Cube Invariants

Explanation:

The cube on the left demonstrates the corner-twist invariant: assign numbers to the stickers as in the solved position. Then, no matter how you scramble the cube, the sum of the numbers on any two opposite faces is always divisible by 3. In particular, the two orange frames show the front and back of the cube, and adding the numbers inside these frames always gives a multiple of 3. Twisting a single corner breaks this rule. This means that if you twist one of the corner cubies of a solved Rubik's Cube, you end up in a configuration that cannot be solved with legal moves.

The cube on the right illustrates the edge-flip invariant: no matter how you scramble the cube, if you count the green squares in the six orange windows, the total will always be even. Flipping a single edge breaks this rule. This means that if you flip one of the edge pieces of a solved Rubik's Cube, you end up in a configuration that cannot be solved with legal moves.

If the overlaid orange frames don't line up properly, adjust your browser window by widening or contracting it slightly.

I've modified the JavaScript Simulator AnimCube3.js to display numbers on the red, blue, and yellow stickers and I've called the modified version AnimCubeInvariant3.js. For more information about the original app, visit the AnimCube website.