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As a student attending a grammar school in the UK ,I have been fortunate to have access to various opportunities to showcase my mathematical abilities. These include participating in maths challenges and engaging in lunchtime clubs dedicated to mathematics, which provide around 35 minutes of additional support. I recognize that these opportunities for mathematical exploration and development may be more limited for students attending state schools in the UK. Lack of support from the school or peer pressure can hinder their mathematical growth, as some students may face stigma or negative stereotypes associated with being interested in mathematics.

However, when comparing the mathematical opportunities available in the UK to those in the United States, it becomes evident that the UK lags behind. The US offers a plethora of mathematical opportunities, particularly through competitions , such as MathsCounts, AMC, and various math leagues. In contrast, the UK seems to have only one prominent maths competition known as the UKMT. Additionally, the US provides further support through courses offered by the Art of Problem Solving (AOPS), which are renowned for their mathematical depth and rigor. It is disheartening to witness this disparity in opportunities.

In my pursuit of mathematical excellence, I have acquired two books from the UKMT. However, I have found them to be somewhat lacking in clear explanations or in-depth exploration of certain topics. Consequently, I am seeking recommendations to address two specific needs:

  1. Preparation for maths olympiads: I am in search of books that provide comprehensive guidance and practice problems specifically tailored to excel in maths olympiads, allowing me to further enhance my mathematical skills and problem-solving abilities.

  2. Interesting maths topics: I am eager to explore captivating areas of mathematics such as topology or the beauty of primes. I seek resources, books, or materials that delve into these topics in a sophisticated and intellectually stimulating manner, helping me appreciate the elegance and depth of these mathematical concepts.

  3. Any courses in preparation for the UKMT challenges

I would greatly appreciate any recommendations or guidance from individuals who can help me find suitable resources to fulfill these needs. Thank you for your assistance in this matter.

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    $\begingroup$ In case it helps others, "grammar schools" in the US mean something different than "grammar schools" in the UK. $\endgroup$
    – user1815
    Commented May 28, 2023 at 20:48
  • $\begingroup$ @A.Goodier Yes sorry I will edit it $\endgroup$
    – Shay_18
    Commented May 29, 2023 at 12:14
  • $\begingroup$ Are you at GCSE or A Level? Sounds like you might end up doing maths at Uni. If so, then good luck! It's difficult, but rewarding. It's a great idea to get ahead. I'll respond to your second question. Honestly I think (asking and answering questions on) the math.stackexchange.com website has helped me more with maths than any other resource. The introductory Springer books are good, but even they require/recommend a first-year undergraduate (university) background. $\endgroup$
    – Adam Rubinson
    Commented Jun 12, 2023 at 18:52
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    $\begingroup$ There are many other good maths books, but in terms of books your (second) question is quite broad to be honest. It really depends what year you're in and what books you have read /courses you have taken previously. Start with first year undergraduate maths and then your second question will have an actually productive answer. $\endgroup$
    – Adam Rubinson
    Commented Jun 12, 2023 at 18:52
  • $\begingroup$ Grammar schools are a type of state school. $\endgroup$
    – tell
    Commented Jan 17 at 21:24

6 Answers 6

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For UKMT competitions, the books and past papers are the best sources of material you can use to prepare. BMO papers dating back to the early 1970s are available on the British Mathematical Olympiad website. Detailed solutions are available in the books by Geoff Smith and Tony Gardiner. If you persevere with the problems in past papers and ask your teacher or the Mathematics Stack Exchange when you get stuck, you will make progress. My experience with the books is not that they are lacking in clear explanations - again, ask your teacher or MSE if there is something that is not clear. You may also find that some of the exercises are designed for you to discover and prove results for yourself.

The books focused on specific topics, such as Bradley's 'Introduction to Number Theory and Inequalities' and Leversha's 'Crossing the Bridge' are structured more like traditional school textbooks with exposition followed by exercises. The olympiad books are designed for the reader to focus more on solving the problems, so much of the exposition comes after you have attempted the problems.

The UKMT also offers other opportunities beyond the challenges and olympiads; there is a UKMT summer school, a mentoring scheme, an olympiad mentoring scheme and training camps for invited students. The team challenges have been paused for the current academic year. The resources from the mentoring scheme (monthly worksheets) are designed to help students progress from just qualifying for the olympiads to performing well. You should be able to access these resources through your school.

Geoff Smith has some excellent advice here about olympiad preparation.

In relation to online courses, AFAIK, you can still access online courses offered by AoPS if you have the means to afford them. Kevin Olding of https://mathsaurus.com/ also offers UKMT-focused courses.

On the second point, I have a few book recommendations:

  • 'The Shape of Space' by Jeff Weeks
  • 'The Music of the Primes' by Marcus du Sautoy
  • 'Euler's Gem' by David Richeson
  • Any introductory number theory book - I can recommend Bradley's book (mentioned above).

You might also like to check out this reading list aimed at sixth form students intending to read mathematics at university.

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  • $\begingroup$ Thank you for your response. I have in fact downloaded most of the AoPS introductory books online. There appears that there is a significant disparity between the two, with the introductory books feeling relatively easy while the intermediate books pose a considerable challenge. Nevertheless, I will continue to work through the introductory textbooks.In parallel to my engagement with the introductory books , I will look into the book recommendations you've provided - most likely 'The Shape of Space' by Jeff Weeks as the realm of topology and the concept of knots intrigues me. $\endgroup$
    – Shay_18
    Commented May 29, 2023 at 12:43
  • $\begingroup$ You might also like 'The Knot Book' by Colin Adams. Knot theory is very accessible for secondary school students - there are few prerequisites. $\endgroup$
    – A. Goodier
    Commented May 29, 2023 at 14:58
  • $\begingroup$ I have actually read 'The Knot Book' by Colin C. Adams, well, most of it. Colin C. Adams is a great author who can explain complex mathematical concepts in an easily understandable manner. $\endgroup$
    – Shay_18
    Commented May 29, 2023 at 18:47
  • $\begingroup$ I came across this website (www.knotplot.com) created by Dr. Robert Glenn Scharein, which features excellent pictures of knots he has created. Amongst them, my personal favourite are hyperbolic knots, due to their intricate and curvilinear nature. I also found a great video on YouTube (youtube.com/watch?v=IrlaVaATiOY) which provides a great explanation of knots. Out of interest, what website do you use to program knots? Most programs or softwares I find online either require money or do not work. $\endgroup$
    – Shay_18
    Commented May 29, 2023 at 18:48
  • $\begingroup$ I do not know of any free software for this. Mathematica has a knot theory package, but it is not free. $\endgroup$
    – A. Goodier
    Commented Jun 1, 2023 at 17:21
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I will address point 2.

I would say there three main accessible areas according to my mathematical knowledge.

  1. Discrete Maths (Susanna Epp, Discrete Mathematics with Applications)
  2. Operation Research (W.L. Winston, Operations Research: Applications and Algorithms)*
  3. Programming (Python) (I prefer using Udemy an online platform rather than textbook. link to course)**

*Some parts of course can be a bit advance. Learning LP(linear programming) and IP(Integer programming) is good goal

** A good goal will be to learn basics of Python, numpy, matplotlib and pandas. Although this may not seem related to math programming play important role in numerical methods and understanding mathematical algorithms as well as data science. Some parts of the course are quite advance however I encourage you try to complete all the sections.

Requisites: Requisites for discrete math and programming are very limited. For operation research you should be comfortable solving a system linear equations.

Although studying can be hard you can find fully worked solutions to most problems using Quizlet.com. If you enroll in a course at Udemy you will have instructor who can answer your questions.

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  • $\begingroup$ thank you for your help - any maths courses u could recommend $\endgroup$
    – Shay_18
    Commented May 28, 2023 at 13:44
  • $\begingroup$ @Shay unfortunately I have not taken any online math courses on an online platform. I think you should be find most of the content easy from the textbooks assuming you have met the requisites. $\endgroup$
    – Tegh
    Commented May 28, 2023 at 15:40
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In Vlaanderen (which is the Dutch speaking part of Belgium), we have a website, containing all questions of the Math Olympiads (https://www.vwo.be/vwo/wedstrijdvragen-per-jaargang-en-ronde/).

The questions are written in Dutch, but a good translating engine should not to be too hard to find.

There also is Köhmal, which is directly written in English (https://www.komal.hu/verseny/feladatok.e.shtml).

Have fun :-)

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Addressing point 2, a few interesting maths topics/unsolved puzzles that you could delve deeper into are:

  1. Oppermann's conjecture, relating to your interest in prime numbers.
  2. Hodge conjecture, relating to your interest in topology and knots.
  3. Euler characteristics, eg. Euler characteristics of Torus/Klein Bottle, also relating to your interest in shapes and space.

Enjoy :)

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    $\begingroup$ The Klein bottle is truly fascinating ! It is great to present the concept of non-orientability . I was thinking of buying one of them but they're so expensive ! $\endgroup$
    – Shay_18
    Commented May 30, 2023 at 8:16
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This responds to comments in @AGoodier's answer. @Shay_18 says,

Out of interest, what website do you use to program knots? Most programs or softwares I find online either require money or do not work.

SageMath is similar to (expensive) Mathematica, but it is free. It has substantial knot theory capabilities:

Sage

They just updated the knot functions this month (Dec 2023).

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  • $\begingroup$ Can you say in what ways Sage is similar to Mathematica. $\endgroup$
    – tell
    Commented Jan 17 at 21:31
  • $\begingroup$ @tell: I am not the best person to make that comparison. I am a regular user of Mathemtica, but have only used Sage a few times. You can find (possibly biased) comparisons on the web, e.g., here. $\endgroup$
    – Joseph O'Rourke
    Commented Jan 17 at 22:50
  • $\begingroup$ But you called it similar. $\endgroup$
    – tell
    Commented Jan 18 at 21:28
  • $\begingroup$ @tell: That is my understanding from a presentation of Sage at JMM. I cannot detail a point-by-point comparison. In any case, I was responding to the knot capabilities of Sage, not a general Sage vs. Mathematica comparison. $\endgroup$
    – Joseph O'Rourke
    Commented Jan 18 at 23:33
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For olympiad preparation, the best book in my opinion is

Arthur Engel, Problem-Solving Strategies.

See this other answer I posted.

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