TRM intern and University of Oxford student Kai Laddiman speaks to St John’s College Computer Scientist Stefan Kiefer about the infamous million-dollar millennium problem: P versus NP.
You can read more about P vs NP here.
TRM intern and University of Oxford student Kai Laddiman speaks to St John’s College Computer Scientist Stefan Kiefer about the infamous million-dollar millennium problem: P versus NP.
You can read more about P vs NP here.
The third video in the fluid dynamics trilogy I made for Numberphile. Rivers contain 80% of pollution which ends up in the ocean, so understanding where the water goes when it leaves the river mouth is of upmost importance in the fight to clean-up our planet.
Watch part 1 on the Navier-Stokes Equations here
Watch part 2 on Reynolds Number here.
A very fun Christmas treat for you all as I team up with my good friend Bobby Seagull for the Funbers Xmas Special – expect fun facts, lots of numbers, and more birds than anyone thought possible… Happy Holidays!!
Looking for some festive fun over the holiday season? Why not try your hand at my 12 Christmas puzzles…
Answers to all puzzles at the bottom of the page.
Puzzle 1: If I set a puzzle every day of the advent period (1-25 December) and spend 1 minute on the first puzzle, 2 minutes on the second, 3 minutes on the third, and so on, with the final one being 25 minutes on the 25th puzzle, what is the total amount of time I will spend writing puzzles?
Puzzle 2: December 6th marked my birthday and to celebrate I travelled to Kiev with 4 friends. If I order a drink on the flight out and then each of my friends orders twice as many as the person before, how many drinks do we order in total?
Puzzle 3: This morning I built a snowman using three spheres of radius 0.5m, 0.4m and 0.2m. However, the sun has since come out and the snowman is starting to melt at a rate of 0.01 m3 per minute. How long will it take for him to disappear completely?
Puzzle 4: Suppose a newly-born pair of elves, one male, one female, are living together at the North pole. Elves are able to mate at the age of one month so that at the end of its second month a female elf can produce another pair of offspring. Suppose that the elves never die, and that the female always produces one new pair (one male, one female) every month from the start of the third month on. After one year, how many pairs of elves will there be?
Puzzle 5: On Christmas day I have 11 people coming to dinner and so I’m working on the seating plan ahead of time. For a round table with exactly 12 chairs, how many different seating plans are possible?
Puzzle 6: My front yard is covered in snow and I need to clear a path connecting my front door to the pavement and then back to the garage. If each square in the diagram is 1m x 1m what is the shortest possible path?
Puzzle 7: The first night of Chanukah is December 22nd when the first candle is lit. If it burns at a rate of 0.05cm per hour, how tall does the candle need to be to last the required 8 days?
Puzzle 8: If you have a square chimney which is 0.7m across, assuming Santa has a round belly what is the maximum waist size that can fit down the chimney?
Puzzle 9: On Christmas Eve Santa needs to visit each country around the world in 24 hours. Assuming time stands still whilst he is travelling, how long can he spend in each country?
Puzzle 10: I got carried away with buying presents this year and now have more than can fit into my stocking. If the stocking has a maximum capacity of 150, and my presents have the following sizes: 16, 27, 37, 65, 52, 42, 95, 59; what is the closest I can get to filling the stocking completely?
(NB: I am not looking for the highest number of presents that will fit, but the largest total that is less than or equal to 150).
Puzzle 11: Santa has 8 reindeer, and each one can pull a weight of 80kg. If Santa weights 90kg, his sleigh 180kg, and each present weighs at least 3kg, what is the maximum number of presents that can be carried in a single trip?
Puzzle 12: To mark the end of the 12 days of Christmas each student at the University of Oxford has kindly decided to donate some money to a charity of their choice. If the first person donates £12 and everyone after donates exactly half the amount of the person before them (rounding down to the nearest penny), how much will be donated in total?
Puzzle 1: 1 + 2 + 3 + … + 25 = 325. There is a faster way to do this which was first discovered by the mathematician Gauss when he was still at school. If you pair each of the numbers in your sum, eg. 0 + 25, 1 + 24, 2 + 23, etc. up to 12 + 13, then you have 13 pairs which each total 25 and so the overall total is 25*13 = 325. The same method works when adding up the first n numbers, with the total always being n(n+1)/2.
Puzzle 2: 1+2+4+8+16 = 31.
Puzzle 3: Volume of a sphere = (4/3)*pi*radius3 and so the total volume of snow = 0.52 + 0.27 + 0.03 = 0.82 m3. Melting at a rate of 0.01 m3 per minute means the snowman will be gone after only 82 minutes!
Puzzle 4: This problem is actually a very famous sequence in disguise…
The first new pair is born at the start of the third month giving 2 pairs after three months. The question tells us that we have to wait one whole month before the new offspring can mate and so only the original pair can give birth during the fourth month which leaves a total of 3 pairs after four months. For the fifth month, both the original pair, and the first-born pair can now produce offspring and so we get two more pairs giving a total of 5 after five months. In month six, the second-born pair can now also produce offspring and so in total we have three offspring-producing pairs, giving 8 pairs after 6 months.
At this point, you may have spotted that the numbers follow the Fibonacci sequence, which is created by adding the previous two numbers together to get the next one along. The first twelve numbers in the sequence are below, which gives an answer of 144 – no wonder Santa is able to make so many toys!
Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
Puzzle 5: I have 12 choices of where to place the first person, 11 for the second, 10 for the third and so on, which gives 12*11*10*9*8*7*6*5*4*3*2*1 = 12! (read as 12 factorial) in total. BUT for any given seating plan we can rotate around the table one place to get the same order, which means we have in fact over counted by a factor of 12. Therefore, the total number is 11! = 39,916,800.
Puzzle 6: Reflect the yard in the pavement and draw a straight line connecting the front door to the edge of the garage closest to the front door (blue). Then add the same line from the ‘reflected’ front door at the top back down to the garage at the bottom (orange). The final shortest path is found by combining both paths for a valid one in the original diagram.
The length is found using Pythagoras’ Theorem. From the door to the pavement we have length
(12 + 22)1/2 = (5)1/2
and from the pavement to the garage the length is
((1.5)2 + 32)1/2 = (11.25)1/2
giving a total length of 2.23 + 3.35 = 5.58m.
Puzzle 7: 8 days = 8*24 hours = 192 hours. 192*0.05 = 9.6cm.
Puzzle 8: Chimney diameter = 0.7m so the maximum circumference (or waist size) that will fit is 0.7*pi = 2.2m or 88 inches!
Puzzle 9: Using the UN list of 193 countries, Santa has 24 * 60 = 1440 minutes total, which means spending only 7.5 minutes in each country!
Puzzle 10: 150 exactly with 16 + 27 + 42 + 65 = 150.
Puzzle 11: We have 8 reindeer each with a capacity of 80kg giving a total of 640kg that can be carried. Subtracting the 90kg for Santa and 180kg for the sleigh leaves 370kg available. Dividing this by 3 gives 123.33 so a maximum of 123 presents can be carried at once.
Puzzle 12: 12 + 6 + 3 + 1.5 + 0.75 + 0.37 + 0.18 + 0.09 + 0.04 + 0.02 + 0.01 + 0 + 0 + 0 + …
The donations stop after the 11th person giving a total of £23.87. Even if we had allowed donations of part of a penny the total would never quite reach £24.00. This is an example of an infinite sum (or Geometric Series) where the total is always two times the first number.
Season 2 comes to a close with stories from my (rather eventful) trip to China, a new video series with BBC Maths Guru Bobby Seagull, and the number of calories needed by a Charizard per day to survive. That’s all on top of the usual puzzle and fun facts about the numbers 0 and 1. Plus, music from the Red Hot Chili Peppers, System of a Down, and Limp Bizkit. This is maths, but not as you know it…
Tom Crawford (Warrington, United Kingdom, 1989) is presented as an atypical math teacher. He teaches mathematics to first and second year students at the University of Oxford (United Kingdom) and carries out an intense dissemination work in which he tries to approach a discipline that is not usually found among the favourites of young students.
In his attempt to popularise science, he does not hesitate to stay in his underpants , using the striptease as a metaphor for his work deepening the meaning of equations such as Navier-Stokes, unveiling them layer by layer, to make something affordable that can result in principle esoteric.
This week, Crawford visited the Student Residence, in Madrid, where, within the Mathematics in Residence cycle organised by the ICMAT, he offered the conference Mathematics of sport . In it, he uses sport as an example of a daily activity that can be better understood and practiced using mathematical equations.
Question. You undress or use sports to make mathematics impose less. Why is it necessary to show that mathematics is fun? I don’t see lawyers or judges, who also deal with very complex issues, trying to present the law as something fun.
Answer. I think it’s because people, for whatever reason, happily admit that they don’t like math, it’s socially acceptable. If you tell someone that you are a lawyer, their default answer is not going to be “I don’t like the law,” and that does happen with math. And it shouldn’t be like that. Everyone should have a basic understanding of math, but many people don’t have it. For me, that is why I want to emphasize that mathematics is fun and accessible. It doesn’t have to be something very hard or something that was taught badly in school.
Q. Do you think mathematics is taught especially badly in school, worse than other subjects?
A. Mathematics has a hard time competing with other subjects in the sense of teaching them through stories. When you learn something, if they can teach you through stories, it is something very powerful, which serves to catch people. And that is easier with literature or history.
A very simple example of how to add stories to mathematics would be trigonometry. The properties of the triangles you learn in high school. If you think about how these functions were discovered or invented, why we invented the sine, the cosine and the tangent, it was the ancient architects who tried to build buildings, churches, pyramids and created those intellectual tools. This is how trigonometry should be taught to me. Imagine they are in ancient Rome and you have to build a concrete building. How would you do it with the technologies available at that time? This prompts you to think about angles and distances and that is where trigonometry is useful and what it was invented for.
Q. A little more than a century ago, in a country like Spain, more than half of the population was illiterate. Do you think it would be possible and desirable to get a large majority of people to be able to handle basic mathematical tools?
A. It is completely possible and I would say that we are already doing it. It depends on what you consider a basic level of mathematics. Most people can, for example, looking at a clock know that the needles return to the same place every 12 hours, it is modular arithmetic, something you don’t study until you get to college. Even being able to calculate changes when they give you a ticket is to do mental arithmetic. Or calculate when you have to leave home if it takes 35 minutes to the station and the train leaves at 12.45. There are many things you do without thinking, but that involve mathematical calculations. So it depends on what you consider a desirable level of mathematics, but a large part of the population already has some capacity to use them.
“You can question whether trying to influence voters is good or bad”
Q. He also talks about the possibilities of mathematics to improve the performance of athletes. There is a movie like Money Ball , which talks about the experience of a baseball coach who uses mathematical analysis to lead a small team to compete against the big ones in the league with much less budget. Do you use math a lot in elite sport?
A. As far as I know, it is an important part of the scout systems of large teams. Today, these scouts, in addition to the classic analysis of a player’s performance, strengths and weaknesses, include teams of mathematicians and data scientists. As in Moneyball , your job is to analyse large amounts of data and detect marginal gains to take advantage of. That works well in baseball, because you have many controllable factors: The pitching of the pitcher, the batter, the race to the base. It is very formulable and they are repetitive behaviours. In football it is more difficult to find those marginal gains because it is less controllable.
The best example I can think of in football is Leicester City, which won the Premiere League in 2016. A big surprise. They had climbed to the first few years before and suddenly they win. In that victory, N’Golo Kanté was very important. He was the star of the season and won the player of the year award. He had been signed by a French second division team because the scout network had identified him among all the defensive midfielders in Europe at any level. As a defensive camper centre, one of your jobs is to stop the attacks of opponents. You can measure this in tickets, but one of the best ways to do this is through interceptions, which has to do with the player’s ability to read a game. It is something very difficult to assess with a number, quite subjective. But interceptions suggest that you are very often in the right place. And from that point of view, their number of interceptions was much higher statistically than the rest of midfielders. If the average of all midfielders in Europe is two, but most of the players are between 1.9 and 2.1 and Kanté is at 3, we see that it is an atypical case. It was not just a statistical analysis, because the human element is valued, but it was a factor to hire him.
Q. Can mathematics tell us what is the limit of human performance in sport? There have already been examples in the past, such as Roger Bannister’s, which went down four minutes on the mile when almost everyone said it was impossible, in which the predictions were completely wrong. Can these limits be accurately identified using mathematics?
R.If you look at the men’s marathon record during the last century, the marks descend, but not at a constant pace. You can estimate, for example, that every 10 years, 10 minutes are trimmed at the beginning, but then, in the 1940s and 1950s, the curve begins to flatten out and already in the 1990s it seems completely flat. So if we had sat here 30 years ago, when the record was around two hours and five minutes, we could have thought we would never run below two hours, because even if it keeps going down, the pace is getting slower. But in recent years, there has been much progress in long-distance races, such as new shoes that can provide 4% more energy. In addition, there is a professionalisation that allows you to train all day and not have a job besides running.
“I could predict with some confidence that the human limit for the marathon would be about an hour and 55 minutes”
So these are new factors that modify our calculations. In the future, in 30 years, new improvements may appear, but it is certain that we will not run a marathon in less than an hour. Given what has happened in the past, I think I could predict with some confidence that the human limit for the marathon would be about an hour and 55 minutes.
Q. Some people, when talking about the possibilities of mathematics to bring humans to the limit of perfection, may think that sports will become more boring, because there will be less and less space for the unexpected.
A. I think that also has to do with the human psychological trait that is nostalgia. But sport evolves and there is always a human factor. If the study allows you to perfect the place where it is better to throw a penalty, the goalkeepers can also work with that information. And then, there are some players who do not shoot at that supposedly perfect space, such as Eden Hazard, of Real Madrid, who when he threw the penalties for Chelsea waited until the last moment to decide where he threw it, a method that goes against what he says The mathematical model. In the end there are many variables in sports.
Q. Can mathematics help us better understand human groups? Does that technology have the potential to improve living together or to make it worse?
A. With all the data available, there are huge technology companies that can make profiles of people. Knowing that you are white, American, that you earn so much money and live in such a state, they can try to predict what you like or what you do and influence your vote in one direction. But this technology could also be used for good and you can also question whether trying to influence voters is good or bad. I think that ultimately we depend on the big companies that have control over these data so that they assume their moral responsibility and use the data well.
In any case, I think that most of the mathematicians working in this field would say that the idea of using mathematical data, algorithms and models to try to predict people’s behaviour is incredibly new and we don’t know exactly what we are doing. Algorithms may be a part of the decision making process, but not the only criteria for making a decision.
You can read the original article on El Pais here.
This week I had the honour of speaking at the Residencia de Estudiantes in Madrid, which has previously hosted Albert Einstein, Marie Curie, Salvador Dali and Igor Stravinsky amongst many, many others.
Ahead of the event I was asked a few questions by the organisers, and here are my answers.
Without revealing all your talk: could you give us an idea about how maths can help to be better at sports?
From calculating the perfect placement of a penalty kick in football to maximise your chance of scoring, to identifying the best location on Earth to try to break a world record, maths can be used to help to improve our performance in almost any sport. The difficultly lies in writing down the correct equations, but once we have them, maths has the answers.
Can you tell us any real example of this maths application?
My favourite example is one that will be featured in my talk: if attempting to break a world record in rowing, the best place to do so is on the equator. This may seem counter-intuitive at first, but as I will explain, by changing the location to the equator you can increase performance by up to 8%, which for an elite athlete is an incredible boost!
In your opinion: what makes maths so useful in different sports context?
Maths can be applied to anything. This is one of the main reasons that I love the subject and travel the world championing its versatility. Given a situation in any sport, you can always use equations to describe what is happening. This might be how a tennis ball moves through the air, or the aerodynamics of a swimmer gliding through the water. Once you have the equations, maths allows you to solve them for the optimal solution, which can then be translated into improved performance by changing your technique appropriately.
You also explain that the mathematical results in sports may vary, how? In which way? What should athletes take into account?
The ideas discussed in my talk are aimed at professional athletes who are already performing at a very high level and therefore need to resort to other approaches to improve performance beyond increased practice. For amateur athletes, whilst the same ideas will still be applicable, they are much more likely to benefit from practice!
What is your personal experience with sports? Have you ever used “math tricks” for optimise your scores?
The idea for the talk came from wanting to combine my two main passions: mathematics and sport. I play football regularly and as the designated penalty taker for my team have ample opportunity to try to hit the mathematically calculated perfect position for a shot. I also run marathons where my knowledge of the history (and mathematically predicted future) of the world record helps me to appreciate my accomplishments in the event.
How did you become a math communicator?
My first taste of maths communication came during my undergraduate degree at Oxford, where I joined the maths outreach group “Marcus’ Marvellous Mathemagicians”. The group was named after Marcus du Sautoy and performed interactive talks and workshops on his behalf in schools across the UK. The next opportunity came during my PhD when I spent two months working with the “Naked Scientists” team in Cambridge to produce a weekly science radio programme for the BBC. I enjoyed the placement so much that I agreed to join the team full-time upon completion of my PhD. After one year of working in radio production, I began to realise that my true calling was in video, and “Tom Rocks Maths” was born.
How are outreach, teaching and research connected in your professional life?
As someone who came from a state school background and worked extremely hard to get to Oxford, I have always had a passion for outreach and the drive to make university accessible to all. My maths communication work is an extension of this, allowing me to not only to visit schools in deprived areas to try to inspire them to consider higher education, but also to encourage the general public to engage more with the subject of maths and to no longer be afraid of numbers.
The teaching role fits perfectly with maths communication as both roles require the ability to be able to explain difficult concepts in ways that can be understood by a given audience. For a public lecture, the mathematical ability of the audience is perhaps less than that of a class of undergraduates, but the need for clear communication remains the same. In this way, I find that each role complements the other perfectly, with many of the topics that my students find difficult providing inspiration for future video ideas.
What do you enjoy most in your outreach talks?
There is nothing I enjoy more than being able to present to a live audience. Whilst I enjoy all aspects of my outreach work – YouTube, television, radio, writing – nothing beats the thrill of speaking to a room full of people who want to hear what you have to say. The small interactions with each individual member of the audience, whether through eye contact or answering a question, remain with me long after the event and act as one of my main motivations to continue with my work.
You are not the speaker one might expect when thinking about a maths communicator, what kind of reactions have you find in this sense? Do you have any anecdote regarding this?
There are two ways of looking at this: first, the notion of a stereotypical mathematician is outdated and from my experience not representative of a large part of the demographic; and second, I hope that by putting myself forward as a public face of mathematics I can help others who may be thinking that they can’t be a mathematician just because of the way that they look.
In terms of anecdotes, I think it best that I point you in the direction of the comments on my YouTube videos…
In particular, what are the reactions with “Equations stripped”? How did you come up with the idea of this series?
The “Equations Stripped” is possibly my favourite of all of the things that I do because it helps to tackle the idea that maths should be serious. The concept of the videos came from thinking about this opinion and trying to come up with what I thought was the best way to present the subject as anything but serious. The result is me talking about maths in my underwear!
My role with the “Naked Scientists” also played a part, as the name would often lead to listeners (or even guests) suggesting that we should all be naked when recording the show, and of course being a radio programme no-one could prove or disprove the theory! I always thought that we should have had more fun with this concept, and when “Tom Rocks Maths” was launched Naked Maths seemed like the way to go!
Over the past 60 years since bird feeders first became commercially available, humans have been changing bird populations across the UK. The overall effect has generally been positive, with an increase in the prevalence of Wood Pigeons, and a shift in the migration pattern of Eurasian Blackcaps, but as with most changes, there is a word of warning… Live interview with BBC Radio Oxford.
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.