Oxbridge admission question: how many paths are there between opposite corners of a cube?

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?

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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.

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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:

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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:

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

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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?

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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.

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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.

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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:

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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.

The original version of this article can be found here.

Vlad Tuchilus

La Razon: Math with Rock

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

El Confidencial Interview

A translation of my interview with Spanish newspaper ‘El Confidencial’ discussing my approach to presenting maths as the solution to everyday problems. The original interview (in Spanish) with Guillermo Cid can be found here.

This teacher knows how to shoot the perfect penalty: “The secret is in the numbers”

Doctor of applied mathematics Tom Crawford has spent years researching and demonstrating how numbers are much more than theory and can be key to our day to day

It is easily seen and is unquestionable. Tom Crawford is not a mathematician, and he knows it perfectly. His image is far and away from those ideas of the typical serious, boring, number-focused expert with squares in all his aspects of life, and it’s not a coincidence. This Englishman, a professor at Oxford University and a doctor from Cambridge University since 2016, is a loose verse in the sector and focuses all his work on proving it . For what? To teach everyone that mathematics is not just theory and paper and that it is present in all aspects of our lives.

With these ideas he has become a famous popularizer in his own country, participating in all kinds of radio programs from stations such as the BBC, and he even has a YouTube channel where he teaches mathematics in a different way. His stage name is Tom Rocks Maths and he is known as ‘the naked mathematician’ because he makes many of his videos without a shirt and even without pants.

This week Crawford is visiting Spain with an event at the Madrid Student Residence where he will talk about one of the aspects that has given him the most success, the relationship between sports and mathematics, and he has been talking with The Confidential on his entire career and, especially, on how the world of sports is intertwined with numbers.

Fan of soccer and of players like N’Golo Kanté or Roberto Firmino, assures that mathematics is leading the human being “to overcome his limits” and that it has been shown that they are a differential point in disciplines such as soccer, but without humans behind it nothing makes sense. “Mathematics is not magic, but a tool that we must know, understand and apply for our benefit.”

Photo: Tom Crawford.

Q: Professor of mathematics at the University of Oxford, doctor of applied mathematics, popularizer … Why have you decided to give a talk on the relationship between mathematics and sport?

A: I love doing sports and following it, and I also love math, so I decided to join both fields. My favorite sports are soccer and running, and in those disciplines I focus research and talk. But well , the main thing is that they are very followed and practiced sports and that they have a clear relationship with the world of numbers. Talking about them, it is very easy to demonstrate how ‘mates’ are present in everything and are very relevant to our day to day. It removes the idea that it is only theoretical and that you learn almost by obligation.

Q: Today we have the cases of Eliud Kipchoge or some soccer teams that are clearly committed to technology and science, with mathematics very present, to improve their brands or achieve greater success. Do you think that there will be a limit in which these disciplines can no longer help us and the human being stops breaking records?

A: It is an interesting matter. For example, if we look at the evolution of athletic records in the last 20 years, we see a graph in which there is a constant and very steep drop in marks. Suddenly, in the early 2000s, disciplines such as mathematics began to come into play and the consequence was that records fell at a dizzying rate, also driven by improvements in training, in nutrition, in scientific research, in the professionalization of the industry … That yes, that occurs until a few years ago, and it is that this fall is stopping.

This, in my view, means that we are also reaching a new limit in progression. Come on, it is already difficult to continue breaking current records and you only have to see the case of Kipchoge and the two hours of the marathon. I do not know how far we can continue to improve, although mathematics could end up giving us a prediction, but I do believe that there will be a time when we will not be able to continue breaking more records. I do not know, it is impossible to think that a person can run 42 kilometers in an hour, for example, no matter how much scientific and technological knowledge is used.

Photo: Tom Crawford.

Q: In football we see more and more teams and clubs that invest millions in ‘big data’ and other knowledge to improve their performance, is this key for a team as well as in athletics?

A: Yes, I think that investment in these areas can be key to improve a team, to study new players, to see the performance of the squad … Of course, without the intervention of a good human team this is useless . The thing is not only to have large volumes of data and good analysis programs, you need people who know how to interpret that information and can also analyze it and make decisions about what they find.

For me a perfect example is that of N’Golo Kanté. The player, who is now at Chelsea, arrived at Leicester City who ended up winning the English league from a French second division team. They signed him because he had stealing and intercepting statistics well above the average in his league, so much so that he made Leicester scouts look at him. But then the team employees had to go to see if he really was a good player, if he fulfilled what they were looking for, if he fit into his system and things went well. The data can give you clues or help you find the player that fits for a position, but then you must do a personal analysis and check what you are looking for. It is not something magical or perfect.

Another good example that demonstrates this is Roberto Firmino. He is a perfect player for the Liverpool system but that was not seen with the data, let’s say, more often like goals or assists, but with other types of records that are more covered but are very important. Who says what data we should look at is a human being who then uses mathematical tools to find just what he is looking for.

Q: In Spain now the use of ‘big data’ has become very fashionable in the sports environment, can a bubble be generated around all this following the case of ‘Moneyball’?

A: Obviously there is a danger and that is that without the correct human vision, without an analysis that makes sense of data and numbers and knows how to analyze them correctly, databases are only millions of numbers. You need a correct interpretation to give value to what you do, otherwise they are useless.

This type of knowledge is not something magical or perfect. They are super useful tools but without a human team that decides what information is important or how we should look at them, the investment will be useless.

N'golo Kante's is one of the cases that Crawford uses as an example.  (Reuters)
N’golo Kante’s is one of the cases that Crawford uses as an example. (Reuters)

Q: One of your most famous sports-related research talks about shooting the perfect penalty. How does mathematics say that you have to shoot that penalty?

A: Yes, the answer is in the numbers. Obviously there is no place that ensures 100% success, but there are two points in the goal that offer you up to 80%. Where are those points? Well, in the corners, as long as the goalkeeper is in the center of the goal.

Studying the speed of the shots and the capacity of the professional goalkeepers, it can be said that the goalkeeper has half a second to react and move from the moment the player shoots until the ball enters the goal. In that time the goalkeeper can move in an arc that does not occupy the entire goal but leaves the sides and especially the corners free, since it is impossible to physically get there from the center.

You have all that leftover area to mark with great security, but the most interesting thing for me is that if we create a circle between the corner that forms the squad and the semicircle that the goalkeeper can reach, we have the perfect point to shoot drawn on the center of that circle. A point as far from the goalkeeper as from the post as from the crossbar. If you are able to shoot at that point you will have thrown a perfect penalty. I think the measurements are something like 1.7 meters high and 0.65 meters measuring from the stick to the inside. Obviously nothing tells you to score because the goalkeeper can move or guess your intentions, but it is the safest place to score.

Q: Math is usually thought of as boring and difficult, and you try to turn this thinking around with this type of research and topic. Do you think that the idea about mathematics can change with these actions?

A: I think there is still a lot to do. It’s not so much that you don’t know what math is but that people don’t understand or are afraid of math. When you are with friends, you don’t hear anyone say let’s not talk about history because I don’t understand history, but you do hear about mathematics. That is what has to change. It can’t be cool to say that you don’t understand math or don’t like math.

But the worst thing is that many do not believe that mathematics is useful and relevant for life. They believe that everything is theory that stays in class and on paper, and that’s why I decided to change this idea by relating this knowledge to real life. Sport is a great example. People are closely related to sports, and even more so to soccer. If you can show people how numbers are being used or can be used in these fields, the message will come much more than simply talking about formulas or theories. Without going any further, we have already discussed the penalty case.

Q: And does ‘Mathematicians naked’ follow this idea?

A: Yes, well, normally people think mathematics is serious and boring, and almost by accident I thought that taking off my shirt and giving a different image could attract users. I created a YouTube channel to teach math and discovered that many people entered when they saw that there was a guy without a shirt in front of the camera. That was not the initial idea but this is how I have managed to get many people who are not related to mathematics to enter this world.

Many people remember math with bad experiences in class, exams and so on and my videos try to change this and leave at least one good experience to at least lose the fear of math and users see that not only are they not scary but they are very important to your life. In addition, in the videos they see that I have tattoos related to formulas and others and that in itself gives you an idea of ​​something positive, ‘cool’.

Q: In Spain we have a paradox with mathematics because while many students do not like them, they have the highest grade to enter university because they have many job opportunities. Do you think that the ‘boom’ in mathematics in the workplace is good for people to get to know this world better?

A: As a mathematician, I think the more mathematicians there are, the better for everyone. There are many sectors where they are needed and the people who make this career are usually graduates who face problems very well, know how to find solutions and have the ability to analyze all kinds of situations. That is why I think that a ‘boom’ in this sector is good for all of society, but I understand that there may be a double reading for this.

If a lot of people get into a race just for work, they will end up being unhappy and have no passion to do their daily work. If the only motivation that leads you to study a career and dedicate yourself to a profession is that there is work, it is very likely that the bad days with cold, with a lot of work, with personal problems or little desire to work end up leaving everything.

LMS Newsletter: Talking Maths in Public

After attending my first Talking Maths in Public conference last August, I was asked by the London Mathematical Society to write a few words about the experience…

“Talking Maths in Public was hands-down the BEST conference I have ever attended. The incredible skill, passion and experience of the attendees was second only to the welcoming and friendly atmosphere across the 3 days. From planning a ‘Maths Cabaret’ show, to the ‘Treasure Punt’ along the River Cam, I enjoyed every minute and cannot wait for the next edition in 2021!

Screenshot 2020-03-10 at 13.56.14
James Grime from Numberphile/Singing Banana

For almost every session that I attended, I found something that I could take away to help to improve my ability to talk maths in public. However, the keynote given by magician Neil Kelso was particularly inspiring. The way in which he was able to control his audience through every little detail of his performance on stage was mesmerising to watch and hearing him break down these movements to explain exactly what role each one played within his show was fascinating. I will certainly be trying to use as many of his tips as possible in my next show!

If you’re thinking about whether maths communication might be for you, my advice is simple: just give it a go! As mathematicians, we are trained to focus on the details and to construct well-thought out and logical proofs, but unfortunately this approach can often be a barrier to trying something new and untested that perhaps feels outside of our comfort zone, like maths communication. My first YouTube video is awkward, its poorly shot and you can tell that I’m not very comfortable in front of a camera. But, fast forward 2 years and being on camera now feels natural, I know how to setup a shot correctly and editing is second nature. This wouldn’t have happened had I not jumped in head-first and just given it a go. No-one expects you to be perfect (or in fact even functional) on your first try, the most important thing to remember is that you learn from experience, so take that first step and hopefully in a few year’s time you can look back with fondness at that first video/performance/article and see just how far you’ve come.”

You can read the full newsletter here.

Maths Week England Puzzle Solution: How many people are in Ada’s class?

Ada’s class were told to get into equal teams to take part in a Maths Week competition. Trouble is, when they got into pairs, she was the only one without a partner. They tried teams of 3 people, but again she was the only one not in a team. They tried to get into teams of 4 people and this time Ada was still the odd one out! Finally, they got into teams of 5 and all was well – Ada was part of a team! How many people are there in Ada’s class?

And here’s a bonus solution – it’s a little more advanced using lowest common multiples and prime factors to give an infinite number of possible solutions!

Recorded for Maths Week England 2019 – more info here.

How to catch a Serial Killer with Hannah Fry

Hannah Fry (UCL) explains how police detectives use maths to help them catch a serial killer.

The second video featuring Hannah discussing the Maths of Data, first part here.

Find out how this method can be used to pinpoint the probable home of ‘Jack the Ripper’ courtesy of Tom Rocks Maths intern and Oxford University student Francesca Lovell-Read here.

Tom Rocks Maths S02 E10

Episode 10 of Tom Rocks Maths on Oxide Radio sees the conclusion of the million-dollar Millennium Problem series with the Hodge Conjecture, a mischievously difficult number puzzle, and the answer to the question on everyone’s lips: how many people have died watching the video of Justin Bieber’s Despacito? Plus, the usual great music from the Prodigy, the Hives and Weezer.

Image credit: Lou Stejskal

Tom Rocks Maths S02 E09

Tom Rocks Maths launches into Hilary Term on Oxide Radio – Oxford University’s student radio station – with the continuation of the million-dollar Millennium Problems series, an explanation of how Tom’s PhD research can be used to help clean-up our oceans, and conspiracy theories aplenty with Funbers 11. Plus, music from Kings of Leon, Biffy Clyro and new Found Glory. This is maths, but not as you know it…

Maths, but not as you know it… (St Edmund Hall Oxford Magazine)

Dr Tom Crawford joined the Hall in October 2018 as a Stipendiary Lecturer in Mathematics, but he is far from your usual mathematician…

Tom’s research investigates where river water goes when it enters the ocean. A simple question, you might first think, but the complexity of the interaction between the lighter freshwater and the heavier saltwater, mixed together by the tides and wind, and pushed ‘right’ along the coast due to the Earth’s rotation, is anything but. The motivation for understanding this process comes from recent attempts to clean-up our oceans. Rivers are the main source of pollution in the ocean, and therefore by understanding where freshwater ends up in the ocean, we can identify the area’s most susceptible to pollution and mitigate for its effects accordingly.

To better understand this process, Tom conducts experiments in the lab and has conducted fieldwork expeditions to places as far-flung as Antarctica. What the southern-most continent lacks in rivers, it makes up for in meltwater from its plethora of ice sheets. The ultimate process is the same – lighter freshwater being discharged into a heavier saltwater ocean – and as the most remote location on Earth the influence of humans is at its least.

If you thought that a mathematician performing experiments and taking part in fieldwork expeditions was unusual, then you haven’t seen anything yet. Tom is also very active in outreach and public engagement as the author of the award-winning website tomrocksmaths.com which looks to entertain, excite and educate about all thing’s maths. The key approach to Tom’s work is to make entertaining content that people want to engage with, without necessarily having an active interest in maths. Questions such as ‘how many ping-pong balls would it take to raise the Titanic from the ocean floor?’ and ‘what is the blast radius of an atomic bomb?’ peak your attention and curiosity meaning you have no choice but to click to find out the answer!

Tom is also the creator of the ‘Funbers’ series which was broadcast on BBC Radio throughout 2018 telling you the ‘fun facts you didn’t realise you’ve secretly always wanted to know’ about a different number every week. From the beauty of the ‘Golden Ratio’ to the world’s unluckiest number (is it really 13?) via the murderous tale of ‘Pythagoras’ Constant’, Funbers is a source of endless entertainment for all ages and mathematical abilities alike.

And now for the big finale. If you are familiar with Tom’s work, you may know where we are heading with this, but if not, strap yourself in for the big reveal. Dr Tom Crawford is the man behind the ‘Naked Mathematician’ (yes you did read that correctly). To try to show that maths isn’t as serious as many people believe, to try to engage a new audience with the subject, and just to have fun, Tom regularly gives maths talks in his underwear! His ‘Equations Stripped’ series on YouTube has reached 250,000 views – that’s a quarter of a million people that have engaged with maths that may otherwise have never done so. His recent tour of UK universities saw several thousand students come to a maths lecture of their own accord to learn about fluid dynamics. It may not be to everyone’s tastes, but our current methods of trying to engage people with maths are failing, so why not try something new? This is maths, but not as you know it.

You can find all of Tom’s work on his award-winning website and you can follow him on FacebookTwitterYouTube and Instagram @tomrocksmaths for the latest updates.

The original article published in the Aularian magazine can be found here.

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