The Mathematics of the Rubik’s Cube – Group Theory

The Rubik’s Cube has 43,252,003,274,489,856,000 possible states (≈ 4.3×10¹⁹). The group of moves has 12 elements? Actually the cube group has size (8! × 3⁷ × 12! × 2¹¹) / 2. Solving the cube involves finding a path from any scrambled state to the identity. God’s number (maximum minimum moves) is 20 (for half‑turn metric) or 26 (quarter‑turn). The mathematics uses conjugacy classes, cosets, and the concept of group presentations. Speedcubers use algorithms derived from group theory to solve in seconds.