 Okay, good. So I haven't got any kind of visual aids or anything, so I'll just shut that off and have a lovely blue screen behind me. So what I'm going to talk about really is kind of taking a hackerish approach to mathematical discovery why you should and how you can with an example from something that I did recently. I don't know how long it's going to take me. I haven't prepared this very thoroughly, but sometime in the reason kind of five to ten minutes probably. So why you should well first of all I mean maybe not everyone agrees with this but I think discovering new mathematical things is one of the most brilliant things that you can do because for a brief period you know something that nobody else knows and that is I think a very exciting feeling. It's not as inaccessible as you might think. There are lots of things that can be discovered that you don't need kind of to spend years and years studying to get to the point where you can understand them and start to think about them and some of them are relatively accessible to being. So of course having discovered something you can't resist the urge to tell people about it so that that kind of private knowledge doesn't last for very long and other people aren't nearly as interested usually as you are because of course it's more fun to have discovered something yourself and hear about something that someone else has discovered. But the other thing is that you know pure mathematicians as a whole they're kind of chalk and blackboard kind of people right? Pencil and paper. It's a pretty old school profession on the whole. I mean you know they do use kind of more applied mathematicians use software and it's starting to creep in a little bit to so the more kind of traditional pure mathematical world but it's still really you know pure mathematicians are mostly not hackers and hackers are mostly speaking not pure mathematicians and that kind of creates opportunities. There are certain kinds of problems that are much more just vulnerable to a sort of hackers approach than to a pencil and paper approach. And so that creates opportunities for people like us to discover things that the pure mathematicians might have missed with their blackboards. So I mean the particular example that I have in mind is something that I did kind of recently like in the last couple of weeks it was a problem that I heard about on Reddit and as far as I know the best kind of theoretical result on this problem prior to this was the proof was written by an anonymous user on 4chan. So it's very atypical for mathematical proofs. It's unique in my experience for mathematical proofs. And the problem is the following one so ambulances right I know like fire engines they go nino nino nino are they going nino or are they going nor knee. Well it's both isn't it they're doing both. And so what's the shorter sequence of knees and nores that you need in order to get both possibilities in there. There you've got it got me nor the first two nor me the second and the third both the possibilities are in there. OK but fire engines you know I don't I'm not an expert on sirens maybe someone here isn't can tell me more details about sirens but I you know when you hear sirens kind of on the streets of London it's not just knees and nores all kinds of wibbles and and other sounds so I mean suppose there were three sounds that a fire engine can make me nor wibble then there are more possible permutations of that you've got OK bear with me knee nor wibble that's one of them knee wibble nor that's number two there should be six so let's see if this works nor knee wibble nor wibble knee wibble knee nor wibble nor knee right six possibilities so so what is the shortest kind of sequence of knees and nores and wibbles that the fire engine can play that will get all those possibilities in there somewhere so I know the answer to this I'll see if I can do it I'm going to have to think about this for a second knee nor wibble knee nor knee wibble nor knee right so so if you think about thank you so I mean if you if you if you think about that you'll you'll see that all the every all six possibilities have been included in there somewhere but there were only nine sounds in that sequence so it's more efficient than just doing all the six possibilities one by one you've you've saved some effort and there's a there's a there's a kind of algorithm that you can use to construct this kind of sequence with different numbers of sounds so super permutations was my ill considered title because someone in about 1993 decided that super permutations were what they wanted to call these sequences because they contain all the possible permutations so they're like super because they contain them all I guess is the idea like a super set so anyway so that you can construct super permutations and it turns out that for four symbols I'm not going to attempt to do this I'm sorry I did can think about it but I'd need to practice lots more than I have to do it but for four symbols it is what's 24 plus 9 it must be 33 33 sounds long is the shortest sequence of five sounds that contains all the possibilities and in general the construction goes it goes do you know about what a factorial is ok so it's it's a sum of factorial so for say for five sounds it would be one factorial plus two factorial plus three factorial plus four factorial plus five factorial and until about the spring of this year it was unknown whether this kind of so if you do one factorial plus three factorial plus three up to five factorial the answer is 153 and so you can construct a sequence of 153 I don't know what the five sounds be it would be like Do Re Mi La So is that right if they were if it was maybe more of a musical kind of fire engine than the traditional ones so that they would be at the sequence of 153 Do Re Mi La So but the the there's a proof as I say that the only written version of this proof that I'm aware of was written by an anonymous 4chan user which proves that the shortest possible sequence would be 152 sounds long so there was this kind of unknown question was it 152 or 153 the shortest one for five sounds and a bloke who whose day job is designing processors for Intel but who obviously is a mathematical hobbyist wrote a C program that used a relatively clever kind of technique to find out all the possible shortest five symbol things and anyway the answer is 153 is the shortest but there are eight different ways of doing it not just the obvious way so then am I out of time another three minutes perfect okay so there's a something called the traveling salesman problem so there's a salesman who has to travel around to lots of different cities to sell his underwear to all the people of these different cities and he wants to do this in the most efficient way possible so he wants the route that has takes the shortest distance that visits all these cities and then returns to where he started and so although maybe it's not obvious this particular problem is kind of the same problem really because your cities are all different permutations and the length of the route is the number of different sounds that you had to do to make the next one so you want to make all the different all the different permutations by the shortest possible route so basically this is an instance of the traveling salesman problem and it also turns out that very very smart people have been working out how ways to solve the traveling salesman problem ever since some guys at Rand Corporation in the 1950s first kind of started tackling this and it's really really well studied and there are really really really fucking good algorithms for doing it which you can download for free and they have a competition every year to see which one's the best and so so the question for the problem for six symbols was totally unknown but I kind of noticed that it was the same as the traveling salesman problem so I downloaded a program called LKH which finds solutions to the traveling salesman problem and I you know worked out its input syntax and fed this thing into it and I left it running overnight and then when I woke up the next morning it had disproved this 20 year old conjecture that the shortest thing for six symbols would be 673 symbols long was the conjecture I think if you added up but it's not true it's not true at all there's one that's 672 symbols long that had gone undiscovered for 20 years that actually you can discover in one night just by kind of reading the manual for a C program and leaving it running overnight so that's really the end of the story thanks. Anyone have any questions for Robin? I uploaded a kind of three page paper to the archive yeah so it's not technically published but good enough I think One right at the back, two at the back. Stop by the door. I did submit an edit to the online encyclopedia of integer sequences yes so that there is a sequence related to this problem which said that it was conjectured to be equal to this other sequence which is the sum of factorials and I submitted an edit saying it was conjectured to be equal to that sequence but actually it isn't and included a reference to that to my thing yeah. And one more question at the back. I've considered it yeah I left it running for a few days but it didn't really seem to be ending anywhere even close to the conjectured band so I think some more intelligent approach is likely to be needed just for seven. So I mean the other thing that's unknown is what the actual minimum is for six. So the program that I used is a heuristic solver it just searches for solutions but it doesn't prove that they're minimal. I tried an exact solver that proves that the solutions are minimal and it ran for a week and then crashed. So I still don't know the answer to that. Thank you.