§15

Other puzzles — same framework, different stages

The cube's success made group theory the standard tool for every twisting puzzle. Each puzzle has its own group, generators, diameter, and open questions. Walking through with the same language — generators, conjugacy classes, subgroup chains, Cayley graphs — you can compare their difficulty, structure, and symmetry on equal footing.

Puzzle	Order	Generators	Diameter
2×2×2 Pocket	3,674,160	⟨U, F, R⟩ (3 faces enough)	11 (HTM)
3×3×3 (this)	4.3 × 10¹⁹	⟨U, D, L, R, F, B⟩	20 (HTM)
Skewb	3,149,280	4 corner cuts	11
Pyraminx	75,582,720	4 tips + 4 axis turns	11 (excluding tips)
4×4×4 Revenge	7.4 × 10⁴⁵	inner + outer slices	unknown (≥ 22, ≤ 36)
5×5×5	2.8 × 10⁷⁴	inner + outer slices	unknown
Megaminx	1.0 × 10⁶⁸	12 pentagonal faces	unknown
Square-1	6.7 × 10¹¹	/ , (1, 0) , (0, 1) etc.	13 (turn metric)

15.1 Closed-form |G| and the n × n × n asymptotic

Each common puzzle's group order is a closed-form "permutations × orientations ÷ parity/centre constraints". Side by side with the 3×3:

2×2×2

\frac{7 ! \cdot 3 ^{6}}{3} = 3, 674, 160

3×3×3 (G)

\frac{8 ! \cdot 12 ! \cdot 3 ^{7} \cdot 2 ^{11}}{2} \approx 4.33 \times 1 0^{19}

4×4×4

\frac{8 ! \cdot 3 ^{7} \cdot 24 ! ^{2}}{4 ! ^{6} \cdot 24} \approx 7.40 \times 1 0^{45}

5×5×5

\frac{8 ! \cdot 3 ^{7} \cdot 24 ! ^{2} \cdot 12 ! \cdot 24 !}{4 ! ^{6} \cdot 2} \approx 2.83 \times 1 0^{74}

Pyraminx

3^{4} \cdot 3^{4} \cdot \frac{6 !}{2} \cdot 3^{4} = 75, 582, 720

Skewb

8! \cdot 3^{4} \cdot 2 = 3, 149, 280

Megaminx

\frac{( 5 ! ) ^{12} \cdot 20 ! \cdot 3 ^{19} \cdot 30 ! \cdot 2 ^{29}}{constraints} \approx 1.01 \times 1 0^{68}

Square-1

\sim 1.78 \times 1 0^{14} (shape-dep. groupoid)

Asymptotic: for general n × n × n the order grows like

∣ G_{n} ∣ = Θ (n!^{O (n^{2})})

, but the God's number (diameter) grows only as

Θ (n^{2} / lo g n)

(Demaine et al. 2018). The state count double-exponentially explodes while the path length grows only polynomially — the precise sense in which the Cayley graph keeps getting fatter without getting much wider.

15.2 Known God's numbers, at a glance

puzzle	HTM	QTM/STM	status
2×2×2	11	14 (QTM)	proven (Reid 1995)
3×3×3	20	26 (QTM)	proven (Rokicki et al. 2010)
4×4×4	[22, 36]	[35, 53]	open — interval slowly tightening
5×5×5	[20, 32]	?	open
Pyraminx	11	11	proven (Cubelovers, 1981)
Skewb	11	11	proven
Megaminx	≈ 45	?	upper bound unproved
Square-1	[31, 35]	[26, 31]	open

A surprising point: the 5×5×5's HTM lower bound is smaller than the 3×3×3's — more degrees of freedom can be affected per turn. Diameter does not grow monotonically with group order; it tracks the covering efficiency of the generating set — a deep graph-theoretic question in its own right.

15.3 Beyond classical permutation groups

Some puzzles' state spaces are not classical groups but weaker algebraic structures — usually because legal states depend on geometric matching. They open three research directions:

AGroupoid (Square-1)

Square-1's top and bottom can rotate at non-integer multiples of 1/12, plus a / (flip) operation. Legal states depend on geometric alignment — not pure permutations. The state space is a groupoid (a categorified group with object types). Each of 12 "shape classes" is an orbit; total state count ≈ 1.78 × 10¹⁴.

BSubmonoid (Bandaged)

In bandaged cubes, some cubies are glued, forbidding certain turns. The result is a proper subgroup of G (smaller generating set), but the solver must be redesigned — Thistlethwaite and Kociemba both assume "all 6 generators available." Many bandage configurations are notoriously hard.

CJumbling (Helicopter)

The Helicopter cube admits jumbling cuts that enter geometric states outside the canonical cube shape. Its "group" effectively has infinite geometric branches; classical permutations don't suffice. Such "jumbling puzzles" are a 21st-century research topic.

"Square-1 is not a counterexample. It is a new example — it tells us that group theory itself needs extending."

— Erik Demaine, on geometric puzzle complexity

These exotic puzzles reveal the elasticity of the framework: once you accept "groups are the language of symmetry," nearly any mechanical puzzle becomes analysable. And when groups don't suffice, we extend into groupoids, submonoids, jumbling manifolds — the mathematics keeps moving forward.

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cuberoot.me · Rubik's Cube as a Group · 2026