IA idea · Games & puzzles
How long does a game of Snakes and Ladders last?
Research question
Using a Markov chain for a particular board, what is the probability that a game finishes within n turns, what are the expected and median game lengths, and which single snake most lengthens the game?
Adapt it: change the place, the data or the comparison until the question is yours.
Why it makes a good exploration
Snakes and Ladders has no decisions, so it is pure probability — and a perfect Markov chain. Powers of a transition matrix give the exact distribution of game length, which you can check by simulation and by playing.
The mathematics you'll need
- Transition matrices (AI HL) for a 101-state chain
- Matrix powers with technology; absorbing states
- Cumulative distribution and median of game length
- Expected value as a sum of tail probabilities
- Simulation to verify
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
Record a board layout (a family board or one you design) and play or simulate games to test predictions.
- 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
- Describe the board and rules (exact finish or not).
- Build the transition matrix, explaining one row in full.
- Compute P(finished by turn n) from matrix powers.
- Find expected and median lengths; verify by simulation.
- Remove each snake in turn to find the one with most effect; reflect on rule variants.
Pitfalls that cost marks
- Building the matrix by hand for all 100 squares (use technology, explain one row).
- Forgetting the rule for overshooting 100.
- Confusing the median with the expected value.
Showing personal engagement
- Use the board you grew up playing.
- Design a board that makes games last about 30 turns.
- Compare your model with games played with family.
See Criterion C: personal engagement for what examiners look for.
Taking it further
Use the fundamental matrix (I − Q)⁻¹ (explain it) to get expected turns from every square.
See it done
Our annotated exemplar How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) explores a question like this one, with an examiner's comment on every criterion.