# Corner Swappings on Rubiks Cube

*Solution:*

As you mentioned, turning just one corner (say, clockwise) results in a position that's not in the Rubik's Cube Group, but turning one clockwise and one counterclockwise is!

If we count a clockwise turn as +1 and a counterclockwise turn as -1, then it turns out that as long as our turns add up to 0 mod 3, then that's a legal permutation. (So turning all eight clockwise is not legal, perhaps you turned one counterclockwise?)

My go-to reference for the Rubik's Cube Group is the book *Adventures in Group Theory* by David Joyner.

If you turn all the corners one step *the same way* (that is, for example, all clockwise), the resulting configuration is solvable if and only if the number of corners you turned is a **multiple of three**.

So turning all eight corners one step clockwise each will not produce a solvable configuration.

More generally, the cube will remain solvable iff the *difference* between how many corners you turned clockwise and how many corners you turned counterclockwise is a multiple of three.

For example, turning equally many corners clockwise as counterclockwise will produce a solvable configuration, because $0$ is a multiple of $3$ (namely, it is $0\times 3$).

It depends on how you turn them. A 1/3 turn clockwise (when looking into the corner from the outside) is called a "quark" and similarly a counter-clockwise rotation is called an "antiquark". In analogy to "real" quarks/antiquarks, you can combine a quark and an antiquark, or you can combine three quarks (or three antiquarks) and still leave the cube in a solvable state. That's IIRC.

Hofstadter's article in Scientific American (ca. 1980) is one reference and David Singmaster's booklet on the cube is another.