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.
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 (23)
- 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
- 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
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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