The "law" of the Rubik's Cube refers to the conditions that determine whether a scrambled cube configuration is solvable. These conditions are not laws in the legal sense but are mathematical constraints related to how the cube's pieces can be rearranged.
Understanding Rubik's Cube Configurations
Not every possible arrangement of the Rubik's Cube's pieces is achievable through legal moves. For a configuration to be solvable, it must adhere to certain mathematical principles. These principles ensure that the cube can be restored to its solved state using only standard rotations of the cube's faces. Reference [1] outlines these rules.
The Three Essential Rules
According to the provided reference [1], a Rubik's Cube configuration is valid (or "legal") if and only if three key conditions are met. These are:
- Even Permutation of Cubies: The arrangement of the cube's individual pieces (cubies) must have an even permutation.
- A permutation refers to the way the pieces are moved around.
- An "even" permutation means that you can achieve the current arrangement by switching pairs of cubies an even number of times. You can't swap any 2 pieces and have a valid configuration.
- In simpler terms, if you could achieve the state by swapping only 2 cube pieces then the configuration isn't valid.
- Zero Sum of Corner Orientations: The sum of the twists (orientations) of the corner pieces must be a multiple of 3 and, in the case of a 3x3 cube, can be considered to be 0.
- Each corner cubie can be twisted in 3 different positions, but a sum of all the twists must be a multiple of 3.
- If some corners are twisted, there must be other corners twisted in the opposite way to balance out.
- Zero Sum of Edge Orientations: The sum of the flips (orientations) of the edge pieces must be zero.
- Each edge cubie has two orientations.
- The flips of all edges must be balanced out by other flips of the other edges in order to be valid.
Breakdown Table
| Condition | Description | Example |
|---|---|---|
| Even Permutation | The pieces' rearrangement is achieved through an even number of swaps. | Swapping two edge pieces makes an impossible state. |
| Corner Orientations Sum to 0 | The total twisting of corner pieces must cancel out. | If one corner is twisted clockwise, there must be a corner twisted counter-clockwise to balance the twist. |
| Edge Orientations Sum to 0 | The total flipping of edge pieces must cancel out. | If one edge is flipped, there must be another flipped edge to make the configuration valid. |
Practical Insights
These rules explain why certain "scrambles" of the Rubik's Cube cannot be solved using standard moves. If a cube is disassembled and put back together randomly, it's likely that it will not meet these three criteria. The cube would be solvable if and only if these rules are met.
Conclusion
In essence, the "law" of the Rubik's Cube dictates the mathematical limits within which it is solvable. A configuration is solvable if the cube's cubies have an even permutation, the corner orientations sum to zero, and the edge orientations sum to zero, as explained in [1]. Any state of the Rubik's Cube that does not satisfy these conditions is impossible to achieve through normal rotations of its faces.
