Rubik's Cube Probability Calculator
Explore the 43 quintillion Rubik's Cube positions.
Calculate random-solve probability 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) / 2 = 43,252,003,274,489,856,000
8! = 40,320 ways to arrange the 8 corners. 3^7 = 2,187 ways to orient them, because the eighth corner’s twist is fixed by the other seven. 12! = 479,001,600 edge arrangements and 2^11 = 2,048 edge orientations, for the same reason on the twelfth edge. The final division by 2 is the parity constraint: you cannot swap a single pair of pieces on a real cube without taking it apart, so exactly half of the arrangements the arithmetic produces are unreachable by turning.
That is the divisor people get wrong most often. Half the pages quoting this formula write /12, which is off by a factor of six and gives an answer no source agrees with.
If you scrambled a cube randomly once per second, and never repeated a position, visiting all of them would take about 1.37 trillion years. The universe is roughly 13.8 billion years old, so that is a hundred lifetimes of the universe to get through one toy.
The maximum number of moves needed to solve any position is 20 in the half-turn metric. Rokicki, Kociemba, Davidson and Dethridge proved it in 2010 using about 35 CPU-years donated by Google. It is called God’s Number, and what makes it interesting is that it is a worst case: almost every position is solvable in 17 or 18, and positions genuinely requiring all 20 are rare.
The probability that a scramble lands on the solved state is 1 in 43,252,003,274,489,856,000. Powerball odds are about 1 in 292 million, so the cube is roughly 148 billion times longer odds than winning a jackpot. Buy the ticket instead.
How the sizes compare
| Cube | Positions | log₁₀ |
|---|---|---|
| 2x2 | 3.67 million | 6.6 |
| 3x3 | 43.3 quintillion | 19.6 |
| 4x4 | 7.4 × 10⁴⁵ | 45.9 |
| 5x5 | 2.8 × 10⁷⁴ | 74.5 |
Even on a log scale the jump from 3x3 to 4x4 is violent, and it is worth being precise about what that top figure means. A 5x5 has about 10⁷⁴ states, and the observable universe holds roughly 10⁸⁰ atoms, so the cube falls short by a factor of about a million. It gets quoted the other way round constantly. The 5x5 does comfortably beat the number of stars, or grains of sand, or seconds since the Big Bang, but atoms is a bigger number than almost any intuition allows for.
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This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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