### The Dame Kathleen Lecture II

Be there or be square. Actually you have to be square now, since you have no chance of being there! It is over now and this was the programme:

- 5.25:
*Sudoku, Latin squares, Geometry, and Hall's condition* - Professor Peter Cameron - Queen Mary, University of London
- Solutions to Sudoku puzzles form a special case of a type of combinatorial designs called gerechte designs, invented in the 1950's for designing experiments in agricultural research. Finding such designs includes such problems as finding a Latin square orthogonal to a given Latin square. Also, the problem of completing such a design from partial information was developed in statistics.

The experience of solving Sudoku puzzles suggests that finding such designs (with some values given) is hard. It is known that completing a partial Latin square is NP-hard. Some similar problems about gerechte designs are unsolved. The problem can be formulated to look like Hall's marriage theorem, but it is not known whether there are conditions analogous to Hall's which are necessary and sufficient for a solution.

Further conditions can be imposed. One variant, due to Robert Connelly, has particularly close connections to several topics in finite geometry: spreads, resolutions, and perfect codes.

Peter Cameron was an undergraduate at the University of Queensland before coming to the UK and obtaining a DPhil from Oxford University in 1971 under the supervision of Peter Neumann. He held a position at Oxford before moving to Queen Mary, University of London, where he is Professor of Mathematics. He is an expert on combinatorics, permutation groups and the structures (for example, designs, graphs, codes and geometries) on which they act. He has authored approximately 250 publications, including several books, and has supervised over 30 doctoral students. In 2003 he was joint winner of the Euler Medal and this year his 60th birthday was marked by an international conference in Ambleside.

He has an Erdos number one too: cool again!

This particular Wednesday I was quite unusually feeling bouncy. (Not in the morning though since I had no breakfast: DAMN HALF TERM!) I laboured through my lectures (apart from the Real Analysis one of course, in which DC improvised quite remarkably! Well he did something different to his notes, but I was always confident that he wouldn't make us cross everything out in case we came to a dodgy conclusion!) I am not getting the material that we are doing now properly, since the bounded business is simply confusing: especially with millions of mini-theorems. But that being said, I was given some sound advice by DC: "Don't spend too much time on this course!" Human nature is such that you tend to spend more time on things that you enjoy, and find fun. Consequently, I have been spending too much time on analysis and algebra. I have spent zero time on stats, and zero amount of time on PDEs too, and a little time on complex and calculus. Then I wonder why in the world the work has been piling up!

That is an honest representation of how I have been working. I just love the analysis and algebra course, because I love the lectures that go with them. There is no question about that. I know DC made a fair point, (as I asked him to have a skim through my proof, and it is not everyday a lecturer tells you to 'study less' for their course!) but it is really difficult for me to sink my teeth into the other stinking courses at the moment. Yeah, that is how I feel about them. Well stats and PDEs especially, they are just confusing me at the moment.

Anyway! After the lectures and some lunch, I had to attend some formal matters. A divergence if I may. Another word that you might possibly want to use to describe me, is energetic or enthusiastic. I would like to think that I am enthusiastic in things that I set my mind to, since that is always a must in my case. Motivation and enthusiasm are directly linked. Because I am motivated in analysis and algebra, I am always found looking forward to them lectures, and go 'Awww' when they are over! However, this trait of mine is also one which many people might find annoying. There are certain degrees to this annoyance, but I make no apologies about being enthusiastic about the things I take part in. As I said before. No matter how unlikely this is, I would be reluctant to employ you if you were not enthusiastic. Actually, before this turns into the whole post, I will pick it up again some other time. A note though: if you find my being annoying, then you do know that you don't have to talk to me? If however I am annoying you for a reason, then please do tell me to stop! I don't get offended by things like that (most times)!

So back to this formal business. It was swings and roundabouts. I seem to be building many responsibilities for myself, and just added another one! So at the moment, on top of my studies (19hrs per week) I have three other responsibilities at university. I think if I manage to bring the studies up to scratch, I will most likely feel better about everything! (Slim chance of that happening though...)

After all this it happened to be 4pm-ish and the the Tweenies united. Well actually they all busied themselves with drinks, whilst I got my badge. However, after I got my badge it was my turn for a drink. From the wide(cough) choices available, I chose.... *drum roll please* TEA! Yes, it was one of the most delightful cups I have had in a while. I could actually accuse someone of 'spiking' my drink, but my jumping around before hand proves me wrong. I was actually quite excited about the lecture in my defence! My friends were rather amused to find me with two milk tubs (which were unfortunately green topped). I then got to work. No dilly dallying with the string: I used the spoon. I have to have my tea made in a certain way, I mixed it and mixed it again until it became the right colour, and then got rid of the tea bag. Then the two sugars went in, and finally the two milk tubs. After the first portion of milk had gone in, Bella exclaimed, "That is enough milk isn't it." But nope, it was still not the right colour, and so in went the second!

I have since had another cup at home, but it just didn't taste quite the same. If you ever have the pleasure (ahem!) of meeting me, and there happens to be a kettle nearby, you will be left wondering 'coffee, what's that!' Or so I hope.

Today I felt no fear. I felt comfortable and was happy at being there. I didn't feel awkward this time round, and didn't feel that I stood out amongst all them mathematicians. This was a welcome feeling. I think having all the Tweenies around was good too, but if I am being completely honest: if I had to go to the lecture on my own, it wouldn't have been a problem. I think it is wrong of me to ask the Tweenies to go in future. Well the reluctant members anyway! Jake and Fizz seemed up for it, but I think Milo and Bella would rather not have been there. I don't want them being there because I asked them to. Well they did have the option to decline... but having them there was better than not having them there! I would rather not 'oblige' them to come, since it is not fair of me to do so. Well maybe next year we will see what happens.

Did I mention how happy I felt? Jake, Fizz and myself had ended up discussing maths and what not, whilst Milo and Bella had found themselves sat at the tables. The hustle and bustle of being there, made me disregard the fact that I was naive undergraduate. Unsurprisingly, on the occasion I wasn't afraid of seeing my lecturers around, I didn't see many! Well I saw them, but didn't manage to speak to any (apart from one or two, who I think just watched as I was unable to stand still and talk at the same time).

There was a talk by someone in the atrium, but I didn't catch much of that. It was about the new building, and I just sighed! I am still sat in the middle about the building you see, and will not be moving for a while. The community feeling (I suppose) was immense. Being there just felt great; I felt like I belonged to something. In this community no weird looks exist, and no one calls you a freak for being there. That is indeed what I love about this community and that of this blog too.

After the five minute talk, we all stood (bounced) in the atrium, talking and having a drink. Eventually we all made our way to one of the larger rooms in the AT building for the lecture. I didn't look for any lecturer to provide cover for me, as I had last year. I think with things like this, there has to obviously be a 'first time' which prepares you for the other times. When everyone had settled, Prof. G invited the speaker and it was here when something changed within me (as I mentioned in the other post). Prof G. mentioned reading Professor Cameron's travel writing. It was in this moment, that yesterday famous mathematicians advice hit home, "To hell with what everyone thinks." echoed in my head. I did go into another cloud for a minute, but I think in the community present, my mathematical passion is not seen as perverse, which is a welcome change.

Onto the lecture now! (Finally I hear.)

Professor Cameron, as has been rightly said about him, is a fantastic lecturer. It was a rather pleasant lecture and very well delivered. My only problem is that had we started at 5ish, we could have gone to 6ish, and the end wouldn't have been slightly rushed. But then again, that want is a selfish want!

The lecture was about how ridiculous newspapers sound when they claim, "Sudoko's need only logic and reasoning. There is no mathematics involved." Quite a ridiculous claim as I am sure you will agree! To be honest, when I wasn't even a mathematician (i.e. college), to persuade my friends to have a go at them, I would say that there is no maths involved, since you can use letters instead - it's all about logic. Maybe that is why I laughed even harder, as I recollected this! Indeed, sudoko consist of or even are Latin squares, so letters would make perfect sense.

The mathematics behind sudoko make them all the more interesting, and magic squares seem to be floating about too. I mean, if I was to go and do a sudoko now I would firstly feel more 'Latin- squred informed', and secondly complete it with more satisfaction because of being slightly aware of the maths. I didn't bother too much which with reading the slides, but instead tried to listen. As soon as I did read a slide I lost track of the lecture (which was rather annoying for me).

In fear of remembering anything incorrectly, I will just try to briefly mention the lecture. I confess that some stuff did go whoosh over my head, and another bad habit of mine resurfaced. Having finally understand numbers in different bases, I became preoccupied 'checking' a square with numbers in base 3 (I think!) I evidently was remembering everything incorrectly because I wasn't getting the same answer. I think it was the square 00 which was said to be equal to 3. Anyway, I am most likely saying incorrect things here since I vaguely remember what it was. (I just remember the 00). If anyone else was at the lecture, please feel free to correct anything incorrect that I write.

However, I had a 'get in there' moment, when the word permutations was mentioned. Now, composition and decomposition of permutations is one thing that I can hopefully do! The word derangement was also mentioned, and it was one of the many words added to my mathematical vocabulary. Other words include: cosets, affine plane, F^4, quasigroups and many more which I will remember later, and hope to actually make sense of. I had the infamous shrek face when them words were mentioned, but I let it be.

It is remarkable how Professor Cameron seemed to be very well statistically informed! He built the story up from the very beginning, and there was mention of Euler and his conjecture too. I particularly liked the phrase Euler Spoilers, which described the people who spoiled Euler's conjecture! (Euler was only correct in his conjectures for n=6(?). ) I can't recall his conjecture I'm afraid. This post is rather disjoint, since different things are coming to me at different stages.

Magic squares are squares where all the rows, columns and diagonals all sum to the same number. Latin squares are sudoko puzzles: well if we take a 3x3 sudoko puzzle then its properties are such that you can only place the number one, once in the first row. It also must appear only once in the second and third rows, but not in the same colums as the other ones! An example would be:

1 2 3

2 3 1

3 1 2

The latin square is also known as the Latin-Greco square, but I am too lazy to write that every time. After Euler the story continued to how other peoples contributions and ideas. The 'just squares' idea was one I liked. I can't remember the mathematician who said it (behren??) but I found the idea of 'just mathematics' amusing. I remember the diagram with the fertilisers, and the idea about it, but can't write it properly in words. This somehow introduced the critical points/set, and a few other definitions.2 3 1

3 1 2

I unfortunately missed a whole slide because of Bella, but all is forgiven! It was about the error code thing, and I honestly didn't catch a word. It seems that the application of these squares is rather impressive, and indeed they themselves are. Especially when the symmetry was considered. When we discussed the affine plane, I got slightly confused with all the coordinate business. I understood mod3, but I didn't really follow the coset and x_3=x_2=0 part and about the affine plane. Although admittedly that is a cool name!

As my jumps continue an example was highlighted, and I a theorem (the 'marriage theorem' Hall?) was discussed. I was amazed at it, because when solving Sudoko I have actually done what was done on the board. If in a sudoko (3, 6,7) can be written in three boxes and (2,3,6) in another. Then it is obvious that the 2 has to go in the fourth box, since 3 and 6 have to go into the first three. Next time I am doing a sudoko I will do so with a mathematical smile on my face!

It is best I stop soon, since the jumps are increasing. There was a whole comfortableness associated with the lecture, and I think the lecturer had a hand in that. The atmosphere was pleasant, and we ended at 6:15. Alarm bells were already ringing in my head, but I stayed until everyone left. (Dame Kathleen even said a few words too). I really would have preferred staying a little longer but had to dash. I actually did dash, and got weird looks from two people in the maths department. (A post grad student and the computer officer as they walked towards the maths building). I think what surprised them was the fact that I was running. If you must know in school sporting events I was the sprinter! I am OK when it comes to a quick dash and enjoy it, but I have let myself go when it comes to the long haul. I could do it at a casual pace, but sprinting it is for now.

Having said that, my legs are wooden but I love running. Last time someone said I should have snapped lecturers who nodded off, and indeed today I laughed to myself when I spotted one! It was only one on this occasion, but for just the smallest of time intervals. There were quite a few people with cameras, and thankfully the annoying woman from last time wasn't here. She had honestly never let go of the button, and the blinding flash had been most annoying. (Is it obvious that I have forgotten something?) Well whatever it is, it can wait another day! I really enjoyed being their today, and maybe that community will soon include another subset of people. The lecture was interesting and I have even got a cool name badge! I wonder whether the slides to the lecture will be available?

I have gone against my own restrictions and finished this post at 1:37am. The recurring words in my mind: who cares what others think. Nobody should make you change what you do. (Well that depends obviously, but on this occasion it holds). As long as you are happy, just keep on doing what it is you do. Hopefully more of these types of lectures will happen this year.

## 3 comments:

Ah I have remembered! I wouldn't mind knowing the number of undergraduate students present at todays lecture. (Say out of curiosity...)

Well, I quite enjoyed the talk; very interesting. I followed most of it, missing a few parts but got the general idea of what was going on for most of it which was probably the idea really.

The x2=x3=0 part was to do with the geometrical interpretation. I can't remember the exact detail but the table was defined as a four dimensional affine space over Z_3 (An affine space being loosely like a vector space without the concept of an origin) where x1 was the cell-row coordinate, x2 the cell-column coordinate, x3, the row-within-cell coordinate and x4 the column-within-cell (I think, may have been different order)

So the x1=x3=0 etc. where equations of lines in the affine space representing certain solutions of the table.In other words, the conditions of solving the table were translated in terms of certain points lying on lines.

I wouldn't worry about there not being many undergrads there, after all, I am sure it was mentioned that one of the aims of the thing (and indeed one of Dame Kathleen's personal interests) was developing the interest in mathematics of younger people. It is now an anachronism but I think that in days of yore, undergrad students attending such events to engender an immersion into the culture of their subject was a lot more common.

Intersting it was indeed, and quite enjoyable too (was a very pleasant time). My problem is when I tend to get caught up in one thing that was said, rather than follow what is being said.

Ah, thanks for that explanation! I followed the lines representing solutions, but didn't know how the co-ordinates were defined and how he had got to them lines.

I don't think it worries me anymore, since I would rather not be part of that culture.

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