Entirely Ridiculously Big Numbers - Numberphile

TL;DR

A mathematician calculates the number of possible scrambles for Rubik's Cubes of increasing size, reaching 10^349 for a 10×10 cube.

Key Points

• 1.The standard 3×3 Rubik's Cube has 43 quintillion (4.3×10¹⁹) possible states. This accounts for identical-looking positions and excludes overcounting due to rotational symmetry.
• 2.The 2×2 cube has ~3.6 million states, calculated as 8! × 3⁷ ÷ 24. The division by 24 removes equivalent whole-cube rotations; the 3⁷ term accounts for cubelet orientations with the last one fixed automatically.
• 3.The 4×4 cube jumps to ~7×10⁴⁵ states by multiplying corner, edge, and center arrangements. It has 24 edge pieces (24! arrangements) and 24 center pieces divided by 4! six times for identical same-color centers.
• 4.The 6×6 cube exceeds the number of atoms in the observable universe at ~1.6×10¹¹⁶ states. Its added complexity comes from two types of edges (central and wing) and four types of center corridors, each contributing independently.
• 5.A general formula for any 2n×2n cube is C × E^(n−1) × K^((n−1)²) ÷ 24. The scramble count grows exponential in n², not just n, which is why numbers explode so rapidly with cube size.
• 6.The 10×10 cube reaches approximately 10^349 possible scrambles. This is so large that replacing every atom in the universe with a whole new universe, repeated four times, still yields fewer atoms than this cube has states.