Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

How many positions does a Rubik's Cube have?

AA HLAA SL Ambitious Also in: Pure maths

Research question

Why does a 3×3×3 Rubik's Cube have exactly 43,252,003,274,489,856,000 reachable positions, and how many does a 2×2×2 cube have?

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Why it makes a good exploration

The count comes from permutations and orientations with three constraints that make most arrangements unreachable. Deriving it is a satisfying exercise in counting principles — and you can check the constraints on a real cube.

The mathematics you'll need

Permutations and factorials
The multiplication principle
Parity and orientation constraints (explain)
Applying the same reasoning to the 2×2×2

Where the data comes from

Use a real cube to demonstrate the constraints (e.g., a single twisted corner cannot be solved).

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

A possible outline

Count arrangements of corners and edges.
Explain each constraint with a physical demonstration.
Derive the final count.
Apply to the 2×2×2.
Reflect on God's number (20) and what it means.

Pitfalls that cost marks

Stating constraints without justification.
Arithmetic errors with huge numbers — use exact arithmetic.
Confusing positions with move sequences.

Showing personal engagement

Demonstrate the constraints with your own cube.
Compare with a 4×4×4 conceptually.
Estimate how long it would take to visit every position.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Fits — ambitious at SL	Permutations and factorials; The multiplication principle
AA HL	Good fit	Permutations and factorials; The multiplication principle
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

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: stating constraints without justification; and arithmetic errors with huge numbers — use exact arithmetic. Say how each one affects your answer.

Use of mathematics (E)

SL: permutations and factorials; the multiplication principle; parity and orientation constraints (explain); applying the same reasoning to the 2×2×2, 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

Show why corner orientations must sum to a multiple of 3 by tracking a single move.

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 HL and AA 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 factorials, the multiplication principle, parity and orientation constraints (explain) and applying the same reasoning to the 2×2×2.

Where can I get data for this IA?

Use a real cube to demonstrate the constraints (e.g., a single twisted corner cannot be solved). Checked sources and tools: OEIS (On-Line Encyclopedia of Integer Sequences).

Turn this idea into your IA

The planner opens with “How many positions does a Rubik's Cube have?” filled in: the question, the maths and the data sources.

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