IA ideas library
Pure maths IA ideas: number, patterns and proof
Not every IA needs data. Pure explorations investigate a pattern, prove it, and extend it — the kind of mathematics that is especially natural for AA, and above all AA HL with its proof by induction, complex numbers and series.
The risk is writing a summary of something you read. The personal engagement has to come from your own conjectures, your own examples and your own attempts, including the ones that failed.
Next step: shortlist ideas with the free IA topic explorer, which filters starting points by course, interest and how easy the data is to get. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.
How pure maths explorations reach the top bands
Personal engagement (C)
Explore special cases yourself before looking up the answer, record your conjectures and false starts, and explain what surprised you.
Reflection (D)
Reflect on what each step of the argument achieves, where it would fail, and what the result does and does not prove.
Use of mathematics (E)
SL: Careful exploration of patterns with sequences, series and algebra, every step justified; a proof of a special case is better than a quoted general result.
HL: Rigorous proof (induction, contradiction), series and limits, complex numbers: HL mathematics used with full rigour, not a summary of someone else's proof.
An HL version
Prove the general result by induction or contradiction, or bound the error of an approximation rigorously.
Pure maths, number & proof ideas (12)
- Beyond Fibonacci: what ratio do other recurrence sequences approach?AA HLAA SLSolid
- Why does Pascal's triangle make a fractal?AA HLAA SLSolid
- The Collatz conjecture: what can we say about stopping times?AA SLAA HLAI SLAI HLSolid
- Finding formulas for 1ᵏ + 2ᵏ + … + nᵏAA HLAA SLSolid
- The product of a regular polygon's diagonalsAA HLAmbitious
- Why is 355/113 such a good approximation to π?AA HLAA SLSolid
- How many primes are there below n? Testing the prime number theoremAA SLAA HLAI SLSolid
- Why is the main body of the Mandelbrot set a cardioid?AA HLAmbitious
- Why does 1 + 1/4 + 1/9 + … equal π²/6?AA HLAmbitious
- How does RSA encryption work, and how long would it take to break a small key?AA HLAA SLAmbitious
- How long is a coastline? Measuring fractal dimensionAA SLAA HLAI SLAI HLSolid
- How does a calculator work out sin(x)?AA HLAA SLSolid
Also relevant (43)
- How far does a bouncing ball really travel?AA SLAI SLAA HLAI HLAccessible
- How many times can you fold a sheet of paper?AA SLAA HLSolid
- Will the sofa fit round the corner? The longest object through a corridorAA SLAA HLAI SLAI HLSolid
- Does caffeine build up if you drink coffee every few hours?AA SLAA HLAI SLAI HLSolid
- Does Benford's law hold for the populations of the world's towns?AI SLAI HLAA HLSolid
- The birthday problem, extended: what about birthdays within a day?AA SLAA HLAI SLAI HLSolid
- When should you stop looking? The 37% rule for choosing the bestAA HLAA SLAI HLAmbitious
- Estimating π by dropping needles: how fast does the estimate improve?AA HLAA SLAI SLAI HLSolid
- Voronoi diagrams in a grid city: Euclidean or taxicab distance?AI SLAI HLAA HLAmbitious
- Why do flight paths curve on the map?AA SLAA HLAI SLAI HLAmbitious
- Is a longer mortgage ever the better deal?AI SLAI HLAA SLAA HLAccessible
- How much does it cost to start saving ten years later?AI SLAI HLAA SLAA HLAccessible
- Where does e come from? Compounding more and more oftenAA SLAA HLSolid
- How does Google rank pages? PageRank on a small websiteAI HLAA HLAmbitious
- What is the fewest number of exam sessions my school needs?AI HLAA HLAI SLSolid
- Why does a piano have twelve notes per octave?AA SLAA HLAI SLAI HLSolid
- Symmetry in Islamic geometric art: which wallpaper groups appear?AA HLAA SLAmbitious
- How are letters drawn on a screen? Bézier curvesAA HLAA SLAI HLSolid
- Why do sunflower seeds form spirals? The golden angleAA SLAA HLAI SLAI HLSolid
- What curve do straight lines make in string art?AA SLAA HLSolid
- Why are camera f-stops 1.4, 2, 2.8, 4, 5.6?AA SLAI SLAA HLAI HLAccessible
- Spreading beats evenly: Euclid's algorithm and musical rhythmAA SLAA HLSolid
- How many positions does a Rubik's Cube have?AA HLAA SLAmbitious
- How many different games of noughts and crosses are there?AA SLAA HLAI SLAI HLSolid
- Cutting shelves from planks with the least wasteAA SLAA HLAI SLAI HLSolid
- Square or hexagonal? Packing tins into a boxAA SLAA HLAI SLSolid
- How far does a random walker get? Distance after n stepsAA HLAA SLAI HLSolid
- How many random numbers until the total passes 1? Finding e by chanceAA HLAA SLSolid
- Which typing errors does a check digit catch? ISBN-10 versus ISBN-13AA HLAA SLSolid
- Agreeing a secret in public: the Diffie–Hellman key exchangeAA HLAmbitious
- Which numbers are the sum of two squares?AA HLAA SLAmbitious
- Patterns in the last digits of powersAA SLAA HLSolid
- Perfect numbers and Mersenne primes: proving Euclid's ruleAA HLAmbitious
- Why does the ancient square-root method double the correct digits?AA SLAA HLSolid
- When does Newton–Raphson fail? Cycles, divergence and basins of attractionAA HLAA SLAmbitious
- Press cos again and again: when does fixed-point iteration converge?AA SLAA HLSolid
- How many terms do you need? Fast and slow series for πAA HLAmbitious
- When the computer gets it wrong: rounding error and cancellationAA HLAA SLAI HLAmbitious
- Infinite perimeter, finite area: designing your own snowflake curveAA SLAI SLAA HLAccessible
- Inside or out? Proving the escape radius for Julia setsAA HLAmbitious
- Fold a strip, unfold it: the dragon curveAA SLAA HLSolid
- A sponge with no volume and infinite surface: the Menger spongeAA SLAI SLAA HLAccessible
- Cutting a cake fairly when people value it differentlyAA SLAA HLSolid
Frequently asked questions
Can a pure maths IA score well without real-world data?
Yes. Personal engagement comes from your own exploration — the cases you try, the conjectures you make, the proofs you attempt — not from a real-world context.
How hard should the maths be in a pure IA?
At the level of your course, used correctly and explained so a classmate could follow it. If you cannot explain a step in your own words, leave it out.
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