United Kingdom · Years 9, 10 and 11 · 18 March 2027

Cayley, Hamilton and Maclaurin preparation

Three olympiad papers, sat on the same afternoon, split not by how good a pupil is but by which school year they are in. Cayley is for Year 9 and below, Hamilton for Year 10, Maclaurin for Year 11, and each is six questions in two hours with every answer written out in full. That structure has a consequence worth understanding before March: a pupil who sits Cayley this year will sit Hamilton next year and Maclaurin the year after, against the same people. Nobody gets one attempt at this. What carries between the three is not topics. It is the habit of finishing a question, which at this level almost always means finding every answer rather than an answer.

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Every figure quoted from UKMT · Our own practice problems · Nobody can promise you a paper

In short

The UK Mathematics Trust runs three intermediate olympiads on one day, 18 March 2027: Cayley for "England, Wales and Overseas: Year 9 and below", Hamilton for "Year 10" and Maclaurin for "Year 11", with Scotland sitting S2 or below, S3 and S4 and Northern Ireland one year higher in each case. Each is a two-hour paper of "six Olympiad style problems". Entry is "by invitation based on a qualifying IMC score, or by discretionary entry", from the Intermediate Mathematical Challenge on 27 January 2027, and UKMT says "around 1,800 students qualify from the IMC each year". Candidates "should give full written solutions, including mathematical reasons as to why your method is correct", because "just stating an answer, even a correct one, will earn you very few marks". We teach the mathematics and the writing, live online. The first class is free; a group place is USD 100 a month and one-to-one teaching USD 150.

Where to start

Three courses for an intermediate olympiad candidate

Choose by what a pupil can finish, not by what they can start.

Maths Olympiad & Competition Course course thumbnail

IMK / 01

Olympiad and competition maths

Number theory, combinatorics and geometry in the olympiad idiom, with solutions written to completion.

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High School Mathematics course thumbnail

IMK / 02

High school mathematics

For a Year 10 or 11 pupil whose algebra needs to stop being the obstacle.

Open the syllabus →
Middle School Mathematics Mastery course thumbnail

IMK / 03

Middle school maths mastery

For a Year 9 Cayley candidate still consolidating the ground the paper stands on.

Open the syllabus →

The three papers

One afternoon, three papers, split by year group

Facts read at UKMT's own pages on 20 September 2026.

Who sits which paper, in UKMT's own words
PaperEngland, Wales and OverseasScotlandNorthern Ireland
CayleyYear 9 and belowS2 or belowYear 10 or below
HamiltonYear 10S3Year 11
MaclaurinYear 11S4Year 12
The intermediate route, with the 2026-27 dates UKMT publishes
RoundDateWhat it is
Intermediate Mathematical Challenge27 January 2027The multiple-choice paper everyone sits first
Cayley, Hamilton and Maclaurin18 March 2027Six written questions, two hours, by year group
Grey Kangaroo18 March 2027A multiple-choice follow-on, same afternoon
Pink Kangaroo18 March 2027The older multiple-choice follow-on, same afternoon

Four papers on one afternoon is the thing to get straight with a school. A pupil qualifies for exactly one of them from their Intermediate Challenge score and their year group, and the Kangaroo papers and the olympiad papers are different animals: multiple choice against full written solutions. Preparing for the wrong one is a wasted six weeks.

UKMT says around 1,800 students qualify from the Intermediate Challenge each year, across all three olympiad papers and all four nations. It is a small room.

The split by year group rather than by score is unusual and deliberate. A brilliant Year 9 pupil is not pushed onto the Maclaurin paper; they sit Cayley against other Year 9s. It keeps the papers age-appropriate and it means the ladder repeats: Cayley, then Hamilton, then Maclaurin.

Entry is a school matter. UKMT invites on a qualifying Intermediate Challenge score or accepts a discretionary entry, for which it charges a fee published on its own site in pounds. We do not print fees and we cannot enter anyone.

Sources: UKMT competitions calendar and the Cayley, Hamilton and Maclaurin page, read 20 September 2026. We have no connection with the UK Mathematics Trust, and nothing on this page should be read as one.

The method

A complete answer means all of them

UKMT is explicit: "just stating an answer, even a correct one, will earn you very few marks". Here is what that looks like on a real question.

A question in this style: find all pairs of positive whole numbers a and b for which 1/a + 1/b = 1/6. Most pupils find one within a minute. a = 12 and b = 12 works, because a sixth is two twelfths. That is a correct answer, and on an olympiad paper it is worth almost nothing, because the question said all.

Our run of 20 September 2026: every solution, and where they come from
abWhy it appears
742a − 6 = 1, so b − 6 = 36
824a − 6 = 2, so b − 6 = 18
918a − 6 = 3, so b − 6 = 12
1015a − 6 = 4, so b − 6 = 9
1212a − 6 = 6, so b − 6 = 6
15, 18, 24, 4210, 9, 8, 7The same four pairs the other way round

The move that finds them

Multiply out and rearrange: ab − 6a − 6b = 0, so (a − 6)(b − 6) = 36. Now the question is about the divisors of 36, and there is nothing left to hunt for.

Why nine and not more

36 has exactly nine positive divisors: 1, 2, 3, 4, 6, 9, 12, 18 and 36. Each one gives one ordered pair. That sentence is the proof that the list is complete, and it is where the marks are.

What we checked

We searched every pair of positive integers up to 200 by computer and found exactly those nine. The search is reassurance; the divisor argument is the answer.

The difference between one solution and nine is a minute of work. The difference between nine solutions and a proof that there are only nine is the whole question, and it is the thing a pupil coming from the Intermediate Challenge has never been asked to supply.

The verbal habit that fixes it is small. After finding an answer, ask out loud: could there be another? Then: how do I know there is not? Pupils who ask those two questions automatically pick up marks all over an olympiad paper.

The same trick, turning an equation into a product and then reading off divisors, comes up again and again at this level. It is worth learning as a move rather than as a fact about the number 6.

And notice what the good answer is not: it is not longer. Three lines of algebra, a list of nine divisors and one sentence about why the list is complete is a full-mark solution. Olympiad marking rewards finishing, not volume.

The problem, the algebra and the search are ours, written and run on 20 September 2026. UKMT publishes its own past papers and full solutions free, and none of them is reproduced here.

How to practise

Six weeks, one habit

The Intermediate Challenge is in January and the olympiads are in March. That gap is enough if it is spent on writing rather than on reading.

What earns marks on these papers, and what does not
HabitWhat it is worthThe failure it prevents
Writing "all solutions are" and then proving itOften the majority of a question's marksA correct first answer scoring almost nothing
Stating the method before using itLets a marker follow a long argumentA page of algebra nobody can grade
Checking the boundary casesCatches the solution that was quietly excludedNine solutions written as eight
Finishing three questions rather than starting sixComplete solutions score; fragments rarely doTwo hours spent producing no full marks
Reading UKMT's published solutions for styleShows the expected standard exactlyGuessing how much detail is enough

The commonest pattern we see in a first lesson is a pupil who can do the mathematics and has never once been told what a finished solution looks like. They are not behind. They have been rewarded for answers their whole school career, and the olympiad has changed the currency.

Two hours for six questions means twenty minutes each, and nobody should aim for that. Three questions finished properly is a good paper and a realistic target for a first-time candidate.

For a Year 9 pupil sitting Cayley, the best investment is the one that pays three times. Everything learned about writing complete solutions this March is worth more on the Hamilton paper next year and more again on Maclaurin after that.

A pupil who finds they enjoy this should look at the competitions calendar: the senior rounds and the British Mathematical Olympiad are the same sport played longer.

Getting ready

Four rungs, and a ladder that repeats

The same pupil climbs this three times, in Years 9, 10 and 11. What is learned once counts three times.

From an answer to a complete, proved solution
StageRungThe sign it is secure
Year 8 to 91. Algebra that does not slow you downRearranges and factorises without stopping to think
Year 92. One complete solutionWrites an argument a classmate could follow unaided
Year 103. All solutionsAsks "could there be another?" without being prompted
Year 114. Proof of completenessExplains why the list cannot be longer, in a sentence

If March is close

Write up three problems the pupil has already solved, in full, and have someone read them as a stranger would. That is the highest-return evening available.

Do not start new topics. Intermediate olympiad questions are built from school mathematics used unfamiliarly.

After Maclaurin

The senior rounds follow, and the British Mathematical Olympiad after those, where the same habit is worth far more.

A pupil who enjoys closing a list of cases usually enjoys writing the program that generated it, which is what maths through coding is for.

The full list

What we teach a Year 9, 10 or 11 mathematician

Four groups, chosen by what is currently in the way rather than by year group.

I

Olympiad mathematics

The paper itself

IMK / O / 01

Olympiad and competition maths

Divisors, invariants and arguments taken to the end.

Open the syllabus

IMK / O / 02

Middle school maths mastery

The ground a Cayley candidate stands on.

Open the syllabus

IMK / O / 03

Statistics and probability

Counting arguments, done carefully.

Open the syllabus
II

School mathematics

Years 9 to 11

IMK / S / 01

High school mathematics

The algebra a Maclaurin candidate should not be thinking about.

Open the syllabus

IMK / S / 02

GCSE maths

Foundation and higher, to the school's board.

Open the syllabus

IMK / S / 03

IGCSE maths

For international and independent schools.

Open the syllabus
III

Checking and building

Where maths meets code

IMK / C / 01

Maths through coding

A program that lists candidates, and a proof that closes the list.

Open the syllabus

IMK / C / 02

Python from start to finish

Enough code to search a space of cases by hand.

Open the syllabus

IMK / C / 03

Algorithms and data structures

Counting, searching and proving a method terminates.

Open the syllabus
IV

Looking further

Sixth form and beyond

IMK / N / 01

A-level maths

Pure, mechanics and statistics, taught properly.

Open the syllabus

IMK / N / 03

AI and machine learning for teens

Where the mathematics goes next.

Open the syllabus

How lessons run

A weekly hour, and a reader for the write-up

Teaching is live on video from India, which sits five and a half hours ahead of Britain in winter and four and a half in summer. Slots are agreed and kept in UK time.

Weekday evening

The usual choice for Years 9 to 11.

Weekend morning

For two hours on a single question, properly.

Half term

Short intensives in the run-up to March.

Written solutions marked

Someone reads the argument as a stranger would and says where it stops convincing.

Problems written by us

In the olympiad idiom. UKMT's past papers stay on UKMT's site.

Five to ten pupils

Enough for two different methods to meet, small enough for every write-up to be read.

Year groups respected

Cayley, Hamilton and Maclaurin candidates are taught at the level their paper is set.

One to one when useful

For a pupil far ahead of their year or preparing for a specific paper.

No score promised

We cannot qualify anyone and do not claim to. Nobody honest promises a medal.

Student work

What our students build

Four published projects by students at our school, none of them competition entries. More on the student labs page.

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Intermediate olympiad questions

What pupils and parents ask

Which paper does my child sit?

By year group, not by score. UKMT sets Cayley for Year 9 and below, Hamilton for Year 10 and Maclaurin for Year 11 in England, Wales and overseas; S2 or below, S3 and S4 in Scotland; and Year 10 or below, Year 11 and Year 12 in Northern Ireland.

When are the intermediate olympiads in 2027?

All three papers fall on 18 March 2027, the same afternoon as the Grey and Pink Kangaroo. The Intermediate Mathematical Challenge that qualifies pupils is on 27 January 2027.

How does a pupil qualify?

UKMT says entry is by invitation based on a qualifying Intermediate Mathematical Challenge score, or by discretionary entry, and that around 1,800 students qualify from the IMC each year.

What is the paper like?

Two hours and six olympiad-style problems, with every answer written out in full. UKMT tells candidates to "give full written solutions, including mathematical reasons as to why your method is correct".

Why did a correct answer score so little?

Because UKMT says so plainly: "just stating an answer, even a correct one, will earn you very few marks; also, incomplete or poorly presented solutions will not receive full marks". At this level the argument is the answer.

What does "find all" actually require?

A list and a reason the list is complete. For example, 1/a + 1/b = 1/6 has exactly nine ordered solutions in positive integers, because the equation rearranges to (a minus 6)(b minus 6) = 36 and 36 has nine divisors. The sentence about the divisors is where the marks are.

How many questions should a pupil finish?

Three, well. Two hours for six questions is twenty minutes each, and complete solutions score where fragments do not.

Is the Kangaroo the same thing?

No. The Grey and Pink Kangaroo papers fall on the same afternoon but are multiple choice. Preparing for one is not preparing for the other.

Do you use UKMT past papers in lessons?

No. UKMT publishes its own past papers and full solutions free, which is the right place for them. Our problems are written by us in the same idiom.

What do classes cost?

The first lesson is free. A group place is then USD 100 a month and one-to-one teaching USD 150, with nothing to join and no minimum term.

Related pages

More for olympiad mathematicians

The rung below, the paper beside it, and the whole competition year.

Junior Mathematical Olympiad

The rung below, for Year 8 and under.

UK competitions calendar 2026-27

The whole season in one table, organiser by organiser.

Scottish Mathematical Challenge

Three rounds a year, and a mug for the winners.

British Informatics Olympiad

For pupils who would rather prove things in code.

Coding classes in the UK

The four school systems, and where every UK page sits.

Choosing an online class

Seven things to ask before a card comes out.

Start here

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