A Rubik's Cube has 43 quintillion combinations but always solvable in 20 moves

While a Rubik’s Cube possesses 43,252,003,274,489,856,000 unique permutations, any scrambled position can be returned to its solved state in 20 moves or fewer. This definitive maximum, known by mathematicians as God's Number, represents the absolute limit of the puzzle's complexity when using the half-turn metric.

• Total permutations: Exactly 43,252,003,274,489,856,000 combinations.
• The Limit: No configuration requires more than 20 moves to solve.
• Discovery: Proved in 2010 using Google’s computing infrastructure.
• Metric: Based on the "Half-Turn Metric" where any 180-degree rotation counts as one move.

The scale of 43 quintillion

To grasp the magnitude of the Rubik's Cube, one must look at the sheer volume of possible states. If you had one cube for every possible permutation, you could cover the entire surface of the Earth including the oceans 273 times over.

Despite this astronomical variety, the internal logic of the cube is remarkably tight. Mathematical group theory treats the cube as a set of permutations. This means that while the number of states is high, the "distance" between them is surprisingly short.

Defining God's Number

For decades, mathematicians searched for the minimum number of moves required to solve the most difficult possible scramble. This theoretical ceiling was dubbed "God's Number." The term implies that an omniscient being would always see the shortest path to the solution.

The quest to find this number began shortly after the cube's international debut in 1980. Early estimates suggested it might be in the high twenties. However, as computing power improved, the number began to shrink.

The 2010 breakthrough

In July 2010, a team of researchers including Morley Davidson, John Dethridge, Herbert Kociemba, and Tomas Rokicki announced they had solved the puzzle of God's Number. Using Google's computing resources, they processed every possible position.

The team did not check all 43 quintillion positions individually, as that would have been computationally impossible even for Google. Instead, they used symmetry and group theory to reduce the workload into 2.2 billion groups of positions.

Distance and moves

The 20-move limit applies specifically to the Half-Turn Metric (HTM). In this system, any rotation of a face—90 degrees or 180 degrees—counts as a single move. If you used the Quarter-Turn Metric, where a 180-degree turn counts as two moves, God's Number would be 26.

According to research published on Cube20.org, the vast majority of positions can actually be solved in 15 to 19 moves. Only a very small fraction of scrambles actually require the full 20 moves.

Why humans can't solve it in 20 moves

While the math proves 20 moves is enough, human "speedcubers" typically use 45 to 60 moves. This is because humans rely on established algorithms and patterns. These methods, like the CFOP (Cross, F2L, OLL, PLL) method, are optimised for human recognition rather than mathematical brevity.

Memorising the exact path for 43 quintillion positions is impossible for a biological brain. Instead, we break the problem into smaller, manageable chunks. This is similar to how we manage mental energy when unfinished tasks stick in our minds, causing us to seek resolution through familiar routines.

The Superflip position

One of the most famous positions that requires exactly 20 moves is called the "superflip." In this state, every corner and edge piece is in its correct location, but every single edge piece is flipped the wrong way.

For years, the superflip was the only known position that required 20 moves. Mathematicians later found other "20-move" positions, but none have ever been discovered that require 21.

Mathematical parallels

The study of the Rubik’s Cube falls under Group Theory, a branch of mathematics that studies symmetry. This same logic applies to molecular chemistry and particle physics.

Just as a cube undergoes a transformation, biological processes like ecdysis involve a structured progression through specific states. In both cases, there is an "optimal" sequence of events dictated by the laws of geometry or biology.

Evolution of the cube

Since Erno Rubik invented the cube in 1974, it has become the world’s best-selling toy. Its complexity has also made it a benchmark for testing computational efficiency.

Related studies in complexity have even touched on space-faring biology. For instance, just as we map the limits of a cube, scientists have discovered that ISS bacteria have evolved into new strains to survive the unique "logic" of a low-gravity environment.

Connections to other fields

• Complexity Theory: Studying how the number of moves scales with the size of the cube (e.g., 4x4 or 5x5).
• Machine Learning: Using AI to find move-optimal solutions without brute-force searching.
• Visual Perception: How our eyes perform a saccade across the cube's faces to identify colours.
• Strategy: Reflecting the Stoic philosophy that you always own the option of having no opinion, cubers must remain detached and calm during a solve.

Practical applications of the logic

The algorithms developed to find God’s Number have practical uses in data encryption and network routing. Finding the "shortest path" is a fundamental problem in computer science, whether you are routing a packet of data or solving a plastic puzzle.

Key Takeaways

• Precise Complexity: There are over 43 quintillion ways to scramble a Rubik's Cube.
• The 20-Move Rule: No matter how scrambled, 20 moves are always sufficient for a solution.
• Computational Proof: This was verified by Google's computers in 2010.
• Human vs Machine: While machines use the absolute shortest path, humans use longer, pattern-based methods for speed.
• Half-Turn Metric: The 20-move limit counts 180-degree turns as a single action.