Maths EE ideas library
107 Maths Extended Essay ideas with research questions
Every idea has a focused research question, the mathematics it needs, one possible line of attack, an honest note on scope and the pitfalls. They are starting points: the strongest essays change the question until it is their own.
Difficulty: Accessible ideas build on DP mathematics; Solid ideas need new mathematics learned independently; Ambitious ideas suit students who are confident reading university-level material.
Number theory
- Which numbers are sums of two squares?Number theorySolid
- Generating every Pythagorean tripleNumber theoryAccessible
- Solving Pell's equation with continued fractionsNumber theoryAmbitious
- When Fermat's little theorem fools usNumber theorySolid
- Perfect numbers and Mersenne primesNumber theorySolid
- Divisibility tests in any baseNumber theoryAccessible
- Patterns in Collatz stopping timesNumber theorySolid
- Farey sequences and Ford circlesNumber theorySolid
- Counting partitions with generating functionsNumber theoryAmbitious
- How long is the repeating block of 1/n?Number theoryAccessible
- The Chinese remainder theorem and calendar cyclesNumber theoryAccessible
- Best rational approximations to πNumber theorySolid
Algebra and abstract structures
- The group theory of a twisty puzzleAlgebraAmbitious
- Classifying frieze patternsAlgebraSolid
- Solving the cubic: Cardano's method and its paradoxAlgebraSolid
- Fibonacci numbers by matrices and eigenvaluesAlgebraSolid
- Rotating in 3D with quaternionsAlgebraAmbitious
- Roots of unity and regular polygonsAlgebraSolid
- Latin squares and group tablesAlgebraSolid
- When solving linear systems goes wrongAlgebraSolid
Analysis and calculus
- Three proofs that Σ1/n² = π²/6Analysis & calculusAmbitious
- How fast do series for π converge?Analysis & calculusSolid
- How good is Stirling's approximation?Analysis & calculusAmbitious
- Trapezium rule vs Simpson's ruleAnalysis & calculusAccessible
- Where Newton's method sends youAnalysis & calculusSolid
- The fastest slide: why a cycloid?Analysis & calculusAmbitious
- Building a square wave from sinesAnalysis & calculusSolid
- Why do all the definitions of e agree?Analysis & calculusSolid
- Proving numbers are irrationalAnalysis & calculusAmbitious
- Curves and solids of constant widthAnalysis & calculusSolid
Geometry and topology
- Triangles on a sphereGeometrySolid
- Two models of hyperbolic geometryGeometryAmbitious
- V − E + F = 2, and when it failsGeometryAccessible
- Why are there only five Platonic solids?GeometryAccessible
- Which pentagons tile the plane?GeometrySolid
- The reflection properties of conicsGeometryAccessible
- Pick's theorem and lattice polygonsGeometryAccessible
- The best shape for a fenceGeometrySolid
- Testing golden-ratio claims mathematicallyGeometryAccessible
Probability
- The gambler's ruin problemProbabilitySolid
- Birthday problems beyond birthdaysProbabilityAccessible
- How long to collect the whole set?ProbabilitySolid
- Buffon's needle and estimating πProbabilitySolid
- Monty Hall and its variantsProbabilityAccessible
- Does a random walk come home?ProbabilityAmbitious
- Long-run behaviour of a board gameProbabilitySolid
- Why Benford's law worksProbabilitySolid
- When to stop looking: the secretary problemProbabilityAmbitious
- Penney's game: beating any coin sequenceProbabilitySolid
Statistics
- Why divide by n − 1?StatisticsAccessible
- Where does the least-squares line come from?StatisticsAccessible
- How quickly does the central limit theorem kick in?StatisticsSolid
- When combining data reverses the resultStatisticsAccessible
- Counting animals you cannot seeStatisticsSolid
- Estimating the largest serial numberStatisticsSolid
- Where does the chi-squared test come from?StatisticsAmbitious
- Spearman vs PearsonStatisticsAccessible
- Can resampling replace formulas?StatisticsAmbitious
- The mathematics of rating playersStatisticsSolid
Applied mathematics and modelling
- What the SIR model predicts about epidemicsModellingSolid
- Predator–prey cyclesModellingAmbitious
- From order to chaos in the logistic mapModellingSolid
- How small is a small angle?ModellingSolid
- Why traffic jams travel backwardsModellingAmbitious
- Is Newton's law of cooling right?ModellingAccessible
- Repeated doses and steady stateModellingAccessible
- The best launch angle with air resistanceModellingAmbitious
- The mathematics of a mortgageModellingAccessible
- Predicting a population with Leslie matricesModellingSolid
Discrete mathematics and graph theory
- Why five colours are enoughGraph theoryAmbitious
- Routes that use every street onceGraph theoryAccessible
- Ramsey numbers: order in any partyGraph theorySolid
- The Catalan numbers everywhereGraph theorySolid
- Derangements and the hat-check problemGraph theoryAccessible
- Colouring graphs to build timetablesGraph theorySolid
- Why greedy works for spanning treesGraph theoryAccessible
- How good is a quick tour?Graph theoryAmbitious
- Stable matchingsGraph theorySolid
- Surprising results from the pigeonhole principleGraph theoryAccessible
Codes, cryptography and algorithms
- Why RSA encryption worksCodes & algorithmsSolid
- Correcting errors with Hamming codesCodes & algorithmsSolid
- Check digits: which errors do they catch?Codes & algorithmsAccessible
- Sharing a secret in publicCodes & algorithmsSolid
- Breaking substitution ciphers with statisticsCodes & algorithmsAccessible
- Why the fast Fourier transform is fastCodes & algorithmsAmbitious
- How many comparisons does sorting need?Codes & algorithmsSolid
- Ranking web pages with eigenvectorsCodes & algorithmsAmbitious
- How random are pseudo-random numbers?Codes & algorithmsSolid
Optimisation and game theory
- Mixed strategies and Nash equilibriaOptimisation & gamesSolid
- Cooperation in the repeated prisoner's dilemmaOptimisation & gamesSolid
- Can any voting system be fair?Optimisation & gamesAmbitious
- Cutting a cake fairlyOptimisation & gamesSolid
- Why the simplex method finds the optimumOptimisation & gamesAmbitious
- Winning at NimOptimisation & gamesSolid
- How much to bet: the Kelly criterionOptimisation & gamesAmbitious
- How long is the queue?Optimisation & gamesSolid
History of mathematics
- Archimedes' bounds for πHistory of mathsAccessible
- How logarithms were first calculatedHistory of mathsSolid
- Egyptian fractionsHistory of mathsAccessible
- Tangents before calculusHistory of mathsSolid
- Pi in ancient ChinaHistory of mathsSolid
- Completing the square, geometricallyHistory of mathsAccessible
- From Königsberg to graph theoryHistory of mathsAccessible
- Cantor and the sizes of infinityHistory of mathsSolid
- The problem of pointsHistory of mathsAccessible
- Newton's method then and nowHistory of mathsSolid
- The rabbits and beyond: Fibonacci's problemsHistory of mathsAccessible
No ideas match those filters. Try fewer.
How to choose
- Pick two or three ideas in an area you enjoy and read a little about each.
- Check you can understand the core mathematics with the time you have (try one worked example).
- Narrow the question: one case, one comparison or one family of examples.
- Check the question is about mathematics, and different from your IA.
- Take your shortlist to your supervisor before you commit.
Then sharpen it with the research-question builder and plan it in the EE planner. Writing a Maths IA too? The IA ideas library is separate.