How Many Combinations Does a 3×3 Rubik’s Cube Have?

One of the most common questions about the Rubik’s Cube is how many combinations it can have. This is a fascinating question that has captured the imagination of many people, from mathematicians to puzzle enthusiasts to casual players. This article will answer this question in detail, delving into the math behind Rubik’s Cube and how we can calculate the possible combination of the Rubik’s cube. After reading this article, you will deeply appreciate the Rubik’s Cube and how it challenges our understanding of logic, math, and problem-solving.

Understanding the Basics of the Rubik’s Cube

To understand the complexity of the Rubik’s Cube, we must first understand its basic structure. A Standard 3×3 Rubik’s Cube is made up of 27 small cubes. Each of these small cubes, or “cubies,” can be moved independently of the others, allowing for a wide range of possible combinations.

The Rubik’s Cube has six faces, each of which is made up of nine cubies. The six faces are typically denoted by the colors of the stickers on each face, which can be white, yellow, blue, green, red, or orange. The goal of Rubik’s Cube is to solve it by rotating the faces and moving the cubies until each face is a solid color.

To rotate the face of the Rubik’s Cube, you must turn it 90 degrees clockwise or counterclockwise. This is known as a “move.” The Rubik’s Cube has 18 possible moves, with three possible rotations (clockwise, counterclockwise, or 180 degrees) for each of the six faces. A “solve” of the Rubik’s Cube refers to a specific combination of moves that results in all six faces being a solid color. This can be a very difficult task, as billions of possible combinations of moves can be made.

Understanding the basics of the Rubik’s Cube is crucial for understanding how to solve it and the math behind its complexity.

How many combinations does a 3×3 Rubik’s cube have?

A standard Rubik’s Cube has 43,252,003,274,489,856,000 (43 quintillions) possible combinations. This number is so massive that it is difficult to comprehend. To put it into perspective, if every person on Earth had a Rubik’s Cube and they all scrambled them randomly, it would take longer than the universe’s age for every combination to be created.

However, not all of these combinations are solvable. Some can be easily solved, while others require advanced strategies and algorithms. According to research, only 7,401,196,841,564,416 (7.4 quintillions) possible combinations can be solved in 20 moves or less.

To understand why the number of combinations is so high, consider that the Rubik’s Cube has 26 pieces that can be arranged in any of the 18,000 possible positions. Additionally, each piece has three possible orientations, meaning there are 18,000 x 3^26 possible combinations.

The Enormity of the Number

The number of combinations that a 3×3 Rubik’s Cube can have is truly mind-boggling. To grasp the enormity of this number, it is often necessary to express it in scientific notation. Scientific notation is a way to represent extremely large numbers more efficiently. It consists of two parts: a coefficient and an exponent of 10.

When calculating the number of combinations for a 3×3 Rubik’s Cube, it is estimated to be around 43 quintillion (43,000,000,000,000,000,000). We can rewrite this number as 4.3 x 10^19 to express it in scientific notation. The coefficient, 4.3, represents the digits before the decimal point, and the exponent, 19, indicates the power of 10.

By using scientific notation, we can better comprehend the immense scale of possibilities that exist within the Rubik’s Cube.

Relating the number of combinations to real-world examples for comparison

To put the number of combinations for a 3×3 Rubik’s Cube into perspective, let’s consider some real-world examples for comparison.

  1. Seconds since the Big Bang: The estimated age of the universe is around 13.8 billion years. If we convert this time span into seconds, it would be approximately 4.35 x 10^17 seconds. Comparing this to the number of combinations of the Rubik’s Cube (4.3 x 10^19), we can see that the cube has significantly more possibilities than the total number of seconds since the Big Bang.
  2. Number of grains of sand on Earth: It is estimated that there are about 7.5 quintillion (7.5 x 10^18) grains of sand on Earth. Even though this is an enormous quantity, it is still roughly six times smaller than the number of combinations of a Rubik’s Cube. This comparison emphasizes the mind-blowing complexity of the cube’s possibilities.
  3. Number of stars in the observable universe: The total number of stars in the observable universe is estimated at about 1 sextillion (1 x 10^21). While this is an incredibly large number, it is still much smaller than the number of combinations for a Rubik’s Cube. In fact, the Rubik’s Cube has approximately 50 times more combinations than there are stars in the observable universe.

These examples help us comprehend the immense magnitude of the Rubik’s Cube’s possibilities and highlight its exceptional complexity. It is truly remarkable how a simple-looking puzzle can have such an extraordinary number of combinations, making it a challenging and engaging puzzle for enthusiasts around the world.


In conclusion, Rubik’s Cube is an incredibly complex puzzle game with a mind-boggling number of possible combinations. The challenge of solving it has captivated generations of people and inspired them to think creatively and logically. The Rubik’s Cube remains a timeless classic and proof of the power of human creativity and curiosity.

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