Classifying the Bountiful Groups

Posted on November 13, 2025 by michaelng126

Some time ago, I was discussing the Groups, Rings and Modules course with a friend, and this inspired me to conjecture and investigate a result in group theory.

The text is accessible to any reader with a sound grasp of standard group theory results (e.g. group actions, classification of finite abelian groups). The proofs are explained in particular detail, for the purpose of a not too taxing, enjoyable read!

Introduction

Groups encapsulate the idea of symmetry, and they hence arise in a plethora of mathematical topics.

Indeed, Cayley’s Theorem (see below) shows that every finite group is isomorphic to a subgroup of a symmetric group – that is to say, given a group $G$ of order $n$ , we can think of $G$ as a set of symmetries on $n$ objects.

But now a natural question to ask: do we really need all $n$ objects? Indeed, $D_6$ , the dihedral group of $6$ elements is in fact isomorphic to $S_3$ – it is the set of symmetries of just three objects, let alone six. But $C_2\times C_2$ is not a subgroup of $S_3$ , thus it can only be embedded in $S_4$ . Which groups share this property – namely where we need all of our $n$ objects?

We define such a group as a Bountiful Group, and explore their properties. In fact, our final result shows that it is possible to classify them all.

Notation and Preliminaries

Cayley’s Theorem. Let $G$ be a group of order $n$ . Then $G$ is isomorphic to a subgroup of $S_n$ .

Cauchy’s Theorem. If $p$ divides the order of a (finite) group $G$ say, then $G$ has an element of order $p$ .

Notation:

$S_n$ denotes the symmetric group of $n$ elements. A finite $p$ -group is a group of order a power of $p$ .
The order of a group $\left| G \right|$ is the number of elements in it. The order of $g \in G$ is the smallest positive integer $k$ such that $g^k = e$ (if it exists). $S_n$ denotes the symmetric group of $n$ elements. A finite $p$ -group is a group of order a power of $p$ .
$G/H$ denotes the set of left cosets of a subgroup $H$ in a group $G$ .

Classification

Definition. A Bountiful Group is a (finite) group of order $n$ that is not isomorphic to a subgroup of $S_k$ for all $k<n$ .

In fact, it is enough that it is not isomorphic to a subgroup of $S_{n-1}$ since $k < l \implies S_k \leq S_l$ , but we choose the above definition as it reflects the motivation of symmetry given in the introduction.

We first note a useful result, and a standard result.

Claim 1. $S_m \times S_n$ is isomorphic to a subgroup of $S_{m+n}$ , where $m,n$ are positive integers.

Proof. Consider $m+n$ objects, and an element $(g_m, g_n)$ of $S_m \times S_n$ . Let $g_m$ act on the first $m$ objects and $g_n$ act on the other $n$ objects. This corresponds to a unique element of $S_{m+n}$ , giving an injective homomorphism $\phi$ say. Thus $S_m \times S_n \cong \mathrm{Im} (\phi)$ by the first isomorphism theorem. $\blacksquare$

Claim 2. If all elements of a group $G$ have order two, then $G$ is abelian.

Proof. $xy = (xy)^{-1} = y^{-1}x^{-1} = yx. \ \blacksquare$

We now find a restriction on possible orders of bountiful groups.

Lemma. Let $G$ be a bountiful Group. Then $G$ has order a prime power.

Proof. Suppose not. Let $G$ be a bountiful group of order $n$ . Then there exist two distinct primes $p,q$ that both divide $n$ . By Cauchy’s Theorem, there exist two elements $h,k \in G$ with orders $p,q$ respectively.
Let $H$ denote the cyclic group (of order $p$ ) generated by $h$ , and $K$ similarly for $k$ .
The left regular action of $G$ on $G/H$ induces a homomorphism $\phi_p : G \rightarrow \mathrm{Sym}(G/H)$ . We get $\phi_q : G \rightarrow \mathrm{Sym}(G/K)$ similarly.

Now consider the combined left regular action $*$ of $G$ on $G/H \times G/K$ via:

$\begin{aligned} * \ : \ G \times (G/H \times G/K) & \rightarrow G/H \times G/K \\ g * (g_p H, g_q K) & = (gg_p H, gg_q K) \end{aligned}$ .

This induces a homomorphism $\phi$ as follows:

$\begin{aligned} \phi \ : \ G & \rightarrow \mathrm{Sym}(G/H) \times \mathrm{Sym}(G/K) \\ g & \mapsto (\phi_p(g), \phi_q(g))\end{aligned}$

Crucially, this is stronger than the natural homomorphism to $\mathrm{Sym}(G/H \times G/K)$ .

Consider the kernel of $\phi$ . Suppose $g \in \ker(\phi)$ . Then $g \in H, g \in K$ by construction, i.e. $g \in H \cap K$ . But $H \cap K = \{e\}$ due to the coprime orders of $H$ and $K$ . Hence the kernel is trivial. Thus the homomorphism is injective.

By the first isomorphism theorem, $G$ is isomorphic to a subgroup of $\mathrm{Sym}(G/H) \times \mathrm{Sym}(G/K)$ . But by Claim 1, $\mathrm{Sym}(G/H) \times \mathrm{Sym}(G/K) \leq \mathrm{Sym}(\left| G/H\right| + \left| G/K \right|) = \mathrm{Sym}(\frac{n}{p} + \frac{n}{q})$ .

Since $p,q$ are distinct primes, so $\frac{n}{p} + \frac{n}{q} < n$ (smallest pair $p,q$ is $2,3$ ). $G \leq \mathrm{Sym}(\frac{n}{p} + \frac{n}{q})$ , so $G$ is not a bountiful group, contradiction. $\blacksquare$

We can now prove the main result.

Theorem. Group $G$ is a bountiful group if and only if it is isomorphic to $C_2 \times C_2$ or a $p$ -group with a unique subgroup of order $p$ .

Proof.

Forward Direction.
Let $G$ be a bountiful group. The lemma from above gives that $G$ is a $p$ -group for some prime, order $p^k$ .

Case $p$ odd: Suppose for contradiction that $G$ does not have a unique subgroup of order $p$ . By Cauchy’s Theorem, there exists at least one subgroup of order $p$ . Thus we can find two non-equal subgroups $H, K$ of order $p$ in $G$ . Then $H \cap K = \{e\}$ , since if $H$ and $K$ shared a non-identity element, then this element would generate both groups, giving $H = K$ .

We now recycle the argument used in the lemma above using $H$ and $K$ . The kernel is trivial by construction of $H \cap K = \{e\}$ . Thus $G \leq \mathrm{Sym}(\frac{n}{p} + \frac{n}{p})$ , but $\frac{n}{p} + \frac{n}{p} = 2p^{k-1} < p^k$ since $p$ odd, so $G$ is not a bountiful group, contradiction.

Case $p=2$ : The argument above fails, for the inequality at the end is not strict. So we need something slightly different.

If all the elements in $G$ have order 2, then $G$ is abelian by Claim 2 above. Thus by the classification of abelian groups, $G \cong C_2 \times C_2 \times \dots \times C_2$ . But by extending the idea of Claim 1, $G \cong S_2 \times S_2 \times \dots \times S_2 \leq S_{2k}$ .

If $k>2$ , then $2k < 2^k$ , so $G$ is not a bountiful group, contradiction. Else either $k=1$ , and $C_2$ indeed has a unique subgroup of order 2, or $k=2$ , yielding $C_2\times C_2$ .

Backward Direction.

$C_2 \times C_2$ is not isomorphic to a subgroup of $S_3$ by orders (4 does not divide 6), so $C_2 \times C_2$ is a bountiful group.
Now let $G$ be a group of order $n = p^k$ , with a unique subgroup of order $p$ , $P$ say.

Claim. Every non-trivial subgroup of $G$ contains $P$ .

Proof. Let $H$ be a non-trivial subgroup of $G$ . Then $p$ divides $\left| H \right|$ , so by Cauchy’s Theorem $H$ contains an element of order $p$ . But then this element generates a subgroup of order $p$ in $H$ , which is also a subgroup of order $p$ in $G$ . So this subgroup is $P$ , hence $P \leq H$ . $\blacksquare$

Now suppose $G$ is isomorphic to a subgroup of $S_{n-1}$ for contradiction. That is to say, there is an injective homomorphism $\phi : G \rightarrow S_{n-1}$ . This then induces an action of $G$ on a set of $n-1$ objects, $X$ say.

By the Orbit-Stabiliser Theorem, given $x \in X$ , $\left| \mathrm{Orb}_G(x)\right| \left| \mathrm{Stab}_G(x) \right| = \left| G \right| = n$ . Orbits partition $X$ , so $\left| \mathrm{Orb}_G(x)\right| \leq n-1$ . Hence $\left| \mathrm{Stab}_G(x)\right| \geq \frac{n}{n-1} > 1$ .

Now $\mathrm{Stab}_G(x)$ is a subgroup of $G$ and is non-trivial from above. So $\forall x \in X, P \leq \mathrm{Stab}_G(x)$ . In other words, each element of $P$ fixes all $x \in X$ , so $P \leq \ker(\phi)$ . But this contradicts that $\phi$ is injective. So $G$ is a bountiful group. $\blacksquare$

It is natural to ask whether we can go further, to find all such groups. By perusing resources such as group order statistics (see Element structure of groups of order 32 for example), we are led to investigate the generalised quaternion groups $Q_{2^k}$ . One can consult Keith Conrad’s excellent Generalised Quaternions article for a definition. In particular, we have a remarkable theorem:

Theorem. If a finite $p$ -group has a unique subgroup of order $p$ , then it is cyclic or generalised quaternion.

Proof. See Keith Conrad’s Generalised Quaternions article, Theorem 4.9.

Thus our final result follows:

Classification Theorem for Bountiful Groups: Let $G$ be a Bountiful Group. Then $G$ is isomorphic to one of:
$C_2 \times C_2$
$C_{p^k}$ for some prime power $p^k$ , or
the generalised quaternion groups $Q_{2^k}$ .

Proof. Follows directly from the two theorems above. $\blacksquare$

Acknowledgements

Many thanks to Professor Imre Leader of Trinity College, Cambridge for his support and advice. Thanks also to two of my good friends Yiannis, for his initial investigations with me, and Weida, with whom a discussion led me to consider this idea.

And to the quaternion group $Q_8$ , for being my canonical example to think about!

AI Sheet Music Page Turner 🎶

Posted on August 6, 2021 by michaelng126

App Store Link: https://apps.apple.com/us/app/ai-page-turner/id1574033597
App Website: https://michaelng126.github.io/ai-page-turner/

Page Turning has been a centuries-old problem for musicians.

Musicians often have both hands occupied, and so cannot turn pages without interrupting their performance. And so I wondered – could we fix this? In fact –

Could we use our faces, to turn sheet music pages?

Now, you can! Find out in the video above, and try it out for yourself!

Behind the Scenes

It was a huge learning curve, and great fun!

300 hours from start to finish, including research and coding
Self-teaching iOS programming – Swift 5 has grown on me! Coding design patterns, unit testing, and so much more to learn. More to come on this soon?

For some more amateur programming adventures, check out EcoScan – scanning food for their carbon footprint. Thanks for reading!

10 Tips for Cambridge Maths Students – what I’d tell myself!

Posted on April 19, 2021 by michaelng126

A few weeks ago, the President of the Adams Society (the Mathematics society of St. John’s College, Cambridge) reached out to me via email, kindly asking whether I’d like to give a short talk to first and second-years. The evening would be themed on giving general advice about the Cambridge Maths Tripos, with an additional focus on revision strategies and exam strategy. Thanks Xuanang for the opportunity!

Reflecting on my four years at Cambridge, I thought back to all those moments when I wished I had known something sooner. I then condensed these into 10 key points due to time constraints, and you can find them below.

These are by no means definitive, but nonetheless the collection of wisdom derived over many weeks spent!

Slides available here:

advice-for-cambridge-maths-students Download

Day in the Life of a Cambridge Math Student – behind the scenes

Posted on February 28, 2020 by michaelng126

Recently, I’ve been seeing ‘vlogs’ of various subjects pop up on YouTube, made by students at Cambridge. It made me think back to the times of UCAS applications and university visits… and upon bringing this up with friends from lower years, some of them even mentioned that these vlogs really helped them with university choices. If you’re looking for more about this, check out this website section on Cambridge advice!

Furthermore, I was surprised to see a lack of Math vlogs – and so with that in mind, I set out to try myself! It turned out to be incredibly enjoyable, and took around 40 hours to edit… more on that below. But first, a link to the finished product:

Day in the Life of a Cambridge Math Student – full video on YouTube:

Behind the Scenes

Walking around with a tripod ostentatiously took more courage than you might expect. The first point of difficulty was in Hall, where they didn’t allow me to record inside the kitchen! Other than that, people didn’t seem to care (and I had to be discreet in lectures, of course).

Another obstacle towards the end was finding background music – namely avoiding any copyright issues. After a full render and upload, I was shocked to find that one of the ‘free music’ tracks was not so free after all. What to do? Some original improvisation instead… (haha).

That’s all for now – enjoy!

How we built EcoScan – HackCambridge 2020

Posted on January 19, 2020 by michaelng126

I teamed up with trusty friend Mukul once again, entering Cambridge University’s official hackathon – HackCambridge – as a team of two. The goal? To develop a novel product in 24 hours.

We built EcoScan – an app where you scan your food, discover your eco footprint. We were delighted to win the BlackRock prize for the most innovative environmental project. You can find the slideshow here, and Github here. Screen shots shown below in all their rookie glory!

(Scan your food, and we’ll display its eco footprint for you!)

What follows is an account of the event, and some behind the scenes material – how we gradually focused on a feasible idea, plus reflections and tips during our development. Intended to be accessible, and somewhat educational (most computer science terms are explained).

As usual, plenty of photos and diagrams to lighten up the read!

Saturday 19th January

The sun shines outside as we make our way to the Corn Exchange, a spacious venue hosting the event. Mukul and I register and pick a seat close to the door (away from the loud speakers! – and close to the snacks). The atmosphere is bustling with anticipation.

The hall fills with over 300 competitors, but it doesn’t feel too cramped – thanks to the able planning of the committee. The sponsors of the event – including BlackRock, the cohost, Microsoft, Avast and others – give brief talks as we wait for the event starting at noon. All too soon, the countdown is upon us and the 24 hours begins!

Ideas (noon – 4pm)

Mukul and I try riffling through our idea brainstorm that we made two days before. But nothing seems to stand out immediately, so we go for lunch – a chorizo baguette – then walk about Cambridge for some inspiration. Ideas involving ‘zen’, ‘creativity spots’ and ‘connecting friends’ pop up as we amble our way around the Burrells’ Gardens in Trinity, but we still struggle to flesh out any convincing ideas.

Walking towards the sun – Burrells’ Field, Trinity College

It’s when we return that we start to realise certain restrictions (e.g. the data that we could find and access), and this guides us to focus on our project idea – an app that scans your food and shows you your carbon footprint.

And much like at any hackathon , this proves to be more difficult and problematic than expected!

Development Begins (4pm – 1am/7am)

Indeed, experience at past hackathons taught me a valuable lesson – know when you’re being too ambitious. It’s all too easy to underestimate just how long a ‘simple’ (simple, haha) task might take, and it’s better to do a smaller project well, than to finish with an unfinished product. This key point proves crucial.

Our initial idea is: ‘take a photo of food items, output the carbon footprint’. Simple concept, and sounds rather innocent.

But this proves to be much too complex.

But no giving up! Diffusion of enthusiasm from a member of the organising committee and close friend

Mukul goes about scraping the data from a website that gives this data for food – but it’s only in the form of a dropdown box selector, and there are hundreds of items, so he needs to automate this (rather than copy and paste each one painstakingly). I remember this kind of work to be extremely fiddly, having used the Python library selenium before – Mukul tackles this using Javascript from console, and it is horrendously difficult.

We attend a workshop given by Microsoft demonstrating how to use their Azure AI services, then armed with this knowledge, I start integrating their Computer Vision API into the project’s backend (using Python). But the issue I run into is that the API detects everything, not just the food. It’s not entirely reliable either. We could use some sort of NLP (natural language processing) to filter this, but then we’d be introducing two layers of unreliability.

The Computer Vision API essentially does this – pretty cool!

We consider scanning receipts instead, so I also try the OCR API (optical character recognition) – but hours in making it work and I realise that it too, is not reliable enough. It detects all items as well, and the text is often too small, prone to being blurred. Too unreliable!

Thus we pivot our idea and simplify. New iteration: ‘take a photo of specific food items, and output their carbon footprint’. In other words, restrict to a small group of foods for the sake of producing a reliable MVP (a fancy computer science term standing for ‘minimum viable product’) – that demonstrates the gist of the idea, but which is also complex enough. Dinner time!

Food is served in the neighbouring Guild Hall – many of the event sponsors set up stalls during the day with ‘free stash’ (as is customary at such events).

The simplification opens up new paths for both of us. Mukul no longer needs to scrape all the data, and can manually obtain it instead. He begins work on the crucial frontend – the design and linking of the browser activity (i.e. all your button presses and food photos) to my backend processing work. He starts by using React JS and uses design ideas from Google’s ‘Material Design‘.

I can finally try the Custom Vision API – where I can upload over 80 manually taken photos of bananas, apples etc., tell the Microsoft AI what they are, and then it ‘learns’ to detect them in future photos. It proves to be somewhat fiddly (there is a complication between detecting one or multiple foods in an image), but I finally manage to integrate it into the project by midnight. 12 hours to go! I go to get some rest, in fear of further worsening the sore throat that I had been developing.

Lots of manually taken photos of some food items. You tell the AI what they are, then it very cleverly ‘learns’ them to distinguish them in future pictures.

Mukul courageously forges on. There is a difficulty linking the frontend with the server – my backend work is in Python whereas the frontend is in React JS, so he needs some way of bridging the difference in languages. He adeptly manages to achieve this using Gatsby JS and GraphQL. Creating a sleek design is by no means a simple task, and he finishes this working through the night – till 7am! (what a team player)

Sunday 19th January (7am – 12pm)

Our foundations are now set, and Mukul passes the baton back to me in the morning to get an hour’s rest! Pressured for time, I work out how to deploy the backend onto an Azure Web App – installing it onto a computer in the cloud that will run the image-processing – with the help of a Microsoft mentor present at the event. There are also minor fixes to be carried out to do with the encoding of the image, connection policies and styling.

Only a few hours are left. Mukul expertly fixes a JSON issue in the receiving of processed data, then gets to work improving and adding the final touches to the design. I begin work on the presentation. Another lesson learnt from past experience is how important the design and presentation are. It’s all too easy to get carried away coding, only to leave a product with an unappealing interface; it’s not much good being unable to communicate your idea and its selling points either!

Minor shock ensues with an hour left, when we try our product after leaving it for a while. This is probably due to the cloud computer sleeping, and thankfully it returns shortly. We race to complete the final bits of code, then submit our code before the 12pm deadline. And the 24 hours is over!

But the hackathon is far from over. For it’s time for the demonstration (but lunch first!).

Demonstration (1pm – ~3pm)

It’s time for each of the 60 teams to set up their exhibition stall. The chaos then begins as the judges make their way around – the teams present and demonstrate their products.

Plenty of excitement bubbles up from the wealth of creativity – ideas include a summarising app to help History students, a game to teach JavaScript, a plant playing chess against the stock market – and lots more.

There are multiple prizes to win – the general competition, and prizes for more specific project types, themed by the sponsors. The finalists for the general competition give further presentations on stage, then the winners are announced! We win the BlackRock prize for environmental projects, and delightedly take home a pair of headphones.

Thanks

Thanks to my brilliant teammate Mukul for his talent and reliability – a shoutout to his blog here https://mukulrathi.com/. And my sincere thanks also to the fantastic HackCambridge committee for organising yet another wonderful event this year!

Promoting Student Welfare – the 2019 Trinity Oriental Society

Posted on September 16, 2019 by michaelng126

Report written as the President of the Trinity Oriental Society this year:

I can say with great pleasure that the Trinity Oriental Society has enjoyed another fantastic year. Looking back through the countless photos from this year’s plethora of events, both old and new, it makes me smile to see the joy of our members delighting in the free food, cultural activities and the heartwarming student welfare that our society brings. And with this fond nostalgia, I invite you to indulge in a summary of this year’s events, and some of the many entertaining anecdotes along the way.

Our first event was the Chaplains’ Squash, a crucial opportunity to attract hungry freshers to sign up. This proved fruitful (perhaps partly due to the authentic snacks from Hong Kong, manually imported by one of the committee members!?) – and our icebreaker event – the ‘Freshers’ Squash’ witnessed a healthy turnout. A fascinatingly large proportion being math students!

Chaplains’ Squash – attracting hungry freshers.

With ambitious targets in mind, we hosted an event almost every week throughout Michaelmas Term. Pepero Day – for which members post humorous photos of others to receive a box of Pepero (Korean chocolate sticks) delivered to their pigeon-hole – was a success, and so were the two Film Nights, including the Japanese anime movie classic ‘Kimi no na wa (Your Name)’!

We decided to bring back ‘Kimbap Making’ this year. (Kimbap is a Korean dish akin to sushi, without raw fish). For true authenticity, we enlisted the help of a good Korean friend (thanks, Gheehyun!) to recommend his recipe, including Korean radish, a special root vegetable, a unique ham and much more. One of the exciting parts of preparation for these events is the sheer scale of the amount of food – there’s nothing quite like getting up earlier to start cooking 12 cups of rice in the morning! There were limited places, and so members had to sign up beforehand. Some of them proved to be rather experienced, producing rather exquisite seaweed rice rolls – indeed an important life skill. The event was a resounding success.

Delicious ingredients, beautifully arranged! Very satisfying.

The exhilarating game ‘Mahjong’ – featured in a scene of ‘Crazy Rich Asians’ for example – is an essential part of Chinese culture, and so we created the ‘Mahjong Night’ – an original event for this year. The president and other experienced players were available to teach members the basics, moving onto simple tips and strategies. The students impressed us with their rapid learning, and some even tried the other game less talked about – that is, the Japanese variant known as ‘Riichi Mahjong’… It was particularly moving to see some of them then being able to teach others.

Fueled by the success of these events, we moved into Lent Term, hosting the Kimbap Making session and Mahjong Night once again. This year, there was a particularly exciting end to the traditional ‘Fire Ramen Challenge’ – in which members try to eat a plate of scorchingly spicy noodles (enjoyed by some as a casual snack in East Asian countries!) as quickly as possible. We recorded the competition, and it was fortunate that we did, for it was a ‘photo finish’ – we had to scroll back frame by frame to see who finished that fateful final mouthful first!

The highlight of the year however, was our flagship event – the Trinity Oriental Society Dumplings Festival, in celebration of Chinese New Year. And this year, we wanted to make it even more magnificent – with the aim of not needing to impose a limit on the number of dumplings each person could eat. The result? Through much forward planning, asking friends for extra pots to cook with, and even running through the snow to collect final pieces of cutlery (who knew that Sainsbury’s could run out of forks!), we managed to cook 1100 dumplings over a few hours – yes, 1100! – of six different fillings, allowing our members to delight in the deliciousness to their hearts’ content. Over a hundred members attended the event, and the dumplings were finished in less than an hour! The most rewarding aspect of the festival were international members (e.g. Jason!) thanking us for making them feel as if they were at home celebrating Chinese New Year.

I am incredibly grateful to my outstanding committee members this year – to Hazel, for being a fantastic Vice-President; to Weida, for her excellent secretarial roles; to Jimmy, for sending entertaining emails and spreading publicity; to Yiannis, for sorting out the funds; to Kelly, our bubbly honorary committee member (haha) and to Rui, for her remarkable reliability in booking all the rooms (and for carrying a Mahjong set from Peterhouse!). I thank my college father and last year’s TOS President Andrey for inspiring me to take on such an exciting and rewarding role. And we all thank Trinity College and our wonderful members for allowing this lovely society to keep growing. I have thoroughly enjoyed every moment of my role this year – and it is with my best wishes that I place the Trinity Oriental Society into the capable hands of Ivan Chan, the President for next year.

Michael Ng
Trinity Oriental Society President 2018-19

IMO 2019 – Coordination! Part 2

Posted on July 28, 2019 by michaelng126

This continues Part 1 – here.

Part 2 – University of Bath, Main IMO Venue

17th July – Marking Day, Move to University of Bath

The morning is rather frenetic, with plenty of paper copies of students’ scripts from over 100 countries. I work diligently through the 19 countries assigned to me for Problem 3, with my trusty coordination partner Lim Jeck (a former perfect scorer at the IMO!).

Lim Jeck – former perfect scorer! At Trinity (of course, haha).

The first challenge – to decipher the scripts. For a large majority of the scripts are in other languages, including Hungarian, Russian, Ukrainian, Bulgarian and more! Fortunately there are language consultants on hand, but the sheer number of scripts (over 100 per pair), and not so clear handwriting proves to be time-consuming. Our preparations end up taking up most of the day.

We move to the University of Bath in the afternoon, the main competition venue. There we see many students playing frisbee, having finished their second exam, and many familiar faces from Cambridge. They chose to be team guides, looking after their assigned country’s team throughout the week. I take this opportunity to take a short break – playing some badminton in the sports hall. There is a talent show in the evening with plenty of music and entertainment.

As the hours draw late, we finish up our initial marks and look towards tomorrow.

Surely someone will agree with me – that this looks like a triple moon from Super Mario Odyssey…?

18th July – Coordination Day 1, Intense Discussion, Board Games Evening

The main feature of coordination – is the process of discussing marks with the countries’ leaders. Each IMO country’s team has a leader (and their underlings), who also mark their students’ scripts in advance. The coordinators and leaders then gracefully dispute to determine the fair mark for each student. This moderation is essential, especially for scripts that may be obscure or incomplete – it is the leader’s job to then explain any further ideas for completion.

Day 1 of this begins early at 9:30. Our first discussion with Kazakhstan takes 45 minutes – but remarkably flies by! The mental challenge is incredibly intense – some of these leaders are distinguished professors at their respective universities, some have over 10 years experience at this activity.

Kazakhstan give us some special chocolate to share! Thank you!

Bulgaria, Italy and Iran all postpone, for our discussion does not fit in half an hour. Passionate arguments ensue, often over the difference between 0, 1 or 2 marks. This may sound trivial, but when there are only 42 marks available (6 problems worth 7 each), 1 mark could be that difference between a Gold Medal or Silver Medal, for that student in their final year. Further, as mathematicians, we highly value correctness and consistency – and this thirst for reaching the correct mark takes us into the evening.

We award our first 7 (out of 7) to a Polish contestant (who later achieves a perfect score!). An Iranian student comes up with a novel solution, but it is unfortunately incomplete. The Iranian leader does a fantastic job of explaining a way to finish it off. I meet the Hungarian leader and deputy once again (having seen them at the Joint UK-Hungary IMO Camp three years ago!) and they are impressed that we point out a flaw in a long Hungarian script immediately. We postpone some discussions till tomorrow.

Chinese Chess
Go! Who’s winning?
Xiangjia – a legend.

We relax by joining the board games night. There is a seasoned Go player from the Chinese team, and many card games in progress. It’s not often that you find a fellow Chinese Chess player, so I’m delighted when Xiangjia finds a set. Chinese Chess is a fantastic game – possibly more interesting than the English counterpart (haha) – check it out!

19th July – Coordination Day 2, 11 Hours of Discussion, Finalisation Jury Meeting

We face a deadline at 8pm to finalise marks – which seems manageable, given that our timetable ends at 3. Little did we know that we would almost struggle to finish! Indeed, most of the scripts are fine (the easiest being both sides agreeing to all zeros), with some highly notable exceptions.

The Bulgarian leader relents and later apologises that he must have been the most persistent! (and no, he wasn’t, haha!). We take a celebratory selfie with him.

As our scheduled end draws near, we have three engrossing scripts left – a Ukrainian, Canadian and Australian one. The problem captain eventually mediates for the Ukrainian script after compelling discussion. It is a similar approach to our first solution, but regrettably misses out a key idea. Time begins to run out, and the coordination team works hard to reallocate tasks.

The Australian script looks promising and defines a curious term – ‘cactus’, but fails to finish off one last step, and it is clear that the student was pressed for time. Even the leader, the legendary Angelo Di Pasquale, requests more time before our discussion! He then returns with a highly admirable and neat summary of the student’s solution, together with a way to complete it. Impressive dedication. We enlist the additional mind-power of other coordinators to give the script a consistent mark.

Such dedication! Turns out Angelo has been doing coordination for many years.

Finally, the Canadian script proves to be fascinating. This student was even more pressed for time and the ideas are sprawled out all over the place – but we see yet another fine display of dedication! The Canadian leader arrives with a detailed structure of the solution and points out the key omissions. It turns out to be a novel approach – and more remarkably, similar to a USA script! After much discussion, we join forces with the coordinators who marked that script, and our work draws to a close. We even get shiny Canadian IMO souvenirs (thank you!).

It’s now 8pm, and as we leave we find the Canadian leader team outside. Of course we take a photo – and it transpires that that Canadian student is in his final year, with his last chance to achieve a Gold Medal. I find it respectable that the leader did not mention this during the coordination!

The excitement of the day is not over. For it is time for the final jury meeting, to iron out any remaining problems, to finalise the results. There is a jury vote to determine the penalty for two students who have committed unfair play, and another to decide between a 6 or 7 for a certain script. I find the elimination process rather intriguing! Finally, the medal boundaries are decided and the results published.

We look with wonder at the two countries tied in first place – USA and China – and third place Korea, only one point behind! It is even more exceptional that all three of these teams achieved six gold medals and two perfect scores – each!

Medal Boundaries
Return of the voting bats

That marks the end of a tremendous coordination experience – both intense and immensely rewarding.

20th July – Bath Visit, Ben Green Lecture, Coordinator Dinner

Everyone is now free to explore, so I wander about to play some badminton. There is a Indian player who used to play professionally – and he lives up to that reputation. I join a group of friends, and we then take this opportunity to visit the city Bath. There is good ice cream (love ice cream) and we visit an abbey. I am horrified to see a tourist leaning on a Steinway grand piano.

I may be smiling, but inside I am horrified. The piano is crying inside. Grand pianos are not to be leaned on…

We also visit the Royal Crescent, then gather in the Forum for the IMO celebratory lecture given by the renowned Ben Green. He expertly uses IMO problems to link into advanced number theory and current research – it is a fantastic talk. When we return to Bath, the coordinators are rewarded with a lavish dinner for our efforts, and the contestants organise karaoke.

Coordinators’ Dinner
Karaoke!

21st July – Stonehenge, Closing Ceremony

It’s the final day of a sensational event. The leaders and coordinators embark on a trip to Stonehenge, one of the excursions that the contestants had organised. I enjoy a conversation with the Bulgarian leader (mentioned above) about his work on optimisation and am fascinated. We spend a few hours exploring the site.

Upon return, it’s time to pack as proceedings draw to a close. The final event is the Closing Ceremony, in which many medals are presented, many speeches delivered, many photos taken. The attendees are moved towards funfair rides to allow food to be set up. And after a few more selfies and expressions of gratitude, it’s through this final heart warming display of post-exam satisfaction and unrestrained enjoyment that I make my exit, marking the close to a phenomenal IMO 2019 experience.

Dr Geoff Smith himself! IMO Chair, past UK IMO Squad Leader!

Thanks

I thank Lim Jeck, my coordinator partner, for his brilliance and sticking with me through the long discussions – thanks also to all the other coordinators, language consultants and problem captains. I thank the leaders and their teams for providing such a rewarding experience. And I am incredibly thankful to Professor Imre Leader for inviting me to coordinate, to Dr Geoff Smith, the IMO 2019 Executive Director Ceri Fiddes and all the others at UKMT, and the sponsors, who made this extraordinary event happen. It is remarkable that all of this is powered by volunteers – indeed, this was a way to give back – and I hope that the UKMT continues to thrive in all of its glory for many years to come.

IMO 2019 – Coordination! Part 1

Posted on July 28, 2019 by michaelng126

The International Math Olympiad is a prestigious worldwide competition for high school students, held each year in a different country. This year, the 60th IMO was held in the UK, and I had the pleasure of being invited to coordinate – to mark the scripts! It was a remarkable experience, and I am highly grateful to all those who made this happen.

What does a IMO Coordinator do? And why do they stay in a five star hotel beforehand? (haha)

For more like this, see the Math Olympiad page.

Part 1 – Stay at Celtic Manor Resort (13th-16th July)

13th July – Arrival, Coordinators’ Dinner

Running to the Cambridge Train Station with a suitcase, I make it in about twenty minutes – and just on time! I happen to find Gheehyun, a fellow second-year at Trinity, and we discuss number theory. We arrive at the luxurious Celtic Manor Resort, a five star resort that hosted the Ryder Cup in 2010.

The coordinators are treated to a lavish dinner overlooking the golf course in the evening – a chance to start socialising and explore the surroundings. I enjoy a pleasant walk on the greens with Renzhi and Oliver.

14th July – Coordinator Introduction, Mark Scheme Creation

The next day marks the beginning of our work. Imre Leader – from Trinity! – is the fantastic chief coordinator, and he delivers an entertaining introduction to coordination – the process of creating mark schemes for the questions, marking the scripts, then most importantly, discussing the final mark with country leaders.

Creating mark schemes is no easy task. The IMO, like most math olympiads, uses a 0+ 7- mark scheme – essentially meaning that completion of a single problem is worth much more than partial work on all of them. It’s eye-opening to see how much dedication is put into specificity of mark schemes and analysis of possible situations to ensure consistency.

After lunch we are assigned to our individual problems – I am fortunate to be given the hardest problem on Day 1, a difficult combinatorics problem about social networks (graph theory), as follows:

Problem 3. A social network has 2019 users, some pairs of whom are friends. Whenever user A is friends with user B, user B is also friends with user A. Events of the following kind may happen repeatedly, one at a time:
Three users A, B, and C such that A is friends with both B and C, but B and C are not friends, change their friendship statuses such that B and C are now friends, but A is no longer friends with B, and no longer friends with C. All other friendship statuses are unchanged.
Initially, 1010 users have 1009 friends each, and 1009 users have 1010 friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.
See the whole paper on the official IMO website.

This was remarkably proposed by Adrian Beker – a fellow second-year at Trinity! Minor spoilers ahead.

In our group of 12 and a problem captain, we spend around two hours trying to solve it and to come up with ideas. The next four hours or so are spent coming up with a mark scheme. The problem proves to be rather difficult in both aspects!

The difficulty in solving it stems from there being many possible initial ideas, only very few of which actually lead to a solution. Eventually two key lines of attack are identified – preserving non-complete, non-even components, or preserving connectivity whilst deleting cycles. The first approach leads to a casework analysis, complicating mark scheme creation, for students may split cases in different ways. For the second approach, we have a lengthy discussion to draw the line between ‘solved’ and ‘unsolved’. It gets late and we postpone till tomorrow.

15th July – Opening Ceremony, Jury Meeting

An eventful day! We finish the mark scheme in the morning, meeting the deadline with little time to spare. Then we travel to the Opening Ceremony of the IMO, held in the Forum in Bath. It’s an honour to meet Po-Shen Loh – the USA Leader – over lunch, and of course we take a selfie:

Lunch for leaders and coordinators. See his website, Expii!

It’s a pleasure to meet the Uzbekistan leader too. The ceremony starts soon after. What happens? A DJ makes an (impressive) attempt to liven up the atmosphere, then we have a speech from the wonderful IMO Chair Geoff Smith! – followed by the country parade. Some countries are adventurous, and it makes me happy to see one team taking a selfie with the audience from the stage.

We finalise the mark scheme when we return. Dinner is a sumptuous buffet, and we try to taste all the desserts (haha). The Jury Meeting happens after, in which the leaders question the mark scheme and raise their (awesome) voting bats to make decisions! Afterwards, we find a piano and various members perform and sing.

Shameless plug for the YouTube Channel! Subscribe to Michael Ng Piano!

Desserts!
Jury Meeting

16th July – Day 1 Paper, Freedom for Coordinators

Students sit the first paper today, so the coordinators are free to roam. Gunnar (one of my supervisors at Cambridge) leads the pack to explore Newport. We play frisbee in a Roman ampitheatre and visit the Roman baths. Other activities after included mini-golf, relaxing in the steam room and sauna. and learning Mandarin pinyin tones on Anki.

We receive the Day 1 scripts in the evening, and start marking. We move to Bath the next day for the main competition venue – where the true excitement begins – see Part 2!

Getting into Cambridge – My Personal Statement

Posted on May 22, 2019 by michaelng126

Whilst participating in the Maths Open Day at the Centre of Mathematical Sciences today, I had the pleasure speaking to a lovely family, the son of which is applying to Cambridge. We discussed college choice, preparation, importance of olympiads – many topics, inspiring me to start writing about applying to the University of Cambridge (for reference, I am currently a second-year Mathematics undergraduate at Trinity College – the best college, haha).

One of those topics was on writing personal statements. The 4000 character monograph detailing your achievements and ambitions; why the university should choose you. And so, I promised to add my own personal statement to my website – here it is in all its glory. Good luck for your application, Tony!

This personal statement was for my 2016 application to Trinity College, Cambridge for Mathematics.

Personal Statement
Mathematics is unique. Not only for its logic and rigour, not only for the satisfaction when one solves a problem, but also for the joy of being able to spread astonishing results.
My experience being selected into the International Mathematical Olympiad Squad last year has broadened my knowledge significantly, due to my work over an extensive range of topics. In fact, ever since I began to tackle difficult problems, my passion for mathematics has never stopped developing. I have always enjoyed taking part in competitions and this year, it culminated in my selection to represent the UK at the Balkan Maths Olympiad. In Tirana, our team of six solved many problems; an interesting one concerned a rectangle of orthocentres, which we solved during a lengthy wait at passport control! In the competition, I solved three out of four problems to achieve a Silver Medal. It was pleasing to see a problem that used the more esoteric Wilson’s Theorem, about which I had fortunately read a few years ago.
Due to my performance in the British Maths Olympiad Round 1, with full marks last year, I was selected to participate in the Joint Hungary-UK IMO Training Camp, working with Hungarian students in problem-solving sessions and lectures. I learnt how to communicate clearly using appropriate, but not excessive, notation when necessary. Indeed, I greatly enjoy teaching and collaborating with others. Having been selected as a moderator on Brilliant.org, a thriving online mathematics community, I answer questions and have contributed over sixty problems, many of which have been featured. At school, I hosted the ‘Maths Club’ for two years, creating olympiad-style problems to broaden the younger students’ knowledge.
In the summer of Year 12, after a sudden interest in Turán’s Theorem, I held a lecture on Graph Theory. I went through seven topics, including basic Ramsey Theory (monochromatic triangle in K6, coloured red and blue) and planar graphs, leading ultimately to the Six (not four!) Colour Theorem. There was an emphasis on accessibility, with the audience being given interactive challenges, such as drawing K3,3 without crossing. Over twenty people attended across the years, and they each received a handout that I created with further problems.
I have had various successes in other Olympiads as well. This year, I achieved a Gold in the Biology and Chemistry Olympiads. I achieved a score in the top five of the country in the Y12 Physics Challenge, and received a ‘Roentgenium’ award in the Y12 Cambridge Chemistry Challenge. I have a passion for computing and was invited to the IOI selection camp due to a score in the top 15 in the UK. It is truly wonderful to see the importance of mathematics.
Being Deputy Head Boy of my school, I have improved my leadership and communication qualities. My commitment to the school is extensive, not only in mathematics, but also in music and House competitions. I greatly enjoy playing the piano and achieved a Diploma at the age of 12. I have played in several concerts, both at school and for charities, including one for the Japanese earthquakes this year. Every year, I have been involved in House Events such as chess and writing; having participated in House Public Speaking for five years, I won the competition in Year 11. I am also a keen player of squash and badminton.
A professor once said to me that ‘maths is a tool’. But for me, mathematics is not simply a tool. It is a pure and beautiful essence of truth, meticulously constructed from its axioms, from which one can even prove incompleteness theorems that demonstrate its limitations. I would greatly relish the opportunity at university to further my knowledge of mathematics.

UKMT Leeds Summer School 2017

Posted on July 17, 2017 by michaelng126

This year, I was fortunate enough to be selected as a Senior Student for the UKMT Leeds summer school. My role involved looking over the Juniors, giving them even more mathematics to think about to their hearts’ content.

The camp started with an icebreaker session upon arrival, building a bridge between two chairs. The effort of all the students proved fruitful, and the design was ultimately simple yet efficient! After the customary farewells to parents and a briefing, we went to dinner in our newly created groups of six, each led by a Senior student. Icebreaker activities followed, including some ‘puzzles’, some of which were pretty nasty! The night ended with a few group games including the likes of Splat and Mafia.

Monday marked the start of the proper schedule, with each following day roughly following a similar format. A lively breakfast with the Juniors, then some geometry in the morning. The masterclass was followed by ‘competitions’; some individual problem solving, followed by some groupwork then a relay. After lunch, the seniors began to prepare the Senior Surprise – the evening entertainment on Wednesday. Much discussion was had, until the theme of polyhedra seemed to prevail. Dinner was pleasant (especially the ice cream), then the Juniors had a talk about Conway’s soldiers. Games into the night, as before.

Dinner included some delicious ice cream.

A rather hollow ‘football’ – with our intuitive way to supply pentagons.

Tuesday began with a surprise – one of my team’s Juniors had to leave the camp, leaving us one short! In the competitions, we had a lot of fun doing more problems. Robin gave a wonderfully explained primitive root theorem talk, and at dinner, we were permitted to sit with people other than those in our teams. And so the seniors formed their own group, with some extended chatting. Not to mention the card games played before dinner…

The UKMT had provided some entertainment for Tuesday night, namely bowling! A short bus journey took us to Shipley Gobowling, and the seniors were allowed two lanes, four people each. It was fantastic fun, and we managed to fit in two games – high score for me was 108! Highest score was 122! Back at the gallery, we organised a concert schedule, and after that I played a bit of piano on the keyboard present. Appassionata, Fantaisie Impromptu, and more! It was tremendously enjoyable.

Wednesday was an eventful day for the seniors. For it was the day of the Senior Surprise! Breakfast followed by morning geometry and competitions, then a wonderful talk given by Oliver about Cayley’s Theorem about counting labelled trees. In the evening, all the seniors gathered to present the Senior Surprise concerning polyhedra. We gave three sessions, the first two of which were introductions, and the last was a proof of the six colour theorem. It was very enjoyable to present.

Thursday morning was largely the same, and in the afternoon James gave a talk on Functional Equations. In the free session that followed, the seniors prepared their second surprise – a performance in the concert! The evening concert was a splendid array of talent, ranging from poetry to musical instruments. There was even a Dedekinds Hotter Cross Buns performance! I played La Campanella. The concert ended with a stunning Senior Surprise performance and the traditional Complex Numbers song. We were permitted to stay up for longer, and so games were played outside.

Friday was the final day, and we enjoyed our final geometry masterclass. There were two more relays, and then everyone moved into the sports classroom for the final round – the mathematical pub quiz. Eight rounds later, and the points were totalled up for all activities throughout the week. And leading by over a hundred points were the Dedekinds (my team hehe)! Fantastic!

Fish and Chips marked the final lunch of the camp, after which people started to disperse on their journeys. Photos were taken, and thanks to the staff were expressed.

Overall, the camp was a wonderful experience, especially as a Senior! Perhaps I may return in the future.