IA idea · Games & puzzles
How many different games of noughts and crosses are there?
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?
Make it yours, then agree it with your teacher.
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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
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.
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.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Good fit | Permutations and the multiplication principle; Counting with restrictions (games ending early) |
| AA HL | Good fit | Permutations and the multiplication principle; Counting with restrictions (games ending early) |
| AI SL | Good fit | Permutations and the multiplication principle; Counting with restrictions (games ending early) |
| AI HL | Good fit | Permutations and the multiplication principle; Counting with restrictions (games ending early) |
Level: Solid. Needs some independent work beyond class examples. 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: double-counting positions reached by different orders; and unverified counts. Say how each one affects your answer.
Use of mathematics (E)
SL: permutations and the multiplication principle; counting with restrictions (games ending early); symmetry (rotations and reflections); probabilities of outcomes under random play, 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
Compute the probability that the first player wins under random play and compare with simulation.
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 AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.
What mathematics does this IA use?
Permutations and the multiplication principle, counting with restrictions (games ending early), symmetry (rotations and reflections) and probabilities of outcomes under random play.
Where can I get data for this IA?
Verify counts with a short program or careful case analysis. Checked sources and tools: OEIS (On-Line Encyclopedia of Integer Sequences).
Turn this idea into your IA
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