Algebra and abstract structures · Maths EE idea · Ambitious

The group theory of a twisty puzzle

A research question to start from

How does group theory explain which positions of a 2×2×2 twisty cube are reachable, and how many there are?

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Why it works as a maths EE

A concrete group where you can prove invariants and count the group's size with the orbit-stabiliser idea.

Mathematics you would need

Permutations and parity
Groups, subgroups and generators
Orbit-stabiliser theorem
Invariants

Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.

One possible line of attack

Model moves as permutations of pieces with orientations.
Prove the orientation invariant (twists sum to 0 mod 3).
Count reachable positions and justify every factor.
Discuss what changes for the 3×3×3.

Scope and difficulty

Ambitious. Ambitious but well bounded if you keep to the 2×2×2.

Pitfalls

Solving the puzzle instead of analysing the group.
Counting without justification.

Where to start reading

Search a library catalogue or a university's open lecture notes for: pocket cube group theory; permutation parity orientation invariant. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).

Make it your EE

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