For preparation, I did exactly the same thing as yesterday, because that seemed to help me tremendously. We did nothing for 45 minutes, and then stretched for the last 15 minutes.
The test opened with a geometry — great! However a slight scan at the other two gave me the impression that they are elaborate, complicated Number Theory + Combinatorics questions. That scared me a little bit, but anyways I have to focus on geometry first.
This year IMO took place in the University of Bath, UK, on July 11 to 22. It may not have been an exotic destination compared to recent years, but it a beautiful location nonetheless.
The two 4.5 hour papers took place on Tuesday and Wednesday. All 600 contestants attempted to solve 3 questions each day in a large sports hall in the university. Day one covered algebra, geometry and combinatorics while day two’s problems were about number theory, combinatorics and geometry. You can view the problems at https://www.imo-official.org/year_info.aspx?year=2019. Ireland did relatively well this year, Tianyiwa Xie and I both received honourable mentions, for problem one and four respectively. Lucas Bachmann won Ireland’s second silver medal ever, in over thirty years of competing, which was fantastic!
We could choose one excursion from several options such as Stonehenge and Glastonbury. The Irish team opted to visit Oxford. After being given a brief introduction to the maths course at the university in the magnificent mathematics institute, we were lucky enough to receive a talk from Andrew Wiles, the mathematician who proved Fermat’s Last Theorem. His informative lecture covered finding solutions to equations and briefly explained some of the topics he conducts research into. I found the lecture to be both fascinating, since research maths is something I have little experience with, and quite accessible. After this the group was split up, the Irish team were taken on a tour of Mansfield College. Our tour guide was friendly and informative, while the college itself reminded me of Hogwarts. We then had free time to explore the city. It is a beautiful, historical city with spectacular buildings, but it was difficult to get an inside into the atmosphere of the town during term time since it was mainly full of tourists. Despite the awful weather, it was a great visit and very helpful for anyone considering studying maths at Oxford.
We had a great guide, Tasos, who brought us all around Bath and the surrounding countryside. We went to Bristol and walked across the city to see a suspension bridge with an amazing view. As a team we played card games some evenings. The opening ceremony was short and to the point, speeches were kept to the absolute minimum, with the main event being the team parade. The closing ceremony, which was held in a marquee in the university, was similar. Afterwards there was a funfair, dinner and live music.
Overall I enjoyed IMO. I really enjoyed solving problems both in training and the competition. I loved exploring Bath, it has gorgeous architecture and a rich history, and despite the heavy rain, the trip to Oxford was fantastic. The papers this year were nice and I enjoyed solving the problems. Participating in maths olympiads has really allowed me to develop my problem solving skills and mathematical passion and ability. Thank you to everyone who organised the IMO and who organises the training and enrichment classes in Ireland for making it all possible!
I’m writing this on the flight back from the 17th International Linguistics Olympiad(IOL). IOL 2019 took place in Yongin, South Korea from the 29th of July to the 2nd of August, and it was a fantastic week. Ireland was represented by myself (Laura Cosgrave), Keelan Daye, Flynn Ryan and Páidí Walsh. We were accompanied by the best team leaders, Cara Greene of ADAPT and Harold Somers. IOL is a relatively young Olympiad at international level, so it is much smaller than IMO, with 209 contestants from 53 teams representing 36 countries/regions. The smaller number of participants means that it is much easier to do activities as a group and to get to know people.
Tuesday was the big day – the individual competition. A six hours exam with five tough problems, this was certainly a challenging, yet enjoyable, experience. It was like nothing I’d ever done before, despite being used to 4.5 hour papers from IMO and EGMO. It left us exhausted – but not too exhausted for karaoke! You can try the problems yourself at ioling.org/problems/2019/. My favourites were problem 2, which won solver’s choice, and problem 5.
The team contest took place on Thursday. Each team of four contestants had three hours to solve a challenging problem that requires communication, collaboration, logic and ingenuity. This year teams had to decipher the scoring and writing systems used for rhythmic gymnastics. Although this does not resemble a typical linguistics problem, the system of symbols is actually a writing system, with the main features of one. Teams were provided with laptops to watch videos of certain rhythmic gymnastics sequences. Some teams even tried to preform the moves themselves! The problem was fascinating and we really enjoyed solving it.
We had the opportunity to really explore Korea on Wednesday, with an excursion to Seoul. We learned about the Korean Hangeul writing system, and its history at the National Hangeul Museum, as well as getting an insight into Korean culture, with a concert that creatively combined both traditional and modern Korean music and dancing, including traditional drums, a traditional fan dance, and breakdancing. After this we had the opportunity to explore the Gyeongbokgung palace. Completely different to anything I had seen before, it was beautiful and has an interesting history. It was the main palace of the Joseon dynasty.
We then spent an hour or so exploring tourist shops and buying souvenirs. We visited a Buddhist temple, something that I had never seen before. The temple had gorgeous gardens full of flowers and was very peaceful. The day was finished off with dinner in a buffet restaurant. A lot was packed into one day and we really enjoyed it and learned lots about Korea.
IOL was an unbeatable experience. There was a fantastic program of activities, we were always having fun and meeting people. Between karaoke, games, a machine translation lecture, IOL’s Got Talent, IOL Jeopardy and #LinguisticsMakesFriends(a group activity that involved matching a long list of texts with the correct language), we were never bored. The social side of IOL is fantastic, one of the reasons the Olympiad was founded was to encourage teenagers who love linguistics and problem solving from all over the world to make friends, an aim it is definitely achieving. I met dozens of lovely people from all over the world and made great friends. Of course, solving this year’s problems in the individual and especially the team round was an enjoyable, mind-bending challenge. Linguistics is a fascinating field and I’m excited to learn more about it! From my own experience and from talking to people, it is definitely one of the most enjoyable science Olympiads. Thanks to Minkyu and the other organisers for organising it all and ADAPT and our team leaders for making Ireland’s participation possible.
To learn more about the All Ireland Linguistics Olympiad and to find out how to participate, visit ailo.adaptcentre.ie.
Introduction: European Girls Maths Olympiad (EGMO) is a mathematical contest similar to the International Maths Olympiad. It takes place every year in April and it aims to inspire more girls to participate in Olmpiads and take pleasure in exploring the world of mathematics. This year EGMO took place in Kyiv, Ukraine.
7th of April
So, here we are again, at the 8th rendition of European Girls’ Maths Olympiad!
We set off on the 7th of April, top of the morning. The flight wasn’t too bad, we exchanged at Amsterdam and the total time we spent in the air was no more than 4 and a half hours. We were lucky, compared to other teams from Australia or India where after spending 30 something hours in the air they still have to solve the problem of jet lag before going to solve the problems in the maths contest.
We arrived at Kyiv late that evening, starving and falling asleep. The hotel was marvellous, with 22 floors, which is the tallest building some of us have been in, and the bag we got was amazing and full of goodies. My favourite thing in the bag was a hair band with flowers on it, following the Ukrainian tradition for girls to wear wreaths on their heads.
8th of April
Today we got up late and enjoyed the bright morning sunshine in Kyiv. The opening ceremony in the morning was very enjoyable, the speeches were not long at all and it’s very interesting to see all the countries. After lunch, our guide Halyna, offered to take us to the city centre. She is really kind and incredibly eager to tell us all about Ukraine and learn about Ireland. We also kind of adopted another guide, Ivan, who is a friend of Halyna’s, and he has something to tell us about everything in Ukraine.
We went to a few shopping centres, visited the central square the Maiden, and experienced the deepest underground railways station in Europe. (whole 3 minutes on the fast moving escalator!)
We got back relatively late, around 8, and we are positively exhausted. Which is not necessarily a bad thing, because tomorrow is the exam and we need a lot of sleep.
We packed all our stuff, did a last minute review of some generaal tops, and went to bed at 9:30. A good night’s sleep, that’s what is the most important part of the maths olympiad preparation.
9th of April
Exam day No 1!
Exam starts at 9:30 and last until 2 o’clock in the afternoon. We got assigned breakfast time, and ours is the earlier shift, 7 to 8. I thought this was unfortunate, because it meant that we would get less time to sleep. However as later event turns out, this hour actually has its advantages.
Because of this time to burn, we went back to our room after breakfast and deliberately did nothing but sit there and controlling our breathing. It helped me calm down, empty my mind and oil the machineries in my brain. Later on we did stretches in our rooms, and it calmed my nerves immediately. When I met our leader in the hallways, she said the minute she saw me: “You look more relaxed.” Indeed, I had never been this relaxed while walking into the exam hall, and I think this state of mind actually helped in my exam.
I opened the paper, and found that either the first nor the second problem is a geometry, which in normal circumstances might cause me to freak out a bit, but that day I was relatively calm and just started reading the question.
Question 1 was algebra, my worst area, really. I think the most difficult part of algebra is that the letters and the computations are so abstract it’s often hard to truly grasp what’s happening in the full picture. You’ll have to keep your mind open and try every single possible technique, and along that line it’s easy to lose your way or go further and further down the wrong track. Sometimes by the time you realise that you are on a crooked path, you don’t have much time left.
For the first hour and a half, I was jotting down equations, manipulating them, changing their form by using factorising, observing patterns and symmetries. It’s easy to make a mistake and sometimes small things like writing “c” instead of “a” can have devastating effects. And after all that time I didn’t seem to be making much progress. All calculations I tried led to three symmetric equations that were neat but weird at the same time. They were concise enough, but I was not sure how to use them. And my work quickly became very messy, I had no idea where one led on to the other and it takes me awhile to even find the sheet I was just writing on. That’s when I decided to turn to geometry.
Granted, this was a risky thing to do, maybe it was a bit reckless of me, because the geometry was at the position of a question 3 — the hardest problem in the paper, and it was there for a reason. What gave me the courage, I think, was that geometry is my strongest and I have managed solved a EGMO geometry question 6 before (And I have hung up the draft paper proudly on my wall). So I drew the picture, and jotted down a few ideas that immediately came to my mind.
Geometry in my opinion, is really like a scavenger hunt. (well, all olympiad problems are, but geometry I think is especially like this) You wander about in the unfamiliar landscape noticing things and gathering clues. Once you get something, use that as a landmark and try extract more things from it. Our amazing geometry teacher (also our leader) like to use the metaphor of someone lost and trying to find their way back… you have to look for a clear noticeable thing in your field of vision, like an overhead power line. There would be cities at the end of the line. Then you work your way across the desert and across swamps but keep your eyes on the line.
So I noticed a few things, some obvious, some not so much, and then somehow after 45 minutes I thought I had a solution.
In retrospect, alarm bells should have been going off there—45 minutes for a question designed to challenge the gold medalists? But I guess I was under pressure and didn’t think much. I was happy, of course, thinking that I have one problem under my belt, so even if I don’t solve question 1, it won’t be too bad a result from day 1. Calmer and reassured I went back to solve question 1, this time employing the technique our leader advised us to use, which is using one separate sheet of paper to track all our progress. This is like drawing a map for yourself, tracking all the steps, writing down all possible things to try at each step, and if one doesn’t seem to work, you can use this to find the last place you stopped at and try another. And I also tried to be careful and probe a bit more before diving into one method of manipulation, just in case it won’t work out.
And chocolate, of course! I took one chocolate break, and I can almost feel my energy level pump up, much like the advertisement you see on TV, where someone stuck in a dismal situation takes a bite out of a bar of chocolate and suddenly there’s flowers and summer breezes and there’s golden light and magic and rainbow and he’s completely energised… that is exactly what happened to me.
All things combined my stress level had decreased, and suddenly I looked up from one line of equation and there’s the solution staring right at me. I gave a mental high five, looked up, and it was 45 minutes left.
I wrote out my solution neatly, arranged everything, and went back to looking at question 3. I noticed halfway through that I have made a mistake, classic mistake of too much wishful thinking and assuming something right. Shots of adrenaline went through my bloodstream, and I struggled to write down any other observation that might lead to a solution, that might get me a few marks down the road.
The last 20 minutes went by so fast. I spent it all trying to find a solution to the problem, to no avail. Frankly I walked out feeling quite disappointed with myself, because although solving question 1 was good, I spent way too long in that question.
I was comforted soon enough, by eavesdropping on the contestants sitting around me talking. Israel 1, India 1, … all saying that they got 2, maybe 3. These were world class problem solvers, and if they think that they only got two problems, then me getting 1 and maybe a little more does not sound entirely too pathetic.
Ah well, no one knows for now and we’ll know soon enough. We went into the shopping mall in search of a hot beverage, and slept or read or done nothing for the afternoon.
Around 5 in the afternoon just before dinner, we went down the hall with a box of games our leaders brought for us and went in search of people from other countries to play games with. We found the members of the Australian team and the Danish team, and we started playing the classic game of Avalon together.
Historical note 1: Avalon the game is a variant of game The Resistance, first released in 2012. In Avalon, the game features Arthurian Knights against evil minions of Modred, and it is a game of deception, lies and manipulation. First introduced by our team leader at her house during one of the training camps, it has now become the standard game for all Irish maths olympians and mischief lovers. Study shows that it is also popular among maths enthusiasts all around the world.
It was an amazing bonding experience, we made new friends among the two teams and got to know each other well.
After dinner, a girl from the UK team approached us and invited us to play a game of Irish Snap.
Historical note 2 (according to the UK team): Irish snap, also called Chinese snap, is a poker game involving reflection time and slapping on tables. You also have to be good at remembering rules as new rules keeps appearing and disappearing, and it’s easy to lose track. The game never ends.
Have to say, we have 4 members in the Irish team, with quite a few Chinese descent, none of us have heard anything with relation to the Irish snap/ Chinese snap. Just as well, it was a great game which became heated in the end before we switched to Avalon. We played till 8 in the evening (Avalon can often drag on forever) and at that point we decided to go back to our room. Another day ahead of us!
Throughout this article, I will begin by describing the Zeeman Effect, as well as the history of the discovery of the effect, and different people, notably Thomas Preston, who contributed greatly to the effect and discovered the Anomalous Zeeman Effect. In this article, I will be addressing the scientific side of the effect, rather than the mathematical, so for readers who are seeking a mathematical approach, I am afraid that this is not the article for you.
Some of you may have heard of Zeeman, but may not know what he is famous for and how he stumbled upon this discovery. Well, fear not, for hopefully all your questions are going to be answered. Pieter Zeeman was born in Zonnemaire, a small town in the Netherlands, on 25th May 1865. He studied Physics at the University of Leiden under Hendrik Lorentz, who will be mentioned in more depth later on.
Firstly, before I move onto Zeeman’s findings, I will briefly describe what a spectroscope is and how it works. A spectroscope is an instrument that allows scientists to determine the chemical makeup of a visible source of light. Spectroscopes may also operate over a wide range of non-optical wavelengths, from gamma rays and X-rays into the far infrared. Light is focused into a thin beam of parallel rays by a lense, then passed through a prism or diffraction grating that separates the light into a frequency spectrum, ranging from the smaller frequency to the bigger frequency wavelengths.
In 1896, Zeeman measured the splitting of spectral lines by a strong magnetic field. However, at the time, he did not realize that the lines were splitting, but instead believed that they simply became broader when a strong magnetic field was formed near the light source under which the spectral lines were observed. He also observed polarisation effects that indicated that the line was split in a manner consistent with the electron theory of Lorentz (which posited that in matter there are charged particles, electrons, that conduct electric current and whose oscillations give rise to light), but he did not realize this until later on.
He decided to continue the discovery and under closer inspection found that the spectral lines separated into either doublets or triplets when a strong magnetic force was placed near.
But Zeeman got no further in his discovery. In 1897, he was appointed as a lecturer at the University of Amsterdam and the disruption to his work caused by the move from Leiden was only increased by the inferior facilities he was forced to work with. He now had a smaller spectroscope, which lacked the sophistication and accuracy of his previous one. He printed thirty photographs of the splitting of the lines, but the quality of the photos was so bad that only one was deemed suitable.
However, around this time, an Irishman named Thomas Preston had heard of Zeeman’s discovery and wanted to investigate it in more depth. He decided to conduct the exact
experiment which Zeeman had done in 1896 in order to determine the accuracy of his theory.
First, before I launch into another description, now some background of Preston’s life. Preston was born on 23rd May 1860 in County Armagh. He graduated from Trinity College Dublin in 1885 in Mathematics and Experimental Science.
Preston learnt of the Zeeman Effect through G.F Fitzgerald, a fellow scientist at Trinity. He obtained a spectroscope from the Royal University and a strong electro-magnet, also from the Royal University. (Later he ordered his own magnet which was constructed to his own special design by the Dublin manufacturer Yeates and Co. This is probably why he remained in debt until his untimely death). Please note that these instruments were, in fact, of better quality and standard than the apparatus used in Zeeman’s conduction of the experiment. Preston was also offered use of the laboratories at the Royal University, where he installed his apparatus.
Thus, in December 1897, he presented his experimental results, stating that he had indeed observed the triplet nature of line splitting as reported by Zeeman. However, due to his higher quality apparatus, he had also reported that he had observed four-fold and six-fold splitting for two significant lines respectively. His photographs were of much higher quality than Zeeman’s and the four fold and six fold splitting of the lines was evident in these photos.
However, there was a catch. As this latter observation was something never seen before, Preston admitted that both these splits did not follow any simple law. A more powerful magnet was needed if he was to pursue his declared aim of seeking a law governing the magnetic splittings. After numerous months of trying to produce a hypothesis, Preston came up with his rule, now known as Preston’s rule. This rule states that all the lines of a spectral series have exactly the same pattern. Preston suggested that the Zeeman pattern was the same in all respects for all the corresponding lines of a given series and that this similarity carried over from one element to another where such elements had similar types of series.
This discovery was known was the Anomalous Zeeman effect, as on first observation, it did not follow any simple law. Later on, the six-fold splitting was realized as being the introduction of quantum mechanics but only after Preston’s death, when the concept of electron spin and wave mechanics was introduced.
And what an untimely death he had. In 1900, Preston died of a perforated ulcer just as he was reaching the height of his academic career. In 1902, Zeeman, together with his former mentor, Lorentz, received the Nobel prize in Physics, for the discovery of his effect, which couldn’t have been proved if Preston didn’t step in. Preston was basically forgotten until the 1920s, when his rule sparked the birth of electron spin (the quantum property of electrons, a form of angular momentum that is a fundamental, unvarying property of the electron) ;only then was his contribution to Physics realized.
I’ve been working on my timing for most of my live as musician, but more in a physical way than a strategic way. Hours practicing with a metronome or performing with drummers can certainly make you think about time in a deeper way than someone who simply uses time to mark the passing of their day. There is also the ability to make the most of your time, which is a popular subject among entrepreneurs who tend to work long hours and strive for a work/life balance. However, being productive is very different from knowing when is the best time to do something.
“Time is an illusion, timing is an art” – Stefan Edmunds
Recently I’ve been reading a great book called ‘When’ (The scientific secrets of perfect timing) by Daniel H Pink. A few key insights jumped out to me. Firstly, there is a natural slump in the middle of any activity. This is true for a thirty minute lesson or over a ten week course. Our work and focus naturally wanes as we become comfortable and lose track of our objectives. We become bogged down or distracted. According to the book, the surprising antidote to this is highlighting the midpoint. Scientists did a number of tests and found that when we are told or realise we have reached the halfway point of an activity, we begin to reorganise our approach and take action in order to succeed. Interestingly, this plateau of progress followed by disruption is mirrored throughout the natural world, even in the process of evolution. However, If we set the alarm bells to go off at the midpoint it kick starts the active stage in any process and gives us the best chance of avoiding our middle slump.
Let’s use Rockjam as an example: Our ten week term will start with the activity of choosing songs and writing. During the middle section of the term there will be a natural slow down followed by a frantic last two weeks when we realise we have a performance. A practical way to mitigate for the slump is to mark the midpoint of the term (week 5). We can give it a name and celebrate it. Let’s call it Hump Week. Hump week can be marked in many ways, but something as simple as mentioning it to the class should have the necessary effect. A solution for short lessons might be to stack the more cognitively demanding work for the start and end of a lesson when the students attention is more focused.
The book also contains many great insights into the best time for focus depending on age. The obvious one is that teenagers focus better after 11am, but I also learned that the midpoint of our day holds a natural slump of energy. The author sites examples of Judges deliberations which are more severe after lunch. If you are in the position to do so, it would be wise to organise your day around your own natural rhythms and sleep cycle. If like most, you are at the mercy of someone else’s timetable you can mitigate by adding naps, walks and strategic caffeine breaks to your day.
For anyone interested in how to decide the best time to start any activity I recommend Daniel H Pinks book ‘When’. Personally I intend taking a hard look at the ‘when’ of all my activities.
(A book review & a mimic of the dialogues in the book, borrowing Mr T. and Mr.A.)
Disclaimer: this only shows my partial opinion of an amazing book which has a lot of ideas interwoven within them. There is so much to this book, and to the Godel’s theorem, that it is not quite realistic to capture them all accurately in one article. I just hope to recommend this book and to provide an relaxing and fun read.
Achilles and Tortoise are visiting the Escher Museum in The Hague
Achilles: Why, will you look at these wonderful paintings! They combine different perspectives in one frame and create a paradoxical world. Tangled hierarchy and strange loops are everywhere.
Tortoise: Yes, I can see why Escher is your favourite painter.
Achilles: Speaking of Escher, were you not reading a book a few months back about Escher? You were fascinated by the book and promised to tell me about it.
Tortoise: Why yes. The book is called Godël, Escher, Bach—an Eternal Golden Braid, written by Douglas R. Hofstadter.
Achilles: My my, what a long title! And what a long name the author has!
Tortoise: The book itself is long as well. It is seven hundred pages long. You can see why it took me so long to read.
Achilles: Seven hundred? That definitely won’t make a relaxing read. But as long as it is interesting, I guess it is worth the time.
Tortoise: I assure you it is deeply intriguing. I have never read any book like this.
Achilles: What kind of book is it?
Tortoise: It is a marvellous book.
Achilles: …Mr Tortoise, you are known for your wise remarks and clarity of mind. I beseech you to offer me a more explicit explanation.
Tortoise: I would be willing to do that, but it would take quite a while to explain. Say, why don’t we sit down at the nice little cafe and have a chat?
Achilles: Yes of course! (sitting down in one of the bean bags offered by the cafe) Wow, comfy! And the cafe is playing music by your favorite Baroque composer, J.S.Bach! What a wonderful coincidence.
Tortoise: (straining his ears to listen) Yes… It is the cello suite No 2 in D minor, I always find it particularly calming.
Achilles: Yes, it does have a pleasing air. Waiter! (motioning to a waiter in a smart white shirt) A cup of cappuccino, please. And for you, Mr Tortoise?
Tortoise: Tea, please. No sugar.
Achilles: Now, you were telling me about this book…what IS it really about?
Tortoise: Dear Achilles, you do pose for me a hard question to answer. Even the venerable author himself admits in the preface that, quote, “This question hounded me”. It is a book that explores different layers of interpretations that arise in life and all fields across maths (hence Godël), art (Escher) and music (Bach). It is also about strange loops and paradoxes that we encounter, and how, when used correctly, things that arise from these things will seem…well, how do I put this…seem to have a life of their own. It is as if we are creating animate things from the inanimate.
Achilles: This reminds me of the endeavour of creating Artificial Intelligence.
Tortoise: AI is indeed mentioned with great zeal in the book, as it uses self-reference to improve itself. See, the way AI learn is from the past experiences and the information that it had already processed. It has the ability to self-repair. We haven’t achieved it yet, clearly, but the magic of self-reference is in use in that branch of study. The book also touches on topics like neuroscience, genetics, physics and linguistics. They all involve strange loops and paradoxes. It would seem that we could never do without self-reference. The author used art, music, and more importantly, maths to demonstrate this point. It actually ties in with a new and exciting branch of maths called “Metamathematics”.
Achilles: Very deep and philosophical. I hope this is not one of you maths books. You know how bored I get, reading those books. No offence, Mr T., but the book you gave me the other day on Statistics nearly killed me. (yawn)
Tortoise: (The waiter came on with the cup of coffee) Here, some caffeine will wake you up. No, I assure you, the book is extraordinarily well written, with amazing illustrations and playful dialogues inserted between each chapter. To make things easier for the reader to understand, the writer borrowed two characters from Lewis Carroll, Mr Tortoise and Mr Achilles, to explain the concepts.
Achilles: Who, you?
Tortoise: Why, you!
Achilles: Coincidence upon coincidences! The two characters share our name!
Tortoise: Yes, I wonder why I have never noticed it before. Anyways, the two characters debate in miniature dramas to explain things to the reader. The drama are sometimes a mimic of a piece of music, very likely a Bach piece, a piece of Escher artwork or a maths story.
Achilles: Much like the conversation we are having now?
Tortoise: Yes, just like what we are doing right now. The dialogues are one of the parts I enjoy the most in the entire book. They are ingenious and awe-inspiring… No really, the book is completely readable, the author took his time in explaining the maths, and invented a new system to explain in detail Godël’s theorem. It is not at all hard. Afterall, this is not a book on maths.
Achilles: Still, I fear that the book might be too long and boring for me. I am not the best reader of maths books.
Tortoise: No matter, I personally took delight in the parts that are not about maths as well. You can always skip around the chapters, and that would be totally enlightening too.
Achilles: You mentioned a Doodle theorem, what is that?
Tortoise: You mean Godël theorem. It is a theorem put forward by the logician, mathematician and philosopher Kurt Godël. It is quite intricate and complicated, but in its core, it states that not all theorem can be proven.
Achilles: Not all theorem can be proven? Is that you being pessimistic and cynical again, Mr T.?
Tortoise: Unfortunately, no. this fact has been proven already, fair and square.
Achilles: You mean… there are conjectures that cannot ever be proven no matter how hard and how long we try?
Achilles: (trembling, takes a sip from his cup) This is too frightful! I feel like I am approaching the end of the world at the speed of light. Would you like to give an outline of how he proved it?
Tortoise: Very gladly, but I’m afraid it would take far longer than necessary, and besides, you can find out about it as you go through the book. At the moment I’ll give the basic idea. The proof involves creating a strange loop in any formal system using self reference. An example of this is the sentence “This sentence is false”.
Achilles: Ha! I know this trick. Don’t try to trick me with this. This is the classic paradox, if you say that this is correct, then that means that this is wrong. If it is wrong, then it is correct. Such a decease of a sentence is obviously not relevant and completely trivial.
Tortoise: Maybe, but trivial or not, it is a valid statement. And if it is a valid statement in a formal system, we must be able to decide if it is true. And we would have to prove it, true or false. However, as this sentence is paradoxical, it is undecidable. What Godël did was to create a self-referent sentence like that inside a mathematical system and concluded that this is a theorem that is undecidable—therefore the system is incomplete.
Achilles: Hmm…you do sound convincing. But I still am skeptical. To me, this sentence is not true–nor false– it is pointless. Self reference does not lead to anything and it is better just to not ask this question!
Tortoise: You don’t realise, but you just touched on another part of the book! It is another interpretation and approach to these kind of problems. It is the Zen Buddhism approach to the problem, by “unasking” the question.
Achilles: Unasking? How can you unask a question once it is spoken?
Tortoise: That’s the one of the core ideas of Zen. In order to understand and able to interpret the hole in the mathematical system that creates that paradox, we must ascend to a more powerful level, a “metalevel”, if you will. But then again this level face the same problem as the ordinary level, so it also need a “meta-metalevel” to mend its hole, and so on and so forth. You will need an infinite amount of “meta”s to be able to describe the final level that gives us a complete system without defect. This is impossible to contain in our finite world, so the Zen approach to transcend infinity is to “unask” the problem instead.
Achilles: (trembling, takes a big gulf from his cup) Pft! It is cold. We have talked for so long. I fear my head might burst.
Tortoise: Do relax a little, Achilles, and listen to the cello sonata, it will clear your head immediately.(calmly sips his tea)
We had two days of excursions As such are the traditions So that the teachers could correct our papers With no distractions
After two days of examinations We set the alarm with precision To catch the bus that will take us far away 2 hours to our destinations
Long bus rides were such an exertion And the mountain roads were no relaxation The boredom was the worst part because There ‘ain't any activity of gratification
Normally at times with no occupation We memorise a math olympiad question And reason and calculate in our heads Or lay down a few equations
But now everybody was in a state of exhaustion, Our brains too ill to function Anna sitting on the left side was just too car sick To start a conversation
Why didn’t we play games for recreation? But it’s so hard to reach a decision When you’re in a hot, stuffed car near the state of suffocation
Poker had been a temptation But we do not wish the affliction Of crawling around in a moving car looking for a card scattered in an acceleration
At last we were here, to our alleviation And where, exactly, were the tourist attractions? This time Salt Mine Turda and Alba Iulia Places of fascination
Only, only we were part of the exhibition As people took photos and documentations Of a group of weird people in similar attire Walking around with no sense of direction
All the fame and reputation Have no reason for glorification For they still forbid us to get on the ferris wheel Due to the age restrictions
Despite the various limitations We still found delectation In new bonds formed, new friendships made, During the socialisations
And ice creams made perfection I’ve had several, despite parents’ prohibition: “At most 3 per 2 weeks!” But who cares? When I’m out of their domination
Ah, I’m being a bit unfair there, the excursions were very enjoyable, if I say these two days were a little low point of the entire trip it was because the rest of the days were just too marvellous. The bus rides were indeed a bit tedious, but we did work around it in some ways. We also made loads of new friends, and that’s the best part!
12th of July
Today was the closing ceremony and farewell party. So I should probably mention the results now.
We got our results after the trip to the salt mine. When I got off the bus, I saw our deputy leader at the front door. I could feel my heart skip a beat, and I pasted a smile on my face and went over slowly. I knew that I did not do well in the exam, and I had an expectation of a really really low score in my head. So all of us walked up to her…
And she went: “Ok now you got 5, you got 5… we tried very hard to get you more marks but we couldn’t get it higher than that, we think your answer is definitely worth more, though.. You got 16, well done!! I think you have a good chance of getting a medal…” And now she turned to me: ”You got 3…”
Now what happened was I BEAMED!! I was so happy I was actually jumping up and down. Our leader was very kind and caring, she obviously thought I would be disappointed or sad so she was very gentle and tried to sooth me and comfort me, but after a few seconds I think everybody realised that my lunatic behavior was actually from the heart, not trying to appear strong or something.
Lucas patted me: ”There, there.”
Not realising this as a joke, I said: ”No, I am GENUINELY happy that I didn’t get zero!!”
So that’s that. No more anxieties and worries now that the dust has settled, the only thing left was to enjoy the every last minute of the remaining journey.
This morning, we walked around the city under the sunny sky, fed the pigeons, ate traditional Romanian food, went into bookstores after bookstores…it was bliss. I could not have had a better morning than this.
We had to go back for lunch and the closing ceremony in the afternoon. We all knew what the closing ceremony would be like, so we just sat back and secretly read books.
Afterwards, it was the event we’ve all been waiting for: the farewell banquet! We had some prior knowledge about what will happen— in short, a disco.
We had a beautiful dinner together, sitting together with the TTO team. The main dishes there were a little bit weird to my Asian taste, but the desert was superb…I had two creme buleés.
Meanwhile, a band was busy setting up on a stage and soon there was music. People started gathering at the foot of the stage and moving with the music. I was awkward at first, but soon I got comfortable as more people joined in. We were gathering together and lining up and forming circles all the time— I was constantly pulling people into the dancing, and I tried with every effort to pull the Chinese team in. The teachers were very supportive and loved to take part, and followed me into the crowd, but the teammates were less active and only one of them tried it briefly. The teachers confessed to me later that they were mindblown to see this kind of festivity.
I should give a clarification here: My “dance” basically consisted of jumping up and down as long as I could, and lots of other people on the crowd couldn’t dance either. We were just moving together because of the immense force of energy that was flowing around us.
At one point, I had just come down to have a rest when this guy, with his T-shirt soaked in sweat, came up to me and said: ”I don’t understand, why are you not dancing??? I don’t understand why ARE you not DANCING???” I protested ”But I just came down and I was dancing for the last 3 hours…” But he won’t listen, so he pulled me into the crowd and I pulled Anna into the crowd and Anna pulled the TTO team into the crowd and so we started again.
It was one of the best dancing experience I’ve ever had, dancing came so naturally, and moving was like an instinct. There is something about the math olympiad discos.
I didn’t sleep well yesterday and this morning I was quite a mess. I was on the bus and trying to wake my brain up by doing multiplications. Yet instead of waking up it just caused me to panic when I could not figure out what is 15 time 15 and what is 25 times 25.
After getting to the arena at a little past 8, I just sat there doing nothing. Can’t really do reviews now, as all our things are taken away from us except for some food and our pencil case. The contest was due to start at 9 and a half. I spent the hour trying to wake up and figure out what is 15 times 15. On my desk, there was a bottle of water, a banana, a few snacks, my name tag and the folder for the questions. In the folder, there was a sealed envelope containing today’s questions, in whatever language you like, and three smaller folders that you are supposed to put the answer sheets in. There are two types of paper, one yellow for scratch work, one green for a formal answer. There were 5 cards in the folder for “Water” “More paper” “Toilet” “Questions” and “Help”. If you need anything, raise the corresponding card. Before the test even started, I raised my white card to ask for more paper–10 pages that were given would not be enough, I knew.
4:30:00. The test would be 4 and a half hours long. I opened my envelope as the timer started ticking away, and was relieved to find Q1 to be a geometry–my strongest subject. Nice.
I scanned through the questions and quickly jotted down all thoughts that came to my mind. Geometry, use equal angles, similar triangles or perpendicular lines? The third one (Although the possibility of me even trying the third one is rare) is a combinatorics question—guessing, and induction? Sequences might be involved… I have to use the first half an hour to make sure I understand the problems and have no doubts because, after the first half an hour, you are no longer allowed to ask questions.
I’m quite a beginner in Olympiad, so I am not expecting anything better than an honourable mention. Therefore I would focus all my energy in solving the first problem.
4:00:00 Geometry—get cracking! This one involves circles, isosceles triangles, and parallel lines. Not a common combination, I had to say. But the graph looked easy enough, and given the difficulty of the Q1 problem of the past few years were not high, I was fairly confident.
3:00:00 Wait, wait, wait… I wasn’t going anywhere! This keeps going round and round in circles! I assumed the wrong thing! I took a deep breath and asked for more paper.
2:30:00 I diverted my attention to Q2, an algebra question, hoping a ray of inspiration will hit me as I was busy thinking of something else.
However, my brain kept whispering: Algebra is not your strong suit! This is a Q2! Your best bet is on Q1 and I won’t have you not getting it because you wasted half an hour on a question that you won’t ever get out anyway. A wave of anxiety hit me as I was trying to figure out what should I do.
2:00:00 I was trembling. I raised the green card and escaped the room to go to the toilet. I ran to the toilet, locked myself in, raised my arms to form positive posture that according to Amy Cuddy (For anyone who have not seen her TED talk, super recommend!) will improve my confidence and overall performance. I splashed some water on my face—calmer now. I ran back to the exam hall. No time to lose!
I made up my mind now. Just do Q1— Even if I can’t get it out at least I can say I tried my best.
1:00:00 I drew my 5th graph of the problem and went over my data. Looking back, it’s amazing how fast time past when you’re stuck on a problem.
0:30:00 Although I know I still have hope I’m already accepting the possibility of not getting anything out of day 1. This was a cruel thought, my cheeks were burning and my head was throbbing and my hands are trembling. I told myself—relax, calm down, accept the fact and be happy. There’s nothing you can do about it, you tried your best, so be it. Walk out of here smiling.
0:20:00 I checked my answers, numbered them. I presented all I have, hoping, basically, to not get zero.
0:10:00 I took one last frenzied look at the graph–ray of inspiration?
0:05:00 I wrote my name. Loudspeaker bellowed.
0:01:00 I put the answers into the folders. Arranged them neatly.
0:00:03. 0:00:02. 0:00:01. Time’s up. The timer turned red.
It was not a good feeling, walking out with a bad result. I was trying to be optimistic and telling myself that this is the first day, it was not over and also, it is the experience that matters, not the marks. But still, I did feel very disappointed with myself and it was still quite exhausting for me now, sitting here, half a year after, remembering it.
I stayed relatively calm and joyful after the exam and didn’t freak out. For the entire day afterwards I managed to push the thoughts of the exam into a box in the corner so that I could still focus on the present. We went shopping together…I thought it would be completely weird for us to, out of all the things, walk around a supermarket and not buying anything, but it turned out to be a fun and relaxing time.
After the hustle and turbulence of the day were finished, in the night lying by myself in the dark, I started reflecting. I comforted myself by thinking “I did my best” and of course internally I had doubts—did I actually try my best? Could I have done better? Could I have coped better and explore more and think more instead of just sticking that same point in the same question?
But the optimistic side, of course, was that there’s another day out there, and also—although it is a big cliche already— The marks weren’t everything and I just decided to try my best and go for another day.
10th of July
So this morning…
While walking into the big arena…
I had a eerie feeling that something BBBIIIGGG will happen.
I tribute this to the extra big mug of coffee I had this morning. But anyways, something did happen that morning, so who knows… sixth sense?
We arrived at the arena at around 8 o’clock, like yesterday. Less nervous I walked around the arena chatting to friends. I spent loads of time finding everybody from my team (when I found them, turns out everybody are at the table of the New Zealand team). I also managed to find my friend from Iran, friend from Ecuador, and friend from China.
Apparently, it wasn’t only me, everybody was much more relaxed than yesterday. A contestant was doing push-ups on the floor and his team members were around him counting. (I actually think this was the spark that inflamed everything that happened afterwards )
About 40 minutes to the exam start, I went to the bathroom and even before I re-entered the hall, I could hear roars of applause coming from the arena. Not wanting to miss anything, I speed-walked back and this is what I found (photo source: 2018 Cluj-Napoca facebook page)
Of course, I was shocked to find that a lot of my friends were lying down as if they were just having a relaxing day off in the warm sun on a Hawaiian beach. And the next thing I knew, I was lying down with them on the concrete floor, facing the bright fluorescent light. It really helped with the butterflies in my stomach.
But the next thing that happened made the butterflies flutter even quicker. The committee apparently thought it was a good idea to put on climatic music on the second day of the contest so the arena began filling with dramatic Start War type of music. The music was really tempting for something big to happen, maybe it was just me and my butterflies, but I thought I could even feel the tension building up inside the room.
After a while, everybody started getting up… and started taking a stroll around the hall. I mean, why not? A stroll was very healthy, particularly in an international exam hall with several hundred people including child prodigies and possibly future Field medal winners from all across the world. Soon everybody was in a parade, pulling their friends and motioning to the volunteers to join in. When the atmosphere had built to this extent it was impossible not to start running, so everybody started to jog. Frankly, the speed was quite fast and I was panting and sweating after just a couple of lapses. (Were the people leading this a maths prodigy AND sports champions?) But I felt not at all tired, I felt exhilarated, running alongside my friends, old and new, running in the stream of brilliant minds from all across the world.
After a while, everybody was getting a bit tired and had to stop for a while. Just when the parade seemed to be ending, a boy in a yellow shirt picked up the flag of IMO from the row of flags at the side of the room and started sprinting with the flag flying behind him. Soon, the flags of different countries joined in. The teammates of each country started grabbing their flag and running after it, and soon the entire scene seemed far too surreal. My vision was soon filled with the moving colour and the moving stream of people waving their flags, and I couldn’t pick up all the information that’s flowing around me all at once. Almost if I was travelling the world, almost as if I’m joining a global community. We change posts in leading the flag whenever one is too tired in sprinting. And have I mentioned the music?
We moved to the back the room after a while, and raised the flags over our heads and gathered them together. We raised them on tiptoes, trying to reach as high as we can.
View from below, the flags are shining under the light of the big arena, the flag of IMO being held among them. We hugged our friends, shook hands with strangers and right here the young people from all around the world were connected through one single event. It was maths and knowledge that surpassed the boundaries of countries and brought all these people together under one roof.
We finally put the flags back where they belong and then slowly sat down. It was 10 minutes to the exam. The music slowly died down. But I was still panting and heaving from excitement, and I felt a strong surge of confidence as I looked at the folder with my challenge inside. I turned around, and smiled to the Indian contestant sitting to my left to whom I have never talked to before, and said: ”Good luck!”.
Am I over-romanticizing this? I must admit that I might be, as I thought that this will make world news but later nothing happened. But I was so very deeply affected by this event, I was almost on the brink of tears as I hugged my teammates and my friends from around the world.
Raising the flag of Ireland there in that Arena almost felt like raising the Olympic flame to me, and the resemblance is very strong in my opinion. One of my teammates later joked that this might be the closest we have ever come to world peace…