Another New Year, another resolution to write more blog posts. We'll see.

A mathematician delivering the Royal Institution Christmas Lectures is always something special. Hannah Fry's three lectures this year were rather different from Christopher Zeeman's classics. Rather than one person directly talking to the audience for the full hour each time, she brought in guests for short interviews, showed activities outside the lecture room, and presented a huge range of activities and apparatus with volunteers from the audience (and in one case, a plant, when the audience member asked to solve Rubik's cube turned out to be the nation's champion speedcuber). This was a team effort and everyone who took part performed splendidly: Matt Parker's many contributions deserve special mention.

And it was wonderful. The excitement was palpable. The enthusiasm of the audience, the rush to put hands up whenever a volunteer was wanted - even allowing for possibly selective editing, it was clear that all the students were having a whale of a time. Did one ever expect to see young people so excited by a maths lecture? (Sure, Zeeman was also exciting, but in a very different way.)

I have seen some comments to the effect that there wasn't very much maths in the lectures. I think that is misguided. There was plenty of maths, with the applications shown but without the technical details. I don't have any problem with that. As a kid I was always motivated by the abstract mathematics rather than the applications, but I'm in a minority. And today, a kid wanting to know the details of anything Hannah talked about can just get out their phone. And what a wonderful panorama Hannah presented of the power of mathematics in today's world of data and machine intelligence. (It was nice to see MENACE, the match-box game player, taking its rightful place in the show!)

(The one unfortunate thing was that the first lecture included an upbeat segment about using maths to judge when it was safe to explore volcanoes - although it must have been filmed before the tragedy in New Zealand which could not have been foreseen, that bit should have been edited out or reshot for the broadcast.)

So - Hannah Fry's lectures have inspired schoolkids to take maths seriously. Hopefully some of them will be motivated to study maths at University. And what will happen when they attend their first lecture? If these Christmas Lectures are their first experience of mathematics lectures, they will be expecting wildly interactive sessions with guest speakers introduced every few minutes, lots of demonstrations and fast-moving material. How will they react to a lecturer spending an hour going through a complex pure mathematical proof line by line?

Have the Royal Institution misled their audience by presenting as a lecture something so far removed from a traditional lecture? Possibly, but the Christmas Lectures is their brand, so they cannot be blamed for doing that.

If we as university lecturers are to avoid disappointing our future students, perhaps we need to rethink our lectures. Rather than go through detailed mathematics at a pace which cannot be right for everyone in the room, perhaps we should try to emulate Hannah's RI Lectures. We could (as I'm sure some already do) present many voices (on video if not live), lots of ideas, and as much interactivity as we can manage to keep the audience enthusiastic, leaving the technical details for students to study in their own time. We can provide lecture notes (or, better, screencasts) that they can go through at their own pace, pausing when their brain is full and returning to them later, and going to Youtube or similar when they get stuck, just as we ourselves study from books and papers. We can use our large-class time to build enthusiasm and give the big picture rather than getting lost in detail.

If Hannah's "lectures" help speed up the move to more useful use of students' time than the traditional lecture, that will be another benefit from these remarkable Christmas Lectures.

(As always, I am presenting my own personal views - that is what a blog is for! I don't expect everyone to agree with me.)

## Sunday, 12 January 2020

## Saturday, 15 June 2019

### Looking forward to the Festival of Mathematics

So in ten days time the University is hosting Greenwich Maths Time, the 2019 IMA Festival of Mathematics and its Applications. The Festival takes place on Tuesday 25 and Wednesday 26 June. Everything is free!

The weather forecast is currently fine and we are looking forward to a feast of mathematics. The programme and booking details can be found at the Festival website www.tinyurl.com/imafest19and the Twitter hashtag is #IMAFest19.

The Festival has attracted a wonderful collections of performers and activities. Visitors will have the opportunity to learn about noon-Newtonian fluids by walking on custard! We're grateful to everyone who is taking part and who has worked to make the Festival possible, and especially to the Festival sponsors, the University of Greenwich, the Institute of Mathematics and its Applications, the Advanced Mathematics Support Programme, The OR Society, FDM, the Mathematical Association, and an anonymous individual donor.

The weather forecast is currently fine and we are looking forward to a feast of mathematics. The programme and booking details can be found at the Festival website www.tinyurl.com/imafest19and the Twitter hashtag is #IMAFest19.

The Festival has attracted a wonderful collections of performers and activities. Visitors will have the opportunity to learn about noon-Newtonian fluids by walking on custard! We're grateful to everyone who is taking part and who has worked to make the Festival possible, and especially to the Festival sponsors, the University of Greenwich, the Institute of Mathematics and its Applications, the Advanced Mathematics Support Programme, The OR Society, FDM, the Mathematical Association, and an anonymous individual donor.

## Sunday, 8 July 2018

### The Big Internet MathOff

The Big Internet MathOff is now taking place. Sixteen mathematicians are competing in a knock-out tournament to present the most interesting piece of mathematics. With the final two matches of the first round now taking place, there has already been a wealth of fascinating mathematics presented. Christian Lawson-Perfect deserves huge thanks for creating this wonderful showcase of exciting maths.

But not only are the contestants' pitches fascinating. There is also interesting maths involved in the background.

First, Each match runs for 48 hours. As a contestant, my most pressing question today is whether I will win my first round match against Jo Morgan, in which case I will need to work urgently on my pitch for round 2. So the question is, how soon can I extrapolate from the early votes to be reasonably sure of the outcome?

I am sure that this has been studied in the context of political elections. But this is harder because the electorate is everybody with access to the internet - several hundred million people (though not all will actually vote).

Now, when I first looked at the score ten minutes after the match started, I was leading by 2 votes to 1, but I think that was a bit too early to be confident of the result. As I write this, four hours into the match, Jo is leading by 58 votes to 29, and my share of the vote has consistently been around 30% for some time. Although time differences may mean that America has yet to vote, my intuition is that I can safely abandon my pitch for round 2 and turn to other things like writing blog posts.

I've already had to find my answer to my second problem. Each contestant has to provide a different piece of fun mathematics for each round so we all had to submit four topics before the tournament started. It's unlikely that each set of four topics were all equally good. So what order do you choose for your topics? Do you save your best one for the final? In that case, you might not get to the final and your best idea is wasted. Or do you use your best ideas in the early rounds, to improve your chances of progress, but risk putting your weakest idea forward in the final?

There may be a similarity with World Cup penalty shoot-outs. With five penalties to take, it's clear that a team should select their best five penalty takers. But in which order should they take the penalties? One suggestion is that you should save your best penalty-takers for the end, when the pressure will be greatest. But that could mean that your team has lost 5-3 and your best taker hasn't taken one of the penalties. The great Ally McCoist, commenting on last night's shoot-out between Russia and Croatia, argued for putting your best takers first, guaranteeing that at least they will take penalties. I'm not convinced that's a sound argument, though if you score your early penalties then the pressure on your opponents will increase.

So what did I do for the MathOff? Did I save my best idea for the final, or bring it out for the first round? Well, there is a further complication. I have no idea how to judge which of my topics would win most votes in an internet poll. Ranking them in order would have been pure guesswork. So, shamefully, I didn't use any game theory or simulation to decide the order of my pitches - I just entered them in the order in which I thought of them. Which may be as good a strategy as any!

But not only are the contestants' pitches fascinating. There is also interesting maths involved in the background.

First, Each match runs for 48 hours. As a contestant, my most pressing question today is whether I will win my first round match against Jo Morgan, in which case I will need to work urgently on my pitch for round 2. So the question is, how soon can I extrapolate from the early votes to be reasonably sure of the outcome?

I am sure that this has been studied in the context of political elections. But this is harder because the electorate is everybody with access to the internet - several hundred million people (though not all will actually vote).

Now, when I first looked at the score ten minutes after the match started, I was leading by 2 votes to 1, but I think that was a bit too early to be confident of the result. As I write this, four hours into the match, Jo is leading by 58 votes to 29, and my share of the vote has consistently been around 30% for some time. Although time differences may mean that America has yet to vote, my intuition is that I can safely abandon my pitch for round 2 and turn to other things like writing blog posts.

I've already had to find my answer to my second problem. Each contestant has to provide a different piece of fun mathematics for each round so we all had to submit four topics before the tournament started. It's unlikely that each set of four topics were all equally good. So what order do you choose for your topics? Do you save your best one for the final? In that case, you might not get to the final and your best idea is wasted. Or do you use your best ideas in the early rounds, to improve your chances of progress, but risk putting your weakest idea forward in the final?

There may be a similarity with World Cup penalty shoot-outs. With five penalties to take, it's clear that a team should select their best five penalty takers. But in which order should they take the penalties? One suggestion is that you should save your best penalty-takers for the end, when the pressure will be greatest. But that could mean that your team has lost 5-3 and your best taker hasn't taken one of the penalties. The great Ally McCoist, commenting on last night's shoot-out between Russia and Croatia, argued for putting your best takers first, guaranteeing that at least they will take penalties. I'm not convinced that's a sound argument, though if you score your early penalties then the pressure on your opponents will increase.

So what did I do for the MathOff? Did I save my best idea for the final, or bring it out for the first round? Well, there is a further complication. I have no idea how to judge which of my topics would win most votes in an internet poll. Ranking them in order would have been pure guesswork. So, shamefully, I didn't use any game theory or simulation to decide the order of my pitches - I just entered them in the order in which I thought of them. Which may be as good a strategy as any!

## Sunday, 20 May 2018

### Mathematical discoveries

I was lucky enough to attend a meeting organised by the British Society for the History of Mathematics on "The History of Cryptography and Coding". It was a quite exceptional meeting - six excellent talks. As one of the other audience members said, I learned something from every talk, and a lot from several talks. (Anyone who goes to these events will know that this isn't always the case.)

The final talk, by Clifford Cocks on the discoveries of the public key cryptography, was fascinating in many respects. (Cocks was one of the people at GCHQ who discovered both the Diffie-Hellman key exchange method and the Rivest-Shamir-Adleman (RSA) algorithm before those after whom the ideas are named, but this wasn't known until GCHQ made it public over 25 years after the event.) Cocks told us about the (different) reasons why the British and American discoverers were looking for these methods. I was particularly struck by his insights into the creative processes that led to the discoveries.

In 1970 James Ellis at GCHQ had the idea of public-key cryptography. Many people at GCHQ tried to find a way to implement it, without success. Cocks suggested that this was because of "tunnel vision" - because Ellis's paper suggested using look-up tables, everyone was focused on that idea. Cocks had just arrived at GCHQ from university, and his mentor mentioned Ellis's problem to him, but described it in general terms without mentioning look-up tables. Without having been led in a wrong direction, Cocks quickly came up with the idea of using factorisation, and the problem was solved. (When Cocks told his colleague and housemate Malcolm Williamson about his paper, Williamson overnight worked up the idea of using the discrete logarithm problem, anticipating Diffie and Hellman.)

Cocks also told us about how Diffie was working on these discoveries having left his academic job, supporting himself on his savings - something which I don't recall knowing.

Then Cocks told us about Rivest, Shamir and Adleman's discovery of RSA. They had tried about 30 ideas, none of which worked. Then after a Passover meal at which alcohol flowed freely, Rivest had the big idea, wrote it down, and checked the next morning to see if it still worked.

I think these stories shed some light on mathematical creativity. It needs hard work, of course, but it also needs flexibility. Cocks (by his modest account) had the advantage over his colleagues that his mind wasn't conditioned by an unproductive idea. Rivest's solution came after a break from thinking about it. Of course, there are many other examples - PoincarĂ©'s inspiration as he was getting on a bus is the standard one - but it is always interesting to hear how great mathematical discoveries came about, and to hear this story from Cocks himself was a wonderful privilege.

The final talk, by Clifford Cocks on the discoveries of the public key cryptography, was fascinating in many respects. (Cocks was one of the people at GCHQ who discovered both the Diffie-Hellman key exchange method and the Rivest-Shamir-Adleman (RSA) algorithm before those after whom the ideas are named, but this wasn't known until GCHQ made it public over 25 years after the event.) Cocks told us about the (different) reasons why the British and American discoverers were looking for these methods. I was particularly struck by his insights into the creative processes that led to the discoveries.

In 1970 James Ellis at GCHQ had the idea of public-key cryptography. Many people at GCHQ tried to find a way to implement it, without success. Cocks suggested that this was because of "tunnel vision" - because Ellis's paper suggested using look-up tables, everyone was focused on that idea. Cocks had just arrived at GCHQ from university, and his mentor mentioned Ellis's problem to him, but described it in general terms without mentioning look-up tables. Without having been led in a wrong direction, Cocks quickly came up with the idea of using factorisation, and the problem was solved. (When Cocks told his colleague and housemate Malcolm Williamson about his paper, Williamson overnight worked up the idea of using the discrete logarithm problem, anticipating Diffie and Hellman.)

Cocks also told us about how Diffie was working on these discoveries having left his academic job, supporting himself on his savings - something which I don't recall knowing.

Then Cocks told us about Rivest, Shamir and Adleman's discovery of RSA. They had tried about 30 ideas, none of which worked. Then after a Passover meal at which alcohol flowed freely, Rivest had the big idea, wrote it down, and checked the next morning to see if it still worked.

I think these stories shed some light on mathematical creativity. It needs hard work, of course, but it also needs flexibility. Cocks (by his modest account) had the advantage over his colleagues that his mind wasn't conditioned by an unproductive idea. Rivest's solution came after a break from thinking about it. Of course, there are many other examples - PoincarĂ©'s inspiration as he was getting on a bus is the standard one - but it is always interesting to hear how great mathematical discoveries came about, and to hear this story from Cocks himself was a wonderful privilege.

## Sunday, 28 January 2018

### London buses, and the use of the mean as an estimate

A couple of weeks ago, I got onto my morning bus, climbed the stairs, holding on as the bus lurched forward, and sat down, to hear for the first time the new announcement "Please hold on: the bus is about to move", introduced by TfL (Transport for London). Over the next few days this announcement was widely ridiculed. It was broadcast after every stop, but often - in my experience almost always - AFTER the bus had started moving, and sometimes when it was slowing down for the next stop, making the announcement appear ridiculous. Occasionally, at busy stops like the railway station, it was broadcast while large numbers of people were still waiting to board, presumably causing consternation to prospective passengers who took it seriously. And on one occasion, while the bus was stationary, I heard "The bus is about to move" followed immediately by the announcement "The driver has been instructed to wait here for a few minutes", flatly contradicting the previous words.

What was happening? TfL explained that they were piloting the announcement for four weeks, to try to reduce the number of injuries sustained by passengers on moving buses - apparently of the order of 5000 each year. The timing of the announcement was based on the average time buses spent at each stop - I suspect by "average" they meant the mean.

The intention is laudable. But the problem with using a mean in situations like this is that it doesn't really tell you how long a particular bus will wait at a given stop. My bus home probably spends longer stopped at the railway station than at all the other stops put together. Just as most people earn less than the mean national salary, which is heavily influenced by the very small number of people earning millions each year, so I imagine most of the time a bus spends less time at a stop than the mean. So a system based on the mean time spent at a stop will result in the announcement usually being played after the us has left the stop, leading to ridicule.

Now, TfL are pretty good at maths - their planning of the transport around London during the 2012 Olympics was a very successful example of operational research in action. So did they really get this wrong? After all, one would think that a few tests would have shown the problem.

Certainly one result of the announcements was a great deal of publicity, which perhaps has made people more aware of the need for care when standing and moving on a bus. The announcements themselves may have a short-term effect, but in fact one very quickly ceases to notice them (or at least I have found that they very rarely impinged on my attention, after the first few instances on the first day). But perhaps the press coverage, and people talking about the announcements, had more impact than the announcements themselves.

But if the announcements are to continue, how can TfL avoid the absurdity of an announcement that the bus is about to move being broadcast after it has moved? The solution TfL have adopted (as well as apparently changing the timing) is simple. The wording of the announcement is now "Please hold on while the bus is moving". The timing no longer offers the possibility of absurdity. The solution to this problem was not mathematical modelling, but thoughtful use of language.

## Sunday, 19 November 2017

### MathsJam

Here is my overdue post on last weekend's MathsJam gathering. MathsJam is always a wonderfully exciting and enjoyable weekend, and this one was particularly good. I can honestly say that I enjoyed every talk, and was entertained, surprised and educated in roughly equal measures.

I'm nervous about selecting highlights, because almost everything was a highlight, and my list inevitably leaves out many excellent talks. There is a list of all the talks on the MathsJam website but here are some I particularly remember (in the order in which they were presented):

And of course the "extra-curricular" puzzles, games and magic, Tiago showing me how to tie a knot with one hand, and the spectacular mathematical cakes.

So once more a memorable MathsJam, with an excellent range of speakers and talks, friendly atmosphere and fascinating and surprising mathematics. The organisers once again did an amazing job!

I'm nervous about selecting highlights, because almost everything was a highlight, and my list inevitably leaves out many excellent talks. There is a list of all the talks on the MathsJam website but here are some I particularly remember (in the order in which they were presented):

- Simon's 3D-printed robot which solves Rubik's cube (time-lapse video shown here with Simon's permission);
- Matt on logical deduction games, which brought back memories of playing Eleusis when I was a student and introduced others I need to find out about;
- Noel-Ann on data and how to it can be represented (and misrepresented);
- Zoe's poem about
*e*, which (understandably) seems to be on everybody's highlight list; - Matthew's amazing recreation of a problem from
*Captain Scarlet*about the bongs of Big Ben; - Andrew's paradoxical balloon monkey, which although made from a single balloon, has an underlying graph which is not semi-Eulerian;
- Angela's poem;
- Rachel on spinning yarn;
- Alison on illogical units, and Dave on illogical scales;
- Will on non-binary cellular automata;
- Miles finding striking similarities between mountaineering and mathematics;
- Glen showing how many holes a constructed object (equivalent to a T-shirt) possessed (most of MathsJam seem to have got it wrong, going for four rather than three!);
- Sue on Ada Lovelace;
- Paolo using a pack of cards to find two numbers from their sum and difference;

And of course the "extra-curricular" puzzles, games and magic, Tiago showing me how to tie a knot with one hand, and the spectacular mathematical cakes.

So once more a memorable MathsJam, with an excellent range of speakers and talks, friendly atmosphere and fascinating and surprising mathematics. The organisers once again did an amazing job!

## Wednesday, 8 November 2017

### Looking forward to MathsJam

With only three days to go till the MathsJam Gathering - the best weekend of the year, I've been thinking of some of my favourite MathsJam discoveries. Sticking to pure mathematics, here are my memories of three gems. I could have chosen many others, but perhaps because these happen to relate to my current teaching, and I showed two of them to my graph theory students immediately upon my return from the gathering, they are the first that come to mind. Since I believe all MathsJam presentations are available online, further details should be readily available.

I could have chosen many more examples: I'm certainly not ranking these presentations or any others. On another day I might have chosen a completely different set! But I'm certainly looking forward to coming across more wonderful mathematics this weekend!

- Colin Wright's amazing talk on graph colouring, which started by asking us to complete a partially-completed 3-colouring of a small graph, and turned into a more-or-less complete proof, within a 5-minute talk, that there is no polynomial-time algorithm for 3-colouring a graph.
- Ross Atkins's talk about Braess's Paradox - a simple situation in which adding an extra road to a network, with no increase in traffic, results in longer average journey times. I should have known about this counter-intuitive result so I'm very glad to have found out about it, and especially with the wonderful demonstration with a network of springs that showed a mechanical realisation of the paradox.
- David Bedford's "What's my polynomial?" I love this because it is arguably what the late Raymond Smullyan called a "monkey trick". David asked you to think of a polynomial
*p*(*x*) with non-negative integer coefficients, and, for a single value of*x*of your choice, greater than any of the coefficients, tell him both*x*and*p*(*x*). He would then tell you your polynomial. Knowing that one needs*n*values to determine a polynomial of degree*n*, I was taken in by this!

I could have chosen many more examples: I'm certainly not ranking these presentations or any others. On another day I might have chosen a completely different set! But I'm certainly looking forward to coming across more wonderful mathematics this weekend!

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