Rubik's Cube Probability Calculator
Explore the 43 quintillion Rubik's Cube positions.
Calculate random-solve probability, time to guess all states, and compare permutation counts from 2x2 to 5x5 cubes.
The 3x3 Rubik’s Cube has exactly 43,252,003,274,489,856,000 possible positions (about 43 quintillion). That number comes from:
(8! x 3^7 x 12! x 2^11) / 12 = 43,252,003,274,489,856,000
8! = 40,320 ways to arrange the 8 corners. 3^7 = 2,187 ways to orient them (the 8th is determined). 12! / 2 = 239,500,800 edge arrangements (the /2 is parity). 2^11 = 2,048 edge orientations.
If you scrambled a cube randomly once per second and each scramble was unique, it would take about 1.37 trillion years to visit every position – roughly 100 times the current age of the universe.
The maximum number of moves needed to solve any position from scratch is 20 (half-turn metric). This was proven in 2010 by Rokicki, Kociemba, Davidson, and Dethridge using 35 CPU-years of computation. This is called “God’s Number” – the minimum worst case.
The probability that a randomly scrambled cube is already in the solved state: 1 in 43,252,003,274,489,856,000. For comparison, Powerball jackpot odds are 1 in 292 million. Your cube has to be scrambled about 148 billion times worse odds than a Powerball win.
The chart below shows how permutation counts scale across cube sizes – even on a log10 scale the jump from 3x3 to 4x4 is dramatic. A 5x5 cube has more possible states than there are atoms in the observable universe (estimated at roughly 10^80).