The Rubik's Cube

The Rubik’s Cube is one of the most recognizable puzzles in the world, loved by enthusiasts and mathematicians alike. Invented in 1974 by Hungarian architect and professor Ernő Rubik, the cube was originally designed as a teaching tool to help students understand three-dimensional geometry. However, it quickly became a global phenomenon, captivating millions with its unique combination of simplicity and complexity. Beneath its colorful stickers lies a deep connection to mathematics, particularly in the fields of group theory, combinatorics, and algorithms.

Structure and Complexity 

The Rubik’s Cube is a 3x3 puzzle with six faces, each composed of nine smaller coloured squares. The goal is to manipulate the cube by twisting its faces until each face is a uniform colour. While the puzzle appears simple at first glance, its complexity becomes evident when you consider the sheer number of possible configurations. 

Counting Configurations 

For the sake of this article, the Rubik’s Cube we will discuss consists of two types of movable pieces: corners and edges. There are 8 corner pieces, each of which can be rotated into 3 orientations, and 12 edge pieces, which have 2 orientations. Additionally, both the corners and edges can be permuted (rearranged) in various ways. To calculate the total number of configurations, we use combinatorics: 

 • The 8 corner pieces can be permuted in 8! ways, and each corner has possible orientations. 

 • The 12 edge pieces can be permuted in 12! ways, and each edge has orientations. 

However, not all these configurations are valid due to the cube’s mechanical constraints. These constraints ensure that only half of the edge orientations, one-third of the corner orientations, and one-twelfth of the permutations are achievable. Taking these restrictions into account, the total number of solvable configurations is: 

 \[\frac{8! \cdot 3^7\cdot 12! \cdot 2 ^{11}}{12} = 43,252,003,274,489,856,000\]

This number, approximately quintillion, demonstrates the staggering complexity of the Rubik’s Cube. 

Group Theory

The Rubik’s Cube is a tangible example of group theory, a branch of mathematics that studies symmetry and transformations. A group is a set of elements combined with an operation that satisfies four properties: closure, associativity, identity, and invertibility.

The Cube as a Group 

Each twist of a face represents an element of the group. The group of the Rubik’s Cube is generated by the six basic face twists: Up (), Down (), Left (), Right (), Front (), and Back (). By combining these moves, any solvable configuration can be reached. 

Key concepts from group theory that apply to the Rubik’s Cube include: 

 • Generators: The six face twists are the generators of the group. Every possible move sequence on the cube can be expressed as a combination of these generators. 

 • Identity Element: The solved state of the cube acts as the identity element. Any sequence of moves followed by its inverse will return the cube to this state. 

 • Commutators: These are sequences of moves where the order of operations matters. For example, performing A, B,A^-1 ,B^-1 (where A^-1 and B^-1 are the inverse operations of A and B) can isolate specific changes on the cube, such as swapping two pieces. 

Algorithms and Solving Methods 

Subgroups and Solving Strategies 

Subgroups of the Rubik’s Cube group play an important role in solving it. A subgroup is a subset of moves that only affects certain parts of the cube. For example, one subgroup might involve moves that manipulate only the edges, leaving the corners unchanged. Solving the cube is often approached by breaking the problem into steps, each of which solves a different subgroup. 

Solving the Rubik’s Cube requires the use of algorithms—predefined sequences of moves designed to achieve specific results, such as positioning or orienting certain pieces without disturbing others. Different solving methods have been developed, ranging from beginner techniques to highly advanced methods used by speedcubers. 

Layer-by-Layer Method 

The most common beginner method is the layer-by-layer approach, which involves solving the cube in stages: 

 1. Cross: Solving one face and aligning the edges with adjacent centres. 

 2. First Two Layers (F2L): Completing the first two layers by solving corner-edge pairs. 

 3. Orientation of the Last Layer (OLL): Ensuring all stickers on the final layer face the same direction. 

 4. Permutation of the Last Layer (PLL): Rearranging the pieces of the last layer to complete the cube. 

Speedcubing Techniques 

Advanced solvers, or speedcubers, use optimized methods to solve the cube as quickly as possible. One popular technique is the CFOP method (Cross, F2L, OLL, PLL), which streamlines the layer-by-layer approach. Other methods include: 

 • Roux Method: Focuses on solving blocks of the cube instead of layers, reducing the number of moves. 

 • ZZ Method: Begins with orienting all edges to make subsequent steps more efficient. 

Speedcubers also rely on finger tricks to execute algorithms rapidly, as well as techniques like lookahead (planning future moves while executing the current ones). 

God’s Number: The Ultimate Challenge 

One of the most intriguing questions about the Rubik’s Cube is: What is the maximum number of moves required to solve the cube from any scrambled state? This number, known as God’s Number, represents the theoretical limit of efficiency. In 2010, researchers conclusively proved that God’s Number for the standard 3x3 Rubik's cube is 20. This means that no matter how scrambled the cube is, it can always be solved in 20 moves or fewer using the most optimal algorithm. 

To understand why the maximum number of moves is exactly 20, we need to look at how the cube is analysed mathematically. Mathematicians and computer scientists divide the problem into two key tasks: 

1. Defining the Search Space: 
   The Rubik’s Cube has approximately 43 quintillion possible configurations. While this number is massive, it is finite, which makes it feasible to analyse all possible positions computationally. The search space includes every valid configuration that can be achieved by twisting the cube’s faces, from the solved state to the most scrambled states. 
2. Counting Moves Efficiently: 
   Moves are measured using the half-turn metric (HTM), where any 90-degree or 180-degree twist of a face counts as one move. For example, turning the top face clockwise  or performing a double turn each count as one move. 

To prove God’s Number, researchers led by Tomas Rokicki and a team of mathematicians used advanced computer algorithms and distributed computing power, including help from Google’s servers. Their strategy involved: 

3. Dividing the Problem into Subgroups: 
   The researchers divided the 43-quintillion configurations into smaller, manageable subsets. They applied group theory principles to identify equivalent positions and reduce redundant calculations. By exploiting symmetries of the cube, they reduced the number of unique positions that needed to be analysed. 
4. Solving Each Subgroup: 
   Each subset of positions was then systematically solved using the fewest possible moves. The researchers used a brute-force search algorithm to determine the shortest solution for every position. These algorithms checked all possible move sequences up to a given length. 
5. Determining the Maximum Moves: 
   After solving all subsets, the researchers identified the most difficult configurations—the ones requiring the maximum number of moves. These positions were confirmed to require exactly 20 moves. 

The proof revealed that the vast majority of configurations require far fewer than 20 moves to solve. However, certain highly scrambled positions—known as God’s Algorithm-hard positions—are designed in such a way that they resist simplification. These positions require a carefully orchestrated sequence of 20 moves to return the cube to its solved state. 

For example, one of the most famous difficult configurations is the "superflip." In this configuration, every edge piece is flipped, while the corners remain in their correct positions. The superflip requires exactly 20 moves to solve when using the most efficient algorithm, confirming that God’s Number applies to even the trickiest states. 

The proof of God’s Number guarantees that no configuration exists that requires more than 20 moves under the half-turn metric. This upper bound comes from systematically verifying every possible configuration and finding that all can be solved in 20 moves or fewer. 

It’s important to note that earlier estimates of God’s Number were higher due to less computational power and incomplete analysis. In the 1980s, mathematicians estimated that the number might be as high as 52. Over time, this estimate was refined as computers became more powerful and algorithms more efficient. By 1995, researchers had reduced the upper bound to 29. Finally, in 2010, the exact number was definitively proven to be 20. 

While God’s Number provides a theoretical maximum for solving the cube, it’s not a practical goal for everyday solvers. Using God’s Algorithm—the shortest possible sequence of moves for a given configuration—requires immense computational effort and is impractical for humans to memorize. Instead, most solvers use methods like CFOP or Roux, which prioritize simplicity and speed over absolute efficiency. These methods typically require 50 to 60 moves but are far easier to execute in real-time. 

The Mathematics of Larger Cubes 

The standard cube is just the beginning. Larger cubes, such as the 4x4 (Rubik’s Revenge) and 5x5 (Professor’s Cube), introduce new layers of complexity. These puzzles require additional algorithms to solve problems unique to larger cubes, such as parity errors, where pieces appear to be swapped in ways impossible on a standard cube. 

The mathematics behind these larger cubes remains rooted in group theory and combinatorics but with expanded challenges due to the increased number of pieces. 

The Rubik’s Cube is far more than a colourful toy—it is a mathematical marvel that embodies the beauty of symmetry, structure, and problem-solving. Whether you approach it as a casual puzzler or a competitive speed-cuber, the cube offers endless opportunities to explore mathematical ideas and develop critical thinking skills. As researchers continue to uncover new insights about its complexity, the Rubik’s Cube remains a timeless symbol of the interplay between mathematics and creativity.