IB maths extended essay topics: 20 ideas with their research question

Choosing well is half the work. Twenty ideas with real mathematics, ready for you to make your own.

The IB extended essay has a limit of 4,000 words and almost always the same problem: it is left until there is no time left to choose well. And choosing well is everything. A topic with real mathematics, data within reach and a question that can be answered leads to a manageable essay; a vague topic leads to months going in circles.

Here are 20 ideas for a maths extended essay, each with its research question, the mathematics it needs and a rough difficulty (relative within this list). They are not topics to copy: they are starting points for you to make your own.

Before you choose: three tests your topic must pass

• Is there real mathematics? Calculations in an interesting context are not enough. There must be a mathematical object you investigate: a proof, a model, an algorithm, a distribution.
• Do you have data or sources within reach? A brilliant topic with no accessible data or bibliography turns into a dead end by month three.
• Does it fit in 4,000 words? If answering it would take a book or a doctorate, narrow it down. A good question is small and deep, not big and shallow.

If you want to know how the work is assessed, the five criteria are summarised on the extended essay help page.

Probability and statistics

1. Benford’s law in Spanish municipalities

To what extent do the populations of Spanish municipalities follow Benford’s law, and what does a deviation from it measure?

Mathematics: logarithms, first-digit distribution, χ² test. Public data from the Spanish statistics office (INE). · Difficulty: medium

2. How many shuffles are enough

How many riffle shuffles are needed for a 52-card deck to be considered well mixed?

Mathematics: permutations, Markov chains, total variation distance. · Difficulty: high

3. The secretary problem

What is the optimal stopping strategy in the secretary problem, and how does it change when there are few candidates?

Mathematics: conditional probability, sums, the limit leading to 1/e. · Difficulty: medium

4. PageRank: how Google ranks pages

How does PageRank compute the importance of a page, and how much does the ranking of a small network change as the damping factor varies?

Mathematics: stochastic matrices, eigenvectors, iteration. · Difficulty: high

5. Penalty kicks and Nash equilibrium

Do penalty takers and goalkeepers behave as the mixed-strategy Nash equilibrium predicts?

Mathematics: game theory, probability, hypothesis testing with real data. · Difficulty: medium-high

6. Monte Carlo estimation of π

How fast does the Monte Carlo method converge when estimating π, and how does it compare with series such as Leibniz’s?

Mathematics: geometric probability, standard error, series. · Difficulty: low-medium

Calculus and analysis

7. The cycloid and the fastest descent

Why is the cycloid the curve of fastest descent, and how much time does it save compared with a straight ramp?

Mathematics: parametric equations, integrals, introductory calculus of variations. · Difficulty: high

8. Catenary or parabola

Is a hanging cable closer to a catenary or to a parabola, and beyond what sag does the parabolic approximation stop being valid?

Mathematics: hyperbolic functions, Taylor series, curve fitting. · Difficulty: medium

9. Newton’s method and fractals

How does the convergence of Newton’s method for z³ − 1 = 0 in the complex plane depend on the starting point?

Mathematics: complex numbers, iteration, convergence, basins of attraction. · Difficulty: high (more natural at HL)

10. Fourier series and sound

How many harmonics of a Fourier series are needed to reproduce a square wave to a given error?

Mathematics: trigonometric series, integrals, the Gibbs phenomenon. · Difficulty: high

11. Computing π with series

Which series for π converges fastest: Leibniz, Machin or Ramanujan, and why?

Mathematics: series, the Taylor expansion of arctangent, convergence. · Difficulty: medium

Algebra and number theory

12. RSA: encryption and factoring

What is the security of RSA encryption based on, and how does the time to factor grow as the key size increases?

Mathematics: modular arithmetic, Euler’s theorem, factoring algorithms. · Difficulty: medium-high

13. Fibonacci and matrices

How is Binet’s formula obtained by diagonalising a matrix, and what happens with more general linear recurrences?

Mathematics: matrices, eigenvalues, induction. · Difficulty: high (more natural at HL)

14. The Collatz sequence

What statistical regularities do Collatz orbits show for the first few million integers?

Mathematics: sequences, descriptive statistics, programming. Note: the goal is not to “prove” the conjecture. · Difficulty: low-medium

15. Hamming codes

How does the Hamming code detect and correct errors, and how many parity bits does it need?

Mathematics: arithmetic mod 2, matrices, combinatorics. · Difficulty: medium

Geometry and discrete mathematics

16. The fractal dimension of the Valencian coastline

What is the fractal dimension of the coast of the Valencian Community, measured with the box-counting method?

Mathematics: logarithms, regression, self-similarity. · Difficulty: medium

17. The travelling salesman problem with Spanish provincial capitals

How far from the optimum is the nearest-neighbour algorithm in the travelling salesman problem with the Spanish provincial capitals?

Mathematics: graphs, combinatorics, algorithm complexity. · Difficulty: medium

18. Conway’s Game of Life

Which small starting configurations survive the most generations in the Game of Life?

Mathematics: combinatorics, cellular automata, counting and symmetry. · Difficulty: medium

Modelling and applied mathematics

19. A SIR model with real data

How is a SIR model fitted to real epidemic data, and how sensitive is its prediction to the basic reproduction number?

Mathematics: differential equations, Euler’s method, least squares. · Difficulty: high

20. Seat allocation: D’Hondt, Sainte-Laguë and Hare

Which seat-allocation method (D’Hondt, Sainte-Laguë or Hare) is most proportional with the real results of the Valencian regional elections?

Mathematics: optimisation, proportionality indices, comparing algorithms. Public data from the regional government. · Difficulty: low-medium

Topics I would avoid

• Purely descriptive ones: “the history of π”, “mathematics in music”. There is no question to answer and no mathematics to do.
• Open problems (Goldbach, Riemann, Navier-Stokes). You won’t solve them and the question becomes impossible to answer.
• Those that stay at lower-secondary level, however well presented: means and percentages of your class marks, Pythagoras applied to a ladder.
• Those that depend on data you can’t get. Check you have it before deciding.

What to do with this list

Pick three that catch your eye, and for each ask yourself two questions: could I get the data or sources this week? and do I already understand the mathematics on the first page? The one that passes both is your candidate. Then you turn it into your question, not mine.

The essay has to be yours and your school supervisor oversees it, as the IB requires. What I do at the academy is help you decide whether the topic holds up, sharpen the question, check the mathematics is right and review your drafts against the criteria. I don’t write for you.

Frequently asked questions

Can I use a topic from the list as it is?

Better to adapt it: change the data, the case or the method. A question that twenty classmates will also choose carries less weight, and your supervisor will want to see your own contribution.

Is a maths extended essay better than one in another subject?

There is no “easier” subject. The maths one needs a topic with real mathematics and a lot of rigour; in return, it depends less on interpreting other people’s sources. What matters is that it interests you and you can sustain it for months.

Which topics are workable at Standard Level?

Plenty: Monte Carlo for π, Benford’s law, seat allocation, the Collatz sequence or the secretary problem. The more advanced ones, such as PageRank, Newton’s method in the complex plane or Fourier series, need more background.