What's the Monkey number of the Rubik's cube?

TL;DR

The video explores various mathematical problems related to Rubik's Cube and other twisty puzzles, including the number of twists required for a sufficiently random position, the average number of twists needed to solve the cube, and the number of configurations.

Transcript

Welcome to another Mathologer video. Today's video should be of interest to everybody who loves twisty puzzles as well as all hardcore Mathologer fans. In 2010, thirty years after the Rubik's Cube rocked the puzzle world it was finally proven that God's number for the Rubik's Cube is 20. What this means is that every single one of the 43 quintillio... Read More

Key Insights

• #️⃣ God's number for the Rubik's Cube is the maximum number of twists required to solve any configuration.
• 🔀 The average number of twists to solve a Rubik's Cube starting from a random configuration is approximately 4.5 million.
• 🧊 The average number of twists to go from a solved cube to a solved cube is equal to the number of configurations.
• 🪘 Random twisting is no longer used in official competitions, and computer programs are used to create truly random configurations.
• ↩️ Different counting metrics (half turn vs. quarter turn) can affect the average number of twists in different scenarios.

Questions & Answers

Q: What is God's number for the Rubik's Cube?

God's number refers to the maximum number of twists required to solve any configuration of the Rubik's Cube, which has been proven to be 20 for the 3x3x3 cube.

Q: How many random twists are needed to sufficiently randomize a Rubik's Cube?

The video mentions that the World Cubing Association no longer uses random twisting to create scrambles for competitions. Instead, they use a computer program called tnoodle, which explodes and reassembles the cube randomly to create a truly random configuration.

Q: What is the average number of twists required to solve a Rubik's Cube starting from a random configuration?

Through computer simulations, it has been found that the average number of twists to solve a Rubik's Cube starting from a random configuration is approximately 4.5 million in the half-turn metric.

Q: How does the average number of twists change when using the quarter turn metric instead of the half turn metric?

The average number of twists from scrambled to solved for the 2x2x2 cube changes from roughly 4.5 million in the half turn metric to 4.6 million in the quarter turn metric. However, the average number of twists from solved to solved remains unchanged.

Summary & Key Takeaways

• The video discusses the concept of God's number for the Rubik's Cube, which is the maximum number of twists required to solve any configuration (20 for the 3x3x3 cube).

• It introduces three related problems: determining the number of twists needed for a sufficiently random position, finding the average number of twists to solve the cube, and calculating the average number of twists to solve the cube starting from a solved position.

• The video presents a proof that the average number of twists for a solved-to-solved cube is equal to the number of configurations.