A maths circle is a small group that meets regularly to work on problems with no syllabus and no test, often staying with one question for several weeks. In the UK, circles and circle-like clubs are run by universities and specialist maths schools, usually free and often with limited places. Modern Age Coders runs a weekly circle-style class online for pupils aged 10 to 18 through our olympiad and competition maths course, as a batch or 1-on-1. We are not part of any university programme. Below: what a circle is, the main UK programmes and who each is for, and a three-week circle arc we wrote.
From the programmes' own pages, read 31 October 2026 · Independent of every university and school named
What is a maths circle, and can my child join one online in the UK?
King's College London Mathematics School, which runs the Axiom circles, defines a maths circle as "a group of students who come together to grapple with intriguing puzzles, discover and explore exciting ideas and learn to think like mathematicians." UK pupils can join several free, partly online programmes: Axiom maths circles from Year 7, with places prioritised by published criteria; the University of Oxford's Online Maths Club, a weekly Thursday livestream open to all; and Cambridge's STEP Support Programme, free online modules for sixth formers. Modern Age Coders, which has no link to any of them, runs a weekly online circle-style class for ages 10 to 18 on our competition maths course, with groups of 15 to 20 pupils or a private teacher.
Where to begin
Three courses that run like a circle
Each card opens a syllabus of problem themes rather than exam topics.
A circle is a community and a pace, not a course. The same pupils meet week after week, a problem may stay on the table for a month, and nobody is graded.
Three things that make a circle
Continuity. The Axiom page puts it plainly: "Axiom is a long-term programme and students who join in year 7 will be guaranteed a place on the programme for their entire school journey." A circle builds over years.
Open problems. Questions are chosen because they lead somewhere, not because they appear on a paper. Peers. Pupils talk to each other more than to the leader, and the leader asks rather than tells.
How it differs from a class or a club
A tuition class moves through a syllabus and checks progress. A school maths club often runs puzzles that end in a session. A masterclass series is a short, intense block. A circle is slow on purpose: ideas get time to go wrong and come back.
King's says the pupils who most enjoy its circles "are high-attainers who love problem-solving and imaginative, creative thinking", which is a description of temperament as much as attainment.
A circle compared with other kinds of maths session, our summary
Which maths circles and university programmes can UK pupils join online?
Several, and most are free, but each has a stage and an entry rule. This table is built from each organiser's own page; check the page before applying, since rules change.
Maths circles and circle-like outreach open to UK pupils, from each organiser, read 31 October 2026
Programme and organiser
Stage
Format
Who can join
Axiom maths circles, King's College London Mathematics School
From Year 7, for the whole school journey
Some online, some in person; Mondays and Thursdays 16:30 to 17:30, all year
Places prioritised under widening-participation criteria the school publishes
GCSE+ Maths, King's College London Mathematics School
Years 9 to 11
"An extensive programme of enrichment"
Apply through the school's outreach pages
Oxford Online Maths Club, University of Oxford
Mainly sixth form
Weekly livestream, Thursdays at 5pm UK time, recorded
Everyone; "there's no registration needed"
Problem Solving Matters, University of Oxford with MEI
Year 12
A course in problem-solving mathematics
State-funded schools and colleges; home-educated students welcomed
STEP Support Programme, University of Cambridge
Year 12 and Year 13
Free online modules with hints and solutions
"The resources are free and open to everyone."
NRICH, University of Cambridge
Primary and secondary
Free problems online; Maths Club webinars
Open online
The year at the Oxford club
Oxford describes its club as "free, interactive, casual, and relaxed". From January to June it covers super-curricular maths; from June to December it hosts the Oxford Maths Admissions Test Livestream, which in 2026 focuses on the TMUA and maths interviews.
It is "mostly aimed at people about to apply to university or about to start a Maths course", so a Year 8 pupil will find much of it hard, but the recordings can be watched at any pace.
The dates King's has published
King's outreach page for autumn 2026 says Axiom Maths Circles and GCSE+ Maths sessions continue from the week beginning 21 September 2026, with Extension Mathematics: University Admissions Prep from 7 September 2026.
King's says all its student events are free. Places on the long-term circles are limited, and the school sets out who is prioritised.
The pattern across the table is plain. Free circles exist, but the ones aimed at younger pupils are few and rationed, while the open-to-all programmes lean towards sixth form and university admissions. A curious Year 6 to Year 10 pupil outside the priority groups is the child most likely to find no circle at all, and that is the gap our weekly class is built for. For the sixth-form end, our STEP preparation page goes further.
What does one problem look like across three circle sessions?
Here is a circle arc we wrote and run, for ages 11 to 15. It is ours, not taken from any programme above. The problem fits on one line; the thinking fills three weeks.
The question. A post office sells only 3p and 5p stamps. Which postage amounts can you not make exactly?
Week 1: explore
Pupils list amounts from 1p upwards. 3, 5, 6, 8, 9 and 10 work; 1, 2, 4 and 7 do not. From 8 onwards every amount works, because 8, 9 and 10 do, and adding a 3p stamp reaches the next three. The largest impossible amount is 7.
Week 2: conjecture
Change the stamps to 4p and 7p and the largest impossible amount is 17. With 2p and 5p it is 3. Someone notices that 3 times 5 minus 3 minus 5 is 7, 4 times 7 minus 4 minus 7 is 17, and 2 times 5 minus 2 minus 5 is 3. The circle has a conjecture.
Week 3: test and explain
Does it always work? Try 4p and 6p: every odd amount is impossible, so there is no largest one. The rule needs stamps with no common factor. The session ends on why the formula holds for such pairs, written in pupils' own words.
Our check: the largest impossible postage for two stamp valuesPython
def largest_impossible(a, b, limit=200):
can = [False] * (limit + 1)
can[0] = True
for n in range(1, limit + 1):
can[n] = (n >= a and can[n - a]) or (n >= b and can[n - b])
missing = [n for n in range(limit + 1) if not can[n]]
return missing[-1] if missing else None
for a, b in [(3, 5), (4, 7), (2, 5), (5, 8)]:
print(a, b, largest_impossible(a, b), a * b - a - b)
Run it and each line prints two equal numbers: 7, 17, 3 and 27. The code does not prove the rule, and a good circle says so out loud; it only makes the conjecture worth proving. That distinction, between evidence and proof, is the most valuable thing a circle teaches, and it takes weeks rather than minutes to sink in.
The problem framing, the three-week plan, the code and the outputs are ours. The underlying result is a classical one in number theory, often called the Frobenius coin problem for two values.
Problems
Circle problems to start at home
Five problems we wrote, each one able to fill a session. Answers and the key idea follow each; resist reading them for at least a quarter of an hour.
A. Locker doors (ages 10 and up)
A corridor has 100 closed lockers. Pupil 1 opens all of them; pupil 2 toggles every second; pupil 3 every third; and so on to pupil 100. Which stay open? Exactly the square numbers, 1, 4, 9 up to 100: only squares have an odd number of factors.
B. Staircase sums (ages 11 and up)
Which numbers are a sum of two or more consecutive whole numbers? 9 is 4 plus 5, and 10 is 1 plus 2 plus 3 plus 4. Test up to 32. The ones that fail are 1, 2, 4, 8, 16 and 32: the powers of two.
C. Knight's reach (ages 11 and up)
On an endless chessboard, what is the fewest knight moves to reach the square one step to the right? Three. Each knight move changes the square colour, so an odd number is needed, and one move cannot do it.
D. Cutting a cake (ages 12 and up)
Three cuts across a round cake: what is the most pieces you can make? Seven. With n straight cuts the maximum is 1 plus n times n plus 1 over 2, so 2 cuts give 4 and 4 cuts give 11.
The circle conversation is why each new cut adds one more piece than the cut before it.
E. Digit reversal (ages 12 and up)
Take any three-digit number with different first and last digits, reverse it, and subtract the smaller from the larger: 742 minus 247 is 495. The answer is always a multiple of 99.
Write the number as 100a plus 10b plus c, and the difference is 99 times the gap between a and c.
All five problems and explanations are written by Modern Age Coders and are not taken from Axiom, NRICH, STEP Support or any other programme.
Choosing well
How do you tell a good online maths circle from a tuition class with a new name?
Ask five questions. A real circle, free or paid, should answer yes to most of them.
Does one problem ever last more than one session?
Do pupils speak to each other during the session, not only to the leader?
Is there no test, no grade and no homework sheet of similar questions?
Are pupils asked to write explanations, and does someone read them?
Does the same group meet for months, so pupils know each other's ideas?
Free first, if your child qualifies
If your child fits the stage and entry rules of a free programme in the table, apply. A long-term free circle with a university or a specialist school behind it is a fine thing to be part of.
Our class is for the many pupils who do not fit those rules, or want an extra weekly session alongside.
What our circle adds
A fixed weekly time with the same pupils and teacher, ages from 10, and a teacher who reads every written explanation and returns comments before the next session.
Sixth formers who want the same habits aimed at admissions can move on to our BMO or STEP preparation.
Stages
Four stages in a circle
Our stages; pupils move up when the habit, not the topic, is secure.
Our circle stages
Stage
Typical age
Secure when
1. Explorer
10 to 12
Lists cases carefully and spots when a list is complete
2. Conjecturer
11 to 14
States a pattern precisely and tries to break it
3. Explainer
13 to 16
Writes a convincing argument a classmate can follow
4. Problem poser
15 to 18
Invents the next question and leads part of a session
Mixed ages work
Circles mix ages comfortably because the problems have many entry points.
We group by stage, so a confident 11-year-old may sit with 13-year-olds.
Competition as a side effect
Circle pupils often enjoy the UKMT olympiad ladder, but we never turn a circle into exam practice.
Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays. Times are set in UK time. Our teachers work from India, so their working day overlaps the UK afternoon and evening comfortably.
Weekday after school
Circle groups for ages 10 to 15.
Weekday evening
Sixth-form problem solving.
Weekend morning
Circles and 1-on-1 sessions.
One problem, many weeks
A problem stays open until the group can explain it.
Breakout pairs
Pupils think in pairs on screen before the whole group meets.
A shared board
Every idea goes up where everyone can see and test it.
Written follow-up
A short explanation after each arc, read and commented on.
Stable groups
15 to 20 pupils who stay together, or 1-on-1.
Code as a lab
Python tests a conjecture when hand cases run out.
Student work
Built by students in our classes
Maths and code projects from our learners. See more in student labs.
AI and ML
NutriLife
An AI nutrition coach that reads what you eat and works you toward a target.
A weather forecasting site with live conditions for any location.
by Krish
Fees
Fees for the online circle
A single monthly fee, priced for UK learners, with no contract for the year, so a pupil can try the circle for a month and decide. The first session is free.
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Maths circle questions
What families ask about maths circles
What is a maths circle?
A small group that meets regularly to work on rich problems with no syllabus and no grades, often staying on one problem for weeks. King's College London Mathematics School describes its Axiom circles as students coming together to grapple with puzzles and learn to think like mathematicians.
Are there maths circles in the UK?
Yes. The Axiom circles run by King's College London Mathematics School are a long-term programme from Year 7, partly online. University outreach such as the Oxford Online Maths Club and Cambridge's STEP Support Programme works in a similar spirit for older pupils.
Can my child join a maths circle online?
Some free programmes run online, but most have an age range or priority rules. The Oxford Online Maths Club needs no registration and suits sixth formers best. Our own weekly circle runs online for ages 10 to 18 and is open to any family.
Is the Oxford Online Maths Club free?
Yes. Oxford says it is free, needs no registration and is recorded. It streams on Thursdays at 5pm UK time, and from June to December it hosts the Oxford Maths Admissions Test Livestream, which now focuses on the TMUA.
Who gets a place on Axiom maths circles?
King's says a pupil who joins in Year 7 keeps a place for their whole school journey. Places are prioritised for students who meet the widening-participation criteria the school publishes on its Axiom page, so read that page before applying.
What age should a child start a maths circle?
Many circles start around age 10 or 11, the Year 6 to Year 7 point, once a child can explain their thinking in words. Younger children do well with reasoning and puzzle lessons first.
Is a maths circle the same as olympiad training?
No. Olympiad training works towards a paper on a fixed date. A circle has no paper at all, although circle pupils often enjoy olympiads because they are used to long, unfamiliar problems.
Is STEP Support a circle?
Not quite. The STEP Support Programme is free online modules, designed to be studied weekly over about six months from Year 12, with hints and full solutions. Working through it with friends gives it some of a circle's feel.
Are you connected with Oxford, Cambridge or King's?
No. We run our own classes and answer to no university or school. We describe these programmes from their own pages so families can find them, and we have no link with any of them.
When does your circle meet?
Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays. Every time is given in UK time.
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