How many shuffles does it take to mix a deck of cards?

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

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

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.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

1. Explain rising sequences and why one shuffle creates at most two.
2. Record real shuffles.
3. Simulate the GSR model.
4. Compare how quickly each approaches randomness.
5. Reflect on human shuffling differences.

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.

Taking it further

Estimate the probability that a card trick based on rising sequences works after k shuffles.