How To Calculate Combinations Of Rubik's Cube

Total Combinations:

\[ 43,252,003,274,489,856,000 \]

Total Combinations:

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1. What Are The Combinations Of Rubik's Cube?

The Rubik's Cube has 43,252,003,274,489,856,000 possible configurations. This astronomical number represents all the different ways the cube can be arranged while maintaining its structural integrity.

2. How Is This Number Calculated?

The calculation considers the arrangement of corners, edges, and centers while accounting for physical constraints that make some positions impossible:

\[ \frac{8! \times 3^7 \times 12! \times 2^{10}}{2} = 43,252,003,274,489,856,000 \]

Where:

\( 8! \) — Permutations of 8 corner pieces
\( 3^7 \) — Orientations of 7 corner pieces (8th is determined)
\( 12! \) — Permutations of 12 edge pieces
\( 2^{10} \) — Orientations of 11 edge pieces (12th is determined)
\( \div 2 \) — Division by 2 accounts for parity constraints

3. Mathematical Formula

Complete Formula: The exact mathematical expression used to calculate the total number of combinations is derived from group theory and combinatorics, considering the cube's physical constraints and permutation properties.

4. Interesting Facts

Details: Despite the enormous number of combinations, any Rubik's Cube can be solved in 20 moves or less (God's Number). This number is so large that if you turned the cube once every second, it would take over 1.4 trillion years to go through all combinations.

5. Frequently Asked Questions (FAQ)

Q1: Why is the number divided by 2?
A: The division by 2 accounts for parity constraints - not all permutations are possible due to the cube's mechanical structure.

Q2: Are all these positions reachable?
A: Yes, all 43 quintillion positions are reachable through legal moves from the solved state.

Q3: How long would it take to try all combinations?
A: At one move per second, it would take over 1.4 trillion years to try all combinations.

Q4: What is God's Number for the Rubik's Cube?
A: God's Number is 20, meaning any scrambled cube can be solved in 20 moves or less.

Q5: Does this number include impossible positions?
A: No, this calculation already excludes positions that are mathematically possible but physically impossible due to the cube's construction.