Resource Guide
Research Project Ideas in Mathematics for High School Students
Specific, explorable questions across pure mathematics, applied modelling, probability, and mathematics in society, with guidance on what a maths project actually is.
How to Use This List
A mathematics project worries students more than most, because they imagine it must mean proving something new. It rarely does. The best school-level mathematics research takes a real piece of mathematics, often just beyond the syllabus, understands it deeply, and explains it clearly, sometimes with the student’s own examples, computations, or small extensions.
Use the questions below as starting points. The goal is depth and clarity, not novelty for its own sake. Our guide to writing a research question helps frame a focus, and good mathematical writing rewards the same care as any other scholarly work.
Choose a Method Before a Topic
The choice that decides whether a project is feasible is the method, not the topic. Every question below is tagged with one of these four routes.
Expository synthesis
Read the primary sources on one result and explain it rigorously in your own words, including the step where the difficulty actually lies. Mathematics rewards this more than any other subject: a genuinely clear exposition of a hard theorem is a real contribution and is exactly what a supervision asks you to do.
The original papers or a graduate text, and the patience to work every line.
Computational experiment
Write code to explore a pattern, test a conjecture on many cases, or generate the data a proof would need. Not a substitute for proof, but it is how most mathematicians decide what to try proving.
Python and a conjecture you can state precisely.
Mathematical modelling
Build a model of a real system, state its assumptions honestly, and test its predictions against data. The mark of a good project is knowing where your model breaks, not making it complicated.
A real system, published data to test against, and stated assumptions.
Applied case analysis
Take an existing piece of theory, game theory, expected value, voting rules, and apply it to one real situation to see whether it predicts what happened. Where mathematics meets evidence.
One well-documented real case and the relevant theory.
Ideas by Sub-Field
Pure mathematics
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What makes the distribution of prime numbers so hard to predict, and what patterns are known?
Expository synthesis Advanced
Output: An account of what is proved, what is conjectured, and where the difficulty lies
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How does knot theory distinguish one knot from another, and why is that hard?
Expository synthesis Advanced
Output: An explanation of one or two knot invariants, worked on examples you compute yourself
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What can be said about a chosen family of geometric shapes or tilings?
Computational experiment Intermediate
Output: A classification of a small family, with the cases enumerated and illustrated
Applied mathematics & modelling
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How can a simple differential-equation model capture the spread of a disease or a rumour?
Mathematical modelling Intermediate
Output: An SIR-style model implemented and compared against published outbreak data
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What does a mathematical model reveal about traffic flow or queueing?
Mathematical modelling Intermediate
Output: A queueing or flow model with predictions tested against timings you observe
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How well does a chosen model predict a real pattern in open data?
Computational experiment Intermediate
Output: A model fitted to open data with the residuals discussed honestly
Probability & statistics
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Why do counter-intuitive results, such as the Monty Hall problem, arise, and what do they teach?
Expository synthesis Accessible
Output: An explanation supported by a simulation you wrote to confirm the result
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How can common statistical claims in the media be checked, and where do they go wrong?
Applied case analysis Accessible
Output: An audit of three published claims against the data they were drawn from
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What does the mathematics of risk reveal about a real-world decision?
Applied case analysis Intermediate
Output: An expected-value analysis of one real decision, with the assumptions stated
Mathematics & society
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How does public-key cryptography use number theory to keep data secure?
Expository synthesis Intermediate
Output: A worked explanation of RSA with a small example computed by hand
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What does game theory predict about a real strategic situation, and does it hold?
Applied case analysis Intermediate
Output: A game-theoretic model of one situation, tested against what actually happened
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How can the mathematics of fairness inform the design of a voting or allocation system?
Applied case analysis Advanced
Output: A comparison of voting rules against named fairness criteria, addressing Arrow
Understanding Deeply, Explaining Clearly
The mark of a strong mathematics project is not difficulty for its own sake but clarity: the ability to take something genuinely hard and make it understood, with rigour intact. A student who can explain why a result is true, not merely state it, has done real mathematical work.
A mentor helps a student choose a topic at the right level, fill the gaps that textbooks leave, and write mathematics the way mathematicians do, precisely, honestly, and with the reader in mind. That skill is exactly what a mathematics interview seeks to draw out.
Taking a Question Further
Mathematics is the language of physics, computer science, and economics, and projects often bridge into them. For the wider context, see our Technology, AI & Engineering field page, our companion guides to physics and computer science, and our broader research project ideas across all six fields. When you are ready to turn a question into a finished project with a mentor who works in the field, the Research Scholar programme is built for exactly that.
Frequently Asked Questions
What does a mathematics research project look like at high school level?
It is rarely about proving a brand-new theorem. More often it is exploring a known area deeply, working through and explaining a result, investigating a conjecture computationally, or applying mathematics to model a real problem. The aim is genuine understanding and a clear written account of it.
Do I need to discover something original?
No. Original results are wonderful but rare at this level. A project that takes a piece of real mathematics, understands it deeply, and explains it clearly, perhaps with the student’s own examples or small extensions, is genuine and valued.
Can a maths project use computing?
Absolutely. Computational exploration, generating data about a conjecture, visualising a structure, or simulating a process, is a powerful way to investigate mathematics, and it pairs naturally with rigorous reasoning.
How is a maths project useful for university applications?
Mathematics and related courses look for students who enjoy thinking about mathematics beyond the syllabus and can communicate it clearly. A project demonstrates exactly that, and gives an applicant something specific to discuss in an interview.
Free PDF
Take the 50 strongest ideas with you
A free PDF with fifty of these questions tagged by method and difficulty, the output each should produce, and the feasibility checklist we apply before recommending any project. We email it to you, along with occasional guidance worth having. No obligation, and your details are not passed to anyone.
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