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?

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

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

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

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.

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

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.

Taking it further

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

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).