IA idea · Games & puzzles
How many shuffles does it take to mix a deck of cards?
Research question
How does the number of rising sequences in a deck change with the number of riffle shuffles, and does my data support the claim that about seven shuffles are needed?
Make it yours, then agree it with your teacher.
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Why it makes a good exploration
Bayer and Diaconis showed in 1992 that about seven riffle shuffles are needed to randomise a deck. Counting rising sequences after real and simulated shuffles tests the claim with your own hands.
The mathematics you'll need
- Permutations and rising sequences (explain)
- The Gilbert–Shannon–Reeds model of a riffle shuffle
- Simulation and expected values
- Comparing distributions
Where the data comes from
Shuffle a numbered deck 1–10 times, recording the order each time (repeat many times); simulate the model.
- Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
A possible outline
- Explain rising sequences and why one shuffle creates at most two.
- Record real shuffles.
- Simulate the GSR model.
- Compare how quickly each approaches randomness.
- Reflect on human shuffling differences.
Pitfalls that cost marks
- Too few repetitions.
- Claiming the full distance-to-randomness result without explanation.
- Inconsistent shuffling technique.
Showing personal engagement
- Compare your shuffles with a friend's.
- Test overhand shuffles too (much slower).
- Explain why magicians exploit rising sequences.
See Criterion C: personal engagement for what examiners look for.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Not a natural fit | The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics. |
| AA HL | Good fit | Permutations and rising sequences (explain); The Gilbert–Shannon–Reeds model of a riffle shuffle |
| AI SL | Fits — ambitious at SL | Permutations and rising sequences (explain); The Gilbert–Shannon–Reeds model of a riffle shuffle |
| AI HL | Good fit | Permutations and rising sequences (explain); The Gilbert–Shannon–Reeds model of a riffle shuffle |
Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.
How this idea reaches the top bands
Reflection (D)
For this idea, start with the weak points: too few repetitions; and claiming the full distance-to-randomness result without explanation. Say how each one affects your answer.
Use of mathematics (E)
SL: permutations and rising sequences (explain); the Gilbert–Shannon–Reeds model of a riffle shuffle; simulation and expected values; comparing distributions, each explained in your own words and carried out correctly.
How games & puzzles explorations reach the top bands in general: Games & puzzles IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
Estimate the probability that a card trick based on rising sequences works after k shuffles.
Extending it for HL
Add an HL technique the question genuinely needs: what an HL games & puzzles exploration adds.
See a complete IA, marked
How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) is a complete games & puzzles exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AI HL, AA HL and AI SL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.
What mathematics does this IA use?
Permutations and rising sequences (explain), the Gilbert–Shannon–Reeds model of a riffle shuffle, simulation and expected values and comparing distributions.
Where can I get data for this IA?
Shuffle a numbered deck 1–10 times, recording the order each time (repeat many times); simulate the model. Checked sources and tools: Desmos graphing calculator.
Turn this idea into your IA
The planner opens with “How many shuffles does it take to mix a deck of cards?” filled in: the question, the maths and the data sources.
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