The Math of the Cube
Warning: This page contains math. If you just want to go fast, turn back now. If you want to understand why the algorithms work, stay.
How many combinations?
A 3x3 cube has 43,252,003,274,489,856,000 possible positions.
- That is 43 quintillion.
- If you had 43 quintillion cubes, you could cover the surface of the Earth 275 times.
God’s Number
What is the maximum number of moves required to solve ANY scramble?
- For years, people thought it was 52, then 42, then 30.
- In 2010, Google computers proved it: 20 moves.
- This means every state can be solved in 20 moves or less (Half-Turn Metric).
Commutators (The Secret Sauce)
Most advanced algorithms are built using Commutators.
A commutator is a sequence in the form: A B A' B'
Why do they work?
Imagine you have a solved cube.
- A: You make a change (e.g., twist a corner). This “breaks” other things.
- B: You move that corner away and replace it with another corner.
- A’: You undo the first move. This “fixes” the broken things, but now applies the reverse twist to the new corner.
- B’: You undo the second move.
Result: Only those two corners are affected. Everything else is solved.
Example: Corner Cycle
Try this: (R U R') D (R U' R') D'
- A:
R U R'(Inserts a corner) - B:
D(Moves the bottom layer) - A’:
R U' R'(Undoes the insertion) - B’:
D'(Undoes the bottom move)
This cycles 3 corners without messing up the rest of the cube. This is the basis of Blindfolded Solving.
Parity (The Math Error)
Parity isn’t a glitch; it’s a mathematical property.
- Even Cubes (4x4): You can swap two edges without swapping corners. This is impossible on a 3x3 because of the way permutations work (even vs odd permutations).
- On a 3x3, every turn swaps a set of corners AND a set of edges. You cannot swap just two pieces.