one way to get to the (1, 0, 1) EXIT
• two ways to get to the (1, 1, 0) EXIT (ah, that pesky purple arrow!)
• two ways to get to the (0, 1, 1) EXIT (either only on the hexagon, or using the shortcut).

In total, with the chosen start, we have 5 ways to get from A to B. But, don’t forget we need to multiply due to the symmetries that we’ve used to simplify the problem. First, multiply by 2 (we can go to (0, 1, 1) instead of (1, 0, 1) starting from (0, 0, 1)) and get 10 ways. And then multiply by 3, the number of ways to choose an ENTRY point. Thus, the actual total is 30.

ANSWER: 30 Ways to get from A to B!

Whilst we take a well-deserved moment to pat ourselves on the back for getting the right answer, let’s look at some things to take away from this problem.

1. Even if you are asked about a cube in a problem, you mustn’t get stuck on the idea of a cube. Maths is flexible that way. You can model your cube into any other structure that keeps the important elements in place. In our case, only the ways in which the vertices were connected mattered, not the angles between the lines or anything. Thus, a graph was a fairly good choice.
2. You can sometimes make use of the symmetry of a problem to make your life easier, as we did with the way we counted the total number of paths. Symmetry plays a very important role in Mathematics, so keep an eye out for it.

I hope you enjoyed my solution and best of luck with your own Maths problems! If you want to discuss something, leave a comment below – this place is always open to interesting solutions or remarks.

Announcing a brand new series with the Maths Appeal duo Bobby Seagull and Susan Okereke where we’ll be exploring the GCSE Maths syllabus to show the world that maths is accessible to everyone!

The Maths Appeal podcast features discussions about key maths topics, maths puzzles and interviews with maths champions from the worlds of tech, entertainment, comedy and education. Available on all platforms here.

I recently took part in a Reddit AMA (Ask Me Anything) on the subject of Mathematics. Here’s what we discussed…

tomrocksmaths

Hi all! I’ll be here throughout the day answering any questions you have about maths – whether related to disease modelling or not – and I’ll try my very best to explain things as simply as possible. This should be fun!!

M3CCA8

I have a question about the Banach-Tarksi paradox. The paradox is that you could make duplicates of equal size to the original by separating the data sets by direction. Basically that the sum of all parts is greater than the whole. However i was shown a 3-dimensional visual and in that context i don’t believe the paradox would hold true because of the nature of up/down/left/right requires a frame of reference. There by ruining the methodology that allows the paradox to occur. I was wondering if you could explain whether or not I’m correct in my thinking.

CarryThe2

Banach Tarski absolutely holds with a frame of reference, that is after all, all that co ordinates systems are!

The gist of banach tarski is thst certain infinite chains of transformations don’t preserve volume, it has no basis in the real world because things aren’t infinitely divisable in reality.

M3CCA8

Thank you

tomrocksmaths

I would agree very much with the idea that infinite chains of transformations aren’t possible in the real world. To get a feel for how weird infinity is, just take the simple example of 0/0. It is equal to absolutely anything you want by just taking the limit in a different way. This is not too far from the kind of ideas that are present with Banach Tarski (or at least how I think about them).

BlazingTrail42

Hey! I’ve recently obtained a place at Oxford to study maths & philosophy, what would you say the best prep would be that I could do in quarantine?

tomrocksmaths

Congratulations!! I hope to see you around next year. I can’t speak for Philosophy, but for Maths we have a set of problems that some colleges send to incoming students here: https://www.maths.ox.ac.uk/study-here/undergraduate-study/practice-problems I’d start there and see how you get on.

BlazingTrail42

Fapitalismm

What’s your favourite maths-based fun fact?

tomrocksmaths

I’d have to say the following formula for pi. Take all of the prime numbers and for each one write it as a fraction: prime/(nearest multiple of 4). If you multiply all of these fractions together you get pi/4. I have no idea why…

CarryThe2

Is there some instinctive reason why pi/4 seems to come up more often than pi?

175gr

Sounds like an Euler product. I’m not sure exactly where pi/4 comes from, but that looks like the central value of a Dirichlet L-function.

If I were asked to prove the pi/4 part, I would probably look at the analytic class number formula.

I like this math fact! The tip of the iceberg for a lot of cool number theory stuff.

Fapitalismm

That’s… ok, I did not know that.

Fun fact indeed! 🙂

aeiouicup

Fav mathematician?

tomrocksmaths

Not very original but has to be Euler. That man was a machine. There isn’t a single course I teach (and I teach everything in first and second year undergraduate maths) that doesn’t have a theorem named after him.

tomrocksmaths

Also not really a mathematician per se, but shout out to Richard Feynman just for the charisma. Interesting people make whatever they are talking about more interesting too.

JKTKops

Euler for results, but Galois for story.

CarryThe2

Galois would absolutely be as prominent as Euler if he’d lived longer.

JKTKops

Agreed. I wonder sometimes how much more we would know.

NerdComplex

What advice would you like to give to a 16 year old interested in mathematics?

tomrocksmaths

Practice, practice, practice. Being able to do algebra without making mistakes, being able to recall definitions from memory, being able to work through a wide variety of different style problems – these are all essential for any mathematician. Beyond this, just keep working hard at school and watch videos on channels such as Numberphile, 3blue1brown etc. so that you are exposed to a wide range of topics. Don’t worry if you don’t follow everything – you’re not expected to – that will all come in time.

NerdComplex

Neat! I also maintain a journal filled with goodies from numberphile,Mathigon and futilitycloset (awfully cool website) to keep me engaged when I’m bored. Thank you so much Dr.Tom! Folks like you really help keep the spirit of curiosity alive!

RhynoD

And /u/tomrocksmaths‘ own channel, of course!

Personally, I am also a fan of standupmaths, vihart, PBS Infinite Series, and Crash Course.

tomrocksmaths

I second all of these suggestions. Also singingbanana and mathologer.

NerdComplex

Of course Tom makes great content,but I’m still in high school and don’t understand that level of fluid mechanics;however his recent ventures like the strings and loops in pi and his math essay competition have been of great amusement to me.

CrashAndSideburns

What role do you think proofs play in the teaching of mathematics? Do all things taught need to be rigorously proven, or will an intuitive understanding of a principle, which forgoes rigour, suffice? Alternatively, is there a necessary balance to be struck between explanations in terms of rigorous proofs and of intuitive understanding? If so, what do you think that balance is?

Really interested to hear your thoughts on this one.

tomrocksmaths

Great question – and one I think to which you will get a different answer depending on whether you ask a ‘pure’ or ‘applied’ mathematician, though ultimately there is a balance.

As an applied mathematician, the intuition often comes first. I’m trying to solve a practical problem relating to fluid mechanics, and so I have some physical intution as to what should happen. For example, for my work on river outflows I expect that when there is an increase in the amount of water leaving a river, the coastal current that forms should be deeper. More water in the system means a ‘bigger’ current. But to prove this mathematically you have to start with a complicated set of governing equations (here Navier-Stokes) and then solve them after making various assumptions relative to the problem at hand. The answer is that yes the current depth increases with the volume of water leaving the river (to the power of 1/2 in fact) and so my intuition was indeed correct… in this instance.

I think there’s a lot to be said for first gaining an intuition for problems by playing around with them – visualisations for example are a great tool here – but ultimately you need a proof to be sure the results are correct mathematically. Just learning proofs without first having any grasp as to WHY certain results should be true isn’t very helpful in my opinion. You want to convince yourself that something is true – be it through intuition, examples, visualisation etc. – and then look for a proof. This is particularly true in research level mathematics where most of the time you don’t actually know if a proof should exist in the first place!

Xefoxmusic

Hey Tom, I had the pleasure of working with you in Shanghai 2 years ago, it’s great to see your continued and well-deserved success!

I know you’re particularly keen on Navier-Stokes, so I won’t ask your favourite Millennium Prize Problem. Instead – which Millennium Prize Problem do you think will be solved next?

tomrocksmaths

Hey – great to hear from you! Hmmm… this is a tricky question. I know of recent progress in most of them, but the one that got me most excited was a lecture by Sir Andrew Wiles 2 years ago when he talked about the Birch and Swinnerton-Dyer Conjecture. Whilst he didn’t admit to be working on it himself, he certainly knew a LOT about what was happening in the field. Given it’s close relationship with elliptic curves and his proof of Fermat’s Last Theorem I wouldn’t bet against him solving that in the near future!

NerdComplex

Do you know of any websites like Mathigon? They’re really killing it!

tomrocksmaths

Mathigon is awesome – shout out to the amazing Philipp for creating it! In terms of similar sites, I honestly don’t know. I tend to watch YouTube videos for my online math content, with the occasional textbook thrown in for good measure.

NerdComplex

What’s your way to celebrate math/nerd culture?

tomrocksmaths

Erm tattoos??? I have 58 currently, 10 of which are maths-themed and about 20 related to video games/cartoons.

NerdComplex

Whoaa can I look?

tomrocksmaths

I’ve just entered a photography exhibition at the BRIDGES Math Art Conference where I hope to display some photographs of them – if the exhibit is accepted I’ll post pics for everyone online 🙂

Splatpope

i’d upvote -1/12 times if I could

n00neperfect

Which field of mathematics, equation haunted you mostly, make you confused or seems unanswered now and then? Thanks

tomrocksmaths

If I’ve understood your question correctly (and I’ve interpreted it as which areas of math do I find most difficult) them for me personally it would be anything that is truly abstract, in the sense that I cannot visualise what is happening. When trying to understand a problem the first thing I do is draw a diagram/picture – and I encourage all of my students to do the same. This even works with abstract areas such as Linear Algebra through mapping diagrams etc. But for some areas – and I’m mainly looking at you higher dimensional topology – visualisation is no longer possible and this is where I sometimes struggle to really understand what is going on. It’s not that everything needs to be visualised to understand it, just that for me, it’s a very powerful tool to help my understanding.

n00neperfect

If I’ve understood your question correctly (and I’ve interpreted it as which areas of math do I find most difficult)

yeah right that’s what i mean….. sure the higher dimensions visualizations will be challenging and uncertain. Thanks for response. 🙂

tomrocksmaths

Well that would be an entire 8-lecture course that I teach to my second year undergraduate students… The key thing with Laplace Transforms are they are a tool to turn ODEs (generally hard to solve, particularly if non-linear) into algebraic equations (much easier to solve). The solution to the original ODE then comes from inverting the Laplace Transform. In other words ODE –> LT –> solve algebra –> invert LT –> ODE solution. It’s a VERY powerful tool.

Thank you very much, that really helps me since I’m a first year undergraduate engineering student coming to grips with Laplace transformations

andbm

What is your opinion on the typically platonic way math is taught, i.e. “math exists in the ideal world, not in the real world”? As a scientist, are you opposed to this dualism?

tomrocksmaths

I’d say that’s simply not true. Sure, there are lots of things being studied now that are very abstract and ideological, but i have no doubt they will have uses in the future. Remember that Einstein’s work on relativity had no practical use for over 100 years until the invention of GPS. Now it’s essential for pretty much any location system, as without relativistic corrections it would only be accurate to 6 miles!

andbm

Thank you for your answer. So you would consider that the applicability of math shows that it exists in reality, e.g. that the perfect circle is not only an abstract object but a real object?

It’s mostly a philosophical question. I’m asking from the perspective of theoretical physics, where math is very obviously useful in describing the real world, but the question is whether mathematical concepts, such as the circle, should be considered “outside” the physical reality, as I would argue it is usually taught, or “inside” the physical reality, which would be the consequence of rejecting dualism.

tomrocksmaths

Ah, I get you now… I suppose you could argue that a ‘perfect circle’ will never exist in reality, just like a ‘perfect right-angle’ will never exist (reference to Navier-Stokes video), as no matter what scale you go down to there will be imperfections. In that case then yes I can see why you might treat such things as only existing in the ‘ideal world’. I interviewed the mathematician and author Prof. Ian Stewart about using maths to communicate with aliens and we touched on some similar concepts – I’ll post the link in case you might find it interesting: https://tomrocksmaths.com/2017/05/18/would-aliens-understand-maths/

GhassanZ

Do you think anyone could develop a mind for numbers? Or would most of us work only to make math tolerable?

tomrocksmaths

Oh no Math is 100% a skill that you can learn – just like any other. And just like any other skill, there may well be a limit to what you can achieve. For example not everyone can play the guitar like Tom Morello, but we can all get to a competent level and beyond with enough practice and dedication. I’m actually making a video with Mike Boyd (Quicklearn on YouTube) where I’m going to teach him how to pass the Oxford Maths entrance exam. Mike learns new skills every day on his channel (its very entertaining) and we both want to show that Maths can be learnt just like any other skill. We are hoping to film over the summer so look out for it if you’re interested.

anooblol

Thoughts on how machine learning / AI will impact or change the landscape of pure math?

tomrocksmaths

Ooooooo this ones a good’un… There are already ‘proof writing’ programs that exist and are used to prove certain results, BUT its quite controversial amongst mathematicians as to whether or not computer-aided proofs should be allowed. I interviewed Thomas Hales a few years ago after his computer-assisted proof for the Kepler Conjecture (what is the best way to stack spheres in a box) was finally accepted. I think it took almost 10 years to be ratified and there are some that don’t think it should be allowed.

In short, AI will certainly lead to new results being proved, but whether or not the community deems them acceptable remains to be seen. I don’t see any problem with them personally – if the proof is correct who cares how it was obtained.

NerdComplex

What are your views on entrance exam based institutions?I think the spirit of learning is lost when approaching with competition in mind.

tomrocksmaths

A valid point, but I also counter with the idea that we have to differentiate between prospective students somehow and perhaps unfortunately testing seems to be the best method we have to do that right now. I would also add that as someone who is involved with maths admissions at Oxford, the test score itself is not the be all and end all of an application. We interview people with very low scores if they show potential elsewhere in their application and there are several examples of students being offered places with test scores way below the average mark. We really do consider the application as a while package, not just the test score. (Though this of course is only valid for Oxford where we interview)

NerdComplex

How did you get to where you are today,Dr.Tom?

tomrocksmaths

I guess all I can do is tell my story… I loved maths at school, worked hard, and got good grades. I applied to Oxford and made sure that I prepared well for the test and interview by doing lots of practice questions. Fortunately they offered me a place. In my second year I took a Fluids course and loved it, so began taking more and more options in fluids. When I decided I wanted to do a PhD I asked my tutor for advice and he told me to go to Cambridge. I looked up some of the projects and working in a lab sounded awesome so I met with Paul Linden who then became my supervisor. After my PhD I worked as a science journalist at the BBC for 12 months which is where I learned how to communicate complex ideas effectively. I then set up my own channel and was offered a teaching position at Oxford to go alongside my outreach work.

babahond

What careers might a math enthusiast hope to get into?

tomrocksmaths

Basically anything you want. I’ve been to countless careers talks where employers just talk about how great mathematicians are to employ. Maths teaches you how to solve problems and this is one of the most valuable skills in almost any industry.

babahond

Thanks for the motivation!

6footstogie

the CIO of the company I work for has a masters in math. a few dbas have b.s. in math. many opportunities out there for you.

babahond

I got hope. Thanks brother.

Choib0i

I’m interested in sports betting and the math that goes into developing models people use to try to find an edge. Where could i learn the statistics skills necessary to begin working on my own model?

tomrocksmaths

You want to start with Probability and Statistics. All predictive models start with probability theory as they will need to incorporate some element of randomness into them. Once you have a good understanding of probability then the statistical models will make much more sense.

bishhhhhhhh

Hi Tom,

For someone who was really poor in math in highschool and actively avoided it in university, but now finds themselves in a finance job dealing in an investments industry, where would you recommend getting started?

I truly believed I didn’t have the mind for math all those years but I think my brain has matured and my ability to concentrate has increased, and I’m looking to get my feet wet again. Both in general and specifically more financial mathematics. Any small pointers are greatly appreciated. 🙂

tomrocksmaths

Hey – I’m afraid this is a story I hear a lot. Math is often taught poorly at schools so if you don’t take to it instantly you very quickly lose interest and then you are too far behind to be able to catch up and everything seems impossible… but it’s great to hear that you want to give it a second go. As far as I can remember (I last did financial math in the third year of my undergraduate degree), most of the equations are based around derivatives of one kind or another, which means you need to go back and learn calculus. Calculus is basically the tool we use to calculate ‘rate of change’ and is essential to almost every model that we use today. I’d suggest starting with 3blue1brown’s excellent ‘Essence of Calculus’ series here: https://www.youtube.com/watch?v=WUvTyaaNkzM and see what piques your interest!

SimplySaturday

So I had had a comment here, but accidentally deleted it when I went to edit my comment, in case you think you’re seeing duplicate. Just some general questions on disease modelling.

Since SIR simplifies the complexities of the disease to a conveniently analyzed/numerically solved form, there are definitely more advanced models to capture real world effects e.g. spatial inhomogeneoties/distributions, time-varying recovery/infection time, age related effects, reinfection, complicance, and stochastic nature of the disease. With a limited, but growing set of data, but with complicated population distributions and government mandated social intervention, in your mathematical experience, which of these real world effects are

(1) the most difficult (choose an uncertainty metric I guess) to model/verify with today’s data and what about our data makes this endeavor so hard,

(2) the most crucial (choose a metric I guess) for capturing timescales of breaching specific critical thresholds (e.g. hospital overload, etc.), and how much “better” (again, whatever metric) is it compared to say how the store-brand (SIR) model does

As a follow-up, since most models are usually only good at onset or in the lategame, how much does the existence of endemic diseases (flu, malaria,…) affect applicability of our model when most of the population is recovering? And since most individual data is scarce, how does substituting general population data (like we have) quantitatively affect most of the modern models that probably focus on individual-ish responses?

Gimbeam

I’m so glad you asked this question because my mind was wandering in the same direction but couldn’t find the right people words to ask the question correctly

tomrocksmaths

(1). Spatial distribution can be modelled reasonably well (by adding diffusion to the systems in the form of spatial derivatives) as can incubation time (we actually do this on one of the problem sheets for the Oxford Math Biology course). Re-infection can also be added – you just have some proportion of your infectives, I, go back and join the susceptibles, S, rather than them all going to the removed category, R. I’m not too familiar with the stochastic nature of disease idea, but that seems by far the most difficult to me. Any method to introduce randomness into a model tends to cause things to get very complicated very quickly.

(2). Timescales are tricky, because they are based on the numbers of infectives, I, reaching some critical threshold which is very hard to define. Let’s say for example, the UK has 12,000 beds in intensive care across the country (I think that was the number mentioned by the BBC recently), then we would want to know when the number of infectives will reach 12,000 at a given point in time. This is similar to calculating I-max in the SIR model I discuss in my video, but crucially is entirely dependent on the value of the contact ratio, q. This value is what is SO difficult to determine, because not only is it affected by the disease itself (which we don’t yet know very much about), but also government interventions and whether people follow them, social norms eg. certain cultures share meals whilst others eat alone, immune system response etc. There are almost too many variables to be able to estimate q accurately, but we have to start somewhere.

(3). I’m not sure even the most up-to-date models can focus on individual responses, everything is applied to an ‘average’ person or specific group of people (eg. age 70+, those with long-term health conditions etc.). I think the key thing with all models is to remember the assumptions upon which they are based. They are only tools to aid with decision-making even when they are used properly – when they are taken out of context they are incredibly dangerous. See Black-Scholes and the 2008 financial crash for a perfect example of the misuse of a model.

Spamakin

Any advice for a budding mathematician who isn’t sure if they want to go into industry or academia? I know picking up programming is an invaluable skill but anything else I should know?

tomrocksmaths

Well, as someone who was in the exact same position during my undergraduate degree I can tell you what I did… First, I worked hard and made sure my grades were good enough that I could apply for PhD’s at top universities where I might want to study. Second, I spent my summer vacations doing internships in various different industries to get a taste of what it might be like to work there. I worked for a bank, an insurance company, a consultancy firm, and a non-academic research institution. Turns out I don’t like having a boss and working hours suck, so academia won. But I only knew that because I tried a few different things – hope that helps!

Spamakin

Do you think a major in applied math + a CS minor would be good for going into industry (in any field)? Also any tips for advertising yourself as a math major when applying for internships at like financial institutions or other such industries when you’re competing with people who are majoring in those specific fields?

tomrocksmaths

Math + CS sounds great to me. Also see below response: https://www.reddit.com/r/explainlikeimfive/comments/fn05i0/ama_mathematics/fl7amch?utm_source=share&utm_medium=web2x

oZOV1_L802B

I have a question about the SIR model you presented in your latest video. In Relation to S_0 is it generally assumed to be the entire population say of a given country minus those already infected / immune? Or does social distancing measures in effect decrease S_0 or does it only decrease the transition rate? Or some combination? Thank you I’m trying to understand the dynamics of the system. (Great Video btw)

tomrocksmaths

Glad you enjoyed the video 🙂

S0 is the initial number of susceptibles in a population. So you can think of this as the entire population minus those who cannot catch the disease for whatever reason. This could be immunity (very unlikely for COVID-19), complete isolation (eg. living on a rmeote island) or some other reason. The key thing is that S0 is fixed at the beginning of the model. This doesn’t necessarily have to be the exact moment the disease first appeared, you can starting point at which to initiate the model, but once it is running then S0 cannot be changed. It is a constant value, but one that is already fixed. The social distancing measures will decrease the transmission rate (labelled r in my model). Lowering r, means the contact ratio q = r / a is also lowered and as we see in the video this means fewer infections at any given point in time, and fewer infections overall.

oZOV1_L802B

Thanks for the response and that makes sense. When I was trying to fit the model to data in China I was getting some problems I thought because of the low ratio from infected to total population being somewhere around 80,000/1.4 billion. So I thought decreasing s_0 would account for that but also a low r would also account for very few infections in relation to the entire population at equilibrium. Thanks for the response!

kathryn_grace

Hi Dr. Crawford, Due to COVID-19 all of my classes have been moved online, including calc II. Unfortunately, I don’t do very well in math /science classes without lecture-style instruction. All my professor is doing is putting his notes online, and my brain needs more than that. Are there any resources you would recommend?

tomrocksmaths

Hey! For calculus you can’t get any better than 3blue1brown’s ‘Essence of Calculus’ series: https://www.youtube.com/watch?v=WUvTyaaNkzM to get a feel for the kind of intuition you need to solve calculus problems. Beyond that, it’s tricky as each university will have its own specific syllabus and therefore lectures will cover different content, or at the very least present it in different ways. Ultimately I think you have to stick to the notes provided by your professor and perhaps try to get inspired by some other topics from online videos that will help to increase you interest in the subject, and hopefully make it easier to study.

Macluawn

What’s a short proof that 1+1=2?

Assuming decimal, for simplicity.

tomrocksmaths

Haha I feel like the word ‘short’ is doing a lot of work here… It depends on what we are allowed to assume in terms of axioms etc. In the Oxford first year undergrad Analysis course we teach a proof that the square root of 2 exists which takes a lot longer and a lot more detail than you might imagine… and that’s about 5 or 6 lectures after starting with the basic axioms for the real numbers defined as a field (if you’re interested I made a video on these axioms here: https://youtu.be/9Efsz2hIpxE). So, in short, I would need to know where we are starting from!

Macluawn

In the Oxford first year undergrad Analysis course we teach a proof that the square root of 2 exists

Are video lectures available online to  non students?

tomrocksmaths

Video lectures I’m afraid not. Though this proof is on my list of upcoming video ideas so watch this space.

raindiioannis

How is Euclid’s 5th postulate not self-evident?

tomrocksmaths

Very glad you asked this because it allows me to point you in the direction of an amazing video one of my students made explaining what happens when you don’t assume it: https://www.youtube.com/watch?v=2dUVCswcYF8

raindiioannis

This is great. Thank you. I’m quite bad at maths even though I do enjoy learning more about it and what all of these processes you learn to do in school are supposed to point towards. I got offered a place for the Linguistics program, so if I see you around Oxford in October I’ll thank you personally.

tomrocksmaths

Amazing – all things being well see you in October!

anais9000

I have a question about Fermat’s Last Theorem. Let me phrase it in terms of the cubic. Say we have a cube and two (imaginary) smaller cubes in the upper left and lower right corners. We expand these two cubes at any rate until they touch, then keep expanding them (they pass through each other) until the volume of the cube of their intersection equals the volume of the interior of the large cube NOT occupied by either of the smaller two cubes. Then obviously the sum of the volumes of the two smaller cubes equals the volume of the large cube. So essentially FLT states that these cubes cannot be measured in any unit such that the lengths will be commensurate. My question is, have many of the failed attempts at “simple” proofs attacked the problem from the angle of incommensurate measure, ie, is this an idiot question, not an insight?

tomrocksmaths

This is a really nice way to think about the problem – and one I’ve never heard before so thanks for sharing. My first question would be how we would know the volume that was left inside the original largest cube? As we increase the volumes of the two smaller ones, they begin to overlap, yes, but I’m not entirely convinced that it would be easy to know what that overlap was and whether it was the same as the volume left in the largest enclosing cube? Because as we increase the size of the two smaller cubes, then the overlapping volume increases, and the remaining space decreases. It seems there might be too many things changing for it to be an easy calculation? Also, how do we control the rate of increase of the size of the smaller cubes? To cover all possibilities you would need to fix the volume of one cube and then gradually increase the size of the other until all possible sizes had been exhausted. You’d then need to re-do this for a slightly different volume of the original first small cube and continue in this way, giving far too many possibilities to check.

I could of course be completely wrong with anything I’m saying above – just my first thoughts on what is an awesome way to think about the problem 🙂

EliteKill

Hi Tom! I’m an undergraduate in Physics/Mathematics, and came across your videos with Numberphile on Navier-Stokes following an experiment I performed. The equations fascinated me, especially after I found out they were a Millennium Problem. What similar fields/problems would you recommend studying to someone like me, who is heavily interested in Navier-Stokes?

And another question – my dream is to do a Masters in Oxford, what advice could you give towards a goal like that? What are the special achievements that get people accepted from overseas?

tomrocksmaths

If you’re interested in solving the Millennium Problem then I’d suggest studying courses on PDEs, but if you are just really interested in fluids then Fluid Dynamics based courses are the way to go. Ideally, I’d say do both!

For postgraduate study there aren’t any special considerations beyond needing a very good undergraduate degree with excellent references from your professors. If possible experience of research or other ways to demonstrate your interest in the subject beyond university courses would be very helpful. The key thing though is to get excellent grades in your undergraduate degree.

wendalls

Is mathematics a human invention, or did we discover it?

tomrocksmaths

Great question – and again let me point you to an interview one of my students recorded with Professor Adrian Moore at the Oxford Philosophy department: https://tomrocksmaths.com/2018/10/31/take-me-to-your-chalkboard/

wendalls

Fantastic- thank you

physicsfreak

Hey Tom! Thanks for doing this. I dropped out of a math degree about 10 years ago, but still maintain a healthy curiosity for the subject. Feel free to cherry pick any of my questions!

Possibly a stupidly esoteric question:

I’m wondering if you might know a sort of top-level overview of what the Langlands program is trying to achieve? I never got much past projective / affine geometry but still find myself ending wikipedia-math-binges on Grothendieks(sp?) work and wondering how in the heck anyone manages to learn it. Is Langlands akin to the classification of finite simple groups?

And just because I think it’d be fun:

Do you have a favorite text / author?

Who are the big players in your field? (past or present)

What are you most excited to learn next?

tomrocksmaths

Hey, you’re welcome – some thoughts below.

Langlands Program – I’ll be completely honest and say that despite interviewing Langlands himself about this (https://www.youtube.com/playlist?list=PLMCRxGutHqfmgC5AjHSYYawu6jRSJFE4W) I’m still not entirely sure I have my head around it. From a simple perspective its all about connections between harmonic analysis and number theory – previously thought to be unrelated areas. Rather than repeating what others had said, I found the following article from Alex Bellos at the Guardian very helpful: https://www.theguardian.com/science/alexs-adventures-in-numberland/2018/mar/20/abel-prize-2018-robert-langlands-wins-for-unified-theory-of-maths

My favourite ‘pop math’ book is The Millennium Problems by Keith Devlin – I first read it when I was 16 and am currently re-reading it and loving it more than ever. Otherwise, An Introduction to Fluid Mechanics by George Batchelor is basically the ‘bible’ in my field. It has all of the answers you could ever want and more.

George Batchelor is often described as the ‘father’ of Fluid Mechanics and is the reason the Cambridge Math Department has a lab for conducting experiments. David Acheson in Oxford is also fantastic – he lectured me as an undergraduate and really helped to steer me towards the subject.

I’m teaching a course next term on Special Relativity which I’m super excited about! I remember doing it as an undergraduate but as I haven’t looked at it for a few years, it almost feels like learning it all over again – and the results are still as surprising as ever.

LoveYourLife33

Hi! My preteen & teen kids say some of the math they are learning is useless! Do you have a tip or two, or some obvious fact that would really explain to them why they need to learn complex math, and not just +-* <= > ? Thanks!

tomrocksmaths

Depending on their interests, I’d try discussing math in the context of their favourite hobby. I have a lot of success explaining the math of sport to people – eg. see here for something on the 2-hour marathon: https://tomrocksmaths.com/2017/05/10/breaking-two/ and the math of video games – eg. see here for something on pokemon: https://tomrocksmaths.com/pokemaths/

Finally, you could try my ‘Funbers’ series that I made for BBC Radio here: https://tomrocksmaths.com/funbers-listen/ The idea is to talk about fin/interesting facts about certain numbers, counting up from 0 to 21. We also throw in a few non-whole numbers too to spice things up!

NerdComplex

Who’s your favorite Pokemon and why?

tomrocksmaths

LOVE THIS. I have both Butterfree and Zapdos tattooed on my leg so those two hands down. Zapdoa because it’s so badass and was my favourite Pokemon card I had as a kid (still got it), Butterfree because it is a great Pokemon to train up early and has the powerful combination of sleep powder plus psychic moves.

NerdComplex

That’s love!

octopisces

Okay look, I completely understand that when you multiply a negative number by a positive number the answer is negative. But my question is, why does the tie always go to the negative? Why wouldn’t the number be positive? I understand the rule, but don’t understand the meaning behind it.

I can understand it like this: 3*(-2)= (-2)+(-2)+(-2)=-6

But I don’t understand this: -2*3=???????=-6

tomrocksmaths

I’m not really sure what you are asking because 3*(-2) = (-2)*3 by what we call the ‘commutativity of real numbers’. It is an axiom that we assume is true: for any two real numbers a, b, a*b = b*a. Assuming this is true the statement follows.

Otherwise, you can interpret the second one as 2 lots of 3 and then multiplied by a minus sign. I’m not sure it makes sense to think of ‘minus 2’ lots of a number… Or by my first point we know the order of multiplication is the same so just change it to 3*(-2) and everything works!

helium_ego

How do you feel about the famous -1/12 proof for the sum of all natural numbers? What does it actually mean + how does this solution actually get used in “real” physics?

tomrocksmaths

HA – I was waiting for this one… The -1/12 value comes from something called ‘Analytic Continuation of the Complex Plane’. It is a valuable mathematical tool, but I don’t think anyone intends for it to be used to ‘prove’ that the sum of all natural numbers is -1/12. Without knowing the full set of assumptions under which the theory of Analytic Continuation is valid, I can’t give specifics, but I imagine at least one of them must be invalidated when extending the Riemann Zeta Function to the negative real axis. And as far as I’m aware, the solution isn’t actually used in practice – it’s more of a fun quirk than anything else.

helium_ego

Since you mentioned the Riemann-Zeta function, I’m curious as to 1. Which Millennium Problem do you believe will be solved next and 2. Which one would you most want to have an answer to?

tomrocksmaths

And for 2. as someone who works in Fluid Dynamics I have to say Navier-Stokes! Though it would also be amazing to see Riemann finally solved after all these years…

helium_ego

As a CS guy, I have to say P vs. NP would be great to have an answer to, but I’m afraid we might never get one.

The SIR model is one of the simplest ways to understand the spread of a disease such as COVID-19 (Coronavirus) through a population. Allowing the movement of populations makes the model slightly more realistic and results in ‘Travelling Wave’ solutions.

In this video, Oxford University Mathematician Dr Tom Crawford explains how including population migration modifies the original SIR model. He then goes on to use the results of the model to answer two important questions:

1. How fast will the disease spread?
2. How severe will the epidemic be?

The answers to these questions are discussed in the context of the current COVID-19 (Coronavirus) outbreak. The model tells us that to reduce the impact of the disease we need to lower the ‘contact ratio’ as much as possible – which is exactly what current social distancing measures are designed to do.

Watch the first video on the basic SIR model here.

Congratulations to Gregory (Grisha) Margulis and Hillel (Harry) Furstenberg on being awarded the 2020 Abel Prize. The prize is one of the most prestigious in mathematics and is presented annually by the Norwegian Academy of Science and Letters.

The official announcement states that Margulis and Furstenberg were awarded the prize “for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics” and their work is described by Hans Munthe-Kaas, chair of the Abel committee, as “bringing down the traditional wall between pure and applied mathematics”. So, who are they?

### Gregory Margulis

Born in Moscow in 1946, Margulis gained international recognition aged only 16 when he received a silver medal at the International Mathematical Olympiad. He began his academic career at Moscow State University and began working towards his PhD under the supervision of 2014 Abel Prize Laureate Yakov Sinai. At the age of 32, he was awarded the 1978 Fields Medal for his work on the ‘arithmeticity and superrigidity theorems’, but was unable to travel to Finland to receive the medal as the soviet authorities refused to provide him with a visa.

Another major result followed in 1984 with his proof of the Oppenheimer Conjecture – a problem in Number Theory first stated in 1929. The ideas he introduced here centred on what is known as ‘ergodic theory’ (more on this later), and have since been used by three recent Fields Medallists: Elon Lindenstrauss, Maryam Mirzakhani and Akshay Venkatesh. In 2008, Pure and Applied Mathematics Quarterly ran an article listing Margulis’s major results which ran to more than 50 pages.

### Hillel Furstenberg

Originally thought to be a pseudonym for a group of mathematicians due to the vast range of ideas published in his early work, Furstenberg is a mathematician with a deep technical knowledge of countless areas of mathematics. He published his first papers as an undergraduate in 1953 and 1955, with the latter giving a topological proof of Euclid’s famous theorem that there are infinitely many prime numbers.

One of his key results came in 1977 when he used methods from ergodic theory to prove a celebrated result by 2012 Abel Prize Laureate Endre Szemerédi on arithmetic progressions of integers. The insights that came from his proof have led to numerous important results, including the recent proof by Ben Green and Terence Tao that the sequence of prime numbers includes arbitrarily large arithmetic progressions.

### So, what is ergodic theory?

Ergodic theory relates to probability and what we call ‘random walks’, best explained by thinking about a dog trying to find some treats buried in a garden…

If you hide some treats in your garden and let your dog try to to find them, it will most likely start sniffing in what seems to be an apparently random pattern. However, after a short period of time, the dog will more often than not successfully find the treats. This method of search might not seem to be systematic, but yet the dog is following its instinct telling it to randomly change its direction at regular intervals to maximise its chance of success. You can think of it as moving one step forwards, then flipping a coin to decide whether you next go left or right for one step, and repeating this indefinitely.

In maths, the dog’s behaviour is encoded in the concept of a random walk. A random walk is a mathematical object that describes a path consisting of a succession of random steps in some mathematical space. There are numerous examples of physical systems that are modelled by random walks: the behaviour of gas molecules, stock markets, the statistical properties of neurons firing in the brain… But, random walks can also be seen as a tool to explore a mathematical object, in the same way that the dog tries to understand the garden. Of course, Hillel Furstenberg and Gregory Margulis are not using random walks to find treats in a garden, they do random walks on graphs or on groups in order to reveal the secrets of these objects.

If the trajectory of the dog is ergodic, this means that the dog will eventually get close to the treat in the long term. In fact, if we were to draw a circle around the treat, of any size (even as small as you can possibly imagine), after some finite amount of time the dog will be sniffing inside the circle, and therefore will probably discover the treat. This is ergodic theory in a nutshell.

More information on the Abel Prize announcement can be found on the website of the Norwegian Academy of Sciences and Letters here or in the official citation here.

Oxford University Mathematician Dr Tom Crawford explains exponential growth in the context of an epidemic such as that for COVID-19/Coronavirus. Beginning with one primary infection we see how the number of cases increases dramatically over a period of only 30 days to more than one thousand. All sources are referenced below.

The value of the reproductive number R0 = 3 is taken from current World Health Organisation estimates. Please see here for more information.

The data for the UK population is from 2018 and is sourced from Statista here.

The data for the COVID-19/Coronavirus death rate is from the Chinese Centre for Disease Control and Prevention. Please see here for more information.

A translation of an article about my work in Spanish newspaper La Razon. You can read the original article here.

Mathematics was, as for so many classmates with little numerical capacity, the coconut of my adolescence. In a twisting mortal with pedagogy, my teacher came to suspend me with a 4.9. I always stayed 0.1 to understand algebra and today I can’t survive without a calculator. I am not proud. I wonder if everything would have gone better with Tom Crawford. This Brit is a professor at Oxford, but he doesn’t wear a herringbone jacket or bottle-butt glasses nor is he older than the polka. Tom is an AC / DC math, the punk kid in the bunch. Unlike the old masters, he does not use the ruler as a throwing weapon but, at most, to measure the meters of cloth that is removed from each lesson. He is a “naked scientist”, not as a nod to precariousness but as a seduction pedagogical strategy. “I want to take the solemnity off the math, make it entertaining,” he says.

That goes through a “look” of a hangover rocker with a given shirt, sucks, piercing, tattoos and hair dye. He calls himself “Tom Rocks Maths.” His profiles on networks and his informative videos, in which he ends up posing in leopard-print briefs, have legions of followers. Will it be the solution to my problems? Be that as it may, Crawford was in Madrid yesterday, for the first time in Spain, to give a talk in his own way about mathematics applied to sport. The event took place at the Student Residence, where in 1923 another weird boy, with more clothes and more hair, Einstein, summed up his theory of relativity in an act presented and translated by Ortega and Gaset. The list of visits to that leading institution is as interesting as that of its well-known students: Lorca, Dalí, Buñuel …

The Residence has long become part of a memorial of what it was, but its teaching program continues far from the spotlight, without neglecting the field of science, which seems to have been overlapped when speaking of the Residence due to talent. creative of the boys of Letters already mentioned. Tom Crawford is the last visit and, although we may feel like a histrion or a secondary actor in “Trainspotting”, we must not forget that this is purely an eminence from Oxford.

Photo: Jesus G. Feria

Pi Day 2020 was the first ever UNESCO International Day of Mathematics. To celebrate we made a worldwide collaborative video on the theme ‘Mathematics is everywhere’. You can watch the full video here (I’m at 3:39) or just my contribution below – enjoy!

Here’s a little something to celebrate Pi Day 2020 – originally written for the St Edmund Hall blog.

March 14th is Pi Day, and as of 2020 is also the official UNESCO International Day of Mathematics. You may be wondering what’s so special about a seemingly random day in the middle of March, and if you’re not from the US then you’re completely right to do so. The key is in the date. March 14th is written as 3-14 using the US system, which just so happens to be the first three digits of the number Pi which is 3.14 to two decimal places. If we use the UK system, then we’d need 31-4, or the 31st April, which unfortunately isn’t a real date in the Gregorian calendar. So, March 14th it is.

But why Pi? Even as a mathematician this might seem a random choice of number to represent the International Day of Mathematics, at least until you ask yourself the following question: which single well-known number best represents the field of mathematics? As an applied mathematician (not too dissimilar to a physicist or engineer) my choice would be the number e – Euler’s number. It’s certainly a great representation of all-things calculus (and therefore pretty much any equation in physics), but well-known outside of the mathematical community? I’m not so sure. And herein lies the reasoning behind the choice of Pi. There may be more important numbers (and please do let me know which one you’d pick if in charge), but better-known I highly doubt it. So, Pi it is.

Now we’ve got a better feel for the day named after the number, let’s talk about Pi itself. You may know it in terms of circles, but it has the rather fantastic knack of cropping up in the most unexpected places… Quantum Theory? check. Einstein’s Theory of Relativity? Check. Newton’s Law of Gravity? Check. Three of the most important theories we use to explain the universe, and each of them has a formula containing the number Pi.

Quantum: Heisenberg’s Uncertainty Principle

Relativity: Einstein Field Equations

Gravity: Newton’s Gravitational Constant

This ‘superhero-like’ ability to appear everywhere means that we can have a lot of fun with how we define the number Pi. The standard definition is to use circles: the perimeter of the circle or circumference, c, is equal to Pi multiplied by the distance across the circle passing through the centre or the diameter, d.

We can make things a little more complicated by thinking about the other circle-based formulas that contain Pi and using them to define its value instead. For example, the volume of a sphere, V, is equal to 4/3 times Pi times the radius, r, cubed. Rearranging, we define Pi as follows:

Now, here’s the best bit. Since Pi appears in the formula for the volume of a sphere, it means that any calculation involving the need to work out the volume of a spherical object will include the number Pi, and that means we can rearrange any such formula to get a new definition. Here’s one for you:

Don’t believe me? Here’s how it works…

We want to know the answer to the question: how many ping-pong balls will it take to lift the Titanic from the ocean floor? The idea being that each ping-pong ball floats, and therefore has a positive buoyancy force (don’t worry too much about exactly what that means other than the fact that objects which float have a positive buoyancy force and those which sink have a negative buoyancy force). If we calculate the buoyancy force on a single ping-pong ball, then that will in fact tell us the amount of weight that each individual ball is able to support before it sinks. Think of doing the following experiment with a boat. If the boat is empty, then it will float. As you start to add weight, it moves down lower and lower into the water until eventually you’ve added so much weight that it is completely submerged and sinks down to the bottom of the lake. The total amount of weight that you added up until the moment before it’s submerged is equal to the maximum amount of weight that the boat is able to support. That weight is the positive buoyancy force of the empty boat.

In our boat experiment we can keep track of the weight we are adding to the boat, but with a ping-pong ball it isn’t quite so simple. Instead, we have to calculate the buoyancy force using a neat idea called ‘Archimedes Principle’. We don’t need to worry about the exact details, just that we have a formula for the buoyancy force courtesy of Archimedes (yes, the guy that ran naked through the streets of Ancient Greece). Archimedes Principle tells us that the buoyancy force of an object is equal to the weight of water displaced by the same object. And what this means in practice, is we just need to multiply the volume of a ping-pong ball by the density of the Atlantic Ocean to get the weight that a single ping-pong ball can support:

Volume = 4/3 x Pi x radius3 = 4/3 x 3.14 x 23 = 33.5 cm3

Density of Atlantic Ocean = 1.027 g/cm3

Weight supported by 1 ping-pong ball = 33.5 x 1.027 = 34.4 g

At this point we must remember to subtract the weight of the ping-pong ball itself, to give the amount that we can ‘add’ before it starts to sink:

Total weight that can be lifted by 1 ping-pong ball = 34.4 – 2.7 = 31.7 g

Finally, the weight of the Titanic is 47,500,000,000 g and so the total number of ping-pong balls required to lift it is given by:

47,500,000,000 / 31.7 = 1.498 billion

(I’m aware I’ve taken you on a slight detour from our original topic of Pi, but I hope you’ve at least been able to follow the main ideas of the calculation. If you would like to see more details of how it all works, then you can watch me run through the solution in full in the video below.)

So, with only 1.498 billion ping-pong balls we can lift the Titanic from the depths of the Atlantic Ocean and display it for all to see in a museum. If only it were so simple… Whilst the calculation is itself entirely correct, there’s something we’ve missed. The idea of using ping-pong balls was a real suggestion put forward by a group of scientists in the 1970’s, which needless to say did not happen. If you think you have an idea why get in touch via the contact form here.

This is no doubt a rather surprising and hopefully interesting result, but remember we only got to ping-pong balls and the Titanic because we were thinking about fun ways to define the number Pi. Looking back through our calculations, you’ll see that Pi appeared when working out the volume of a ping-pong ball, and so by putting all of our calculations together and rearranging the equation (like we did for the volume of a sphere) we end up with the brilliant – and my all-time favourite – definition (shown again because I love it so much):

With Pi Day and the first ever International Day of Mathematics fast approaching, why not celebrate the wonderful world of numbers by creating your very own definition of Pi. As you’ve seen with the example above, all you need is a calculation involving circles or spheres and some rearranging to get the tastiest number all by itself. Give it a go and let me know how you get on! @tomrocksmaths