How many different games of noughts and crosses are there?

AA SLAA HLAI SLAI HL Solid Also in: Pure maths, Probability

Research question

How many possible games of noughts and crosses are there, how many distinct end positions, and how does using symmetry reduce the count?

Adapt it: change the place, the data or the comparison until the question is yours.

Why it makes a good exploration

The famous answer (255,168 games) depends on careful counting of games that end early. Building the count yourself, then using the board's symmetries, is a counting-principles investigation you can check by computer.

The mathematics you'll need

Permutations and the multiplication principle
Counting with restrictions (games ending early)
Symmetry (rotations and reflections)
Probabilities of outcomes under random play

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

Verify counts with a short program or careful case analysis.

OEIS (On-Line Encyclopedia of Integer Sequences) — Check a sequence you have found and read its known formulas and references.

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

Give an upper bound (9!).
Refine by counting games that end early.
Use symmetry to count distinct positions.
Compute outcome probabilities for random play.
Reflect on perfect play (always a draw).

Pitfalls that cost marks

Double-counting positions reached by different orders.
Unverified counts.
Confusing games with positions.

Showing personal engagement

Play random games and compare frequencies with your probabilities.
Extend to 4×4 or connect-three variants.
Explain the winning strategy for the first player to force at least a draw.

See Criterion C: personal engagement for what examiners look for.

Taking it further

Compute the probability that the first player wins under random play and compare with simulation.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).