My brother likes to solve Rubik's cubes. Occasionally I flip a corner or two as an added challenge, and he eventually is able to see where the illegal corner is. The other day, I turned all eight corners and the cube was solvable without any corner turns.
I know that the Rubik's cube is a group and that a corner turn takes the cube out of the group, meaning that we cannot transform the cube back into its fully solved position.
My question is: how many corners need to be turned exactly once for the cube to be a part of the original group that consists of all legal positions? Can I turn just two, or maybe four corners and create a legal position?
Additionally, any good papers or authors to read to learn about the mathematics of the Rubik's Cube?