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I recently came up with a Rubik's configuration which, when applied twice to a 3x3x3 cube, will put it in the super flip (R' F' D R D B2 D F L F' L' F2 R2 U D L2 F2 [17 HTM]). What I found is clearly not the only such square root for the super flip. However, when you look at something simple like R, it seems that a square root is not quite as obvious. In fact, with the following reasoning, I do not believe that there exists a square root for an R turn:

- Consider representing just the position of the corners in an 8x8 permutation matrix. Then, doing an R turn will result in a permutation matrix with a negative determinant. If such a matrix, let's call it A, is the square root of this permutation matrix, then A would need a complex determinant in order for det(A)^2 = -1. Thus, such a square root cannot exist.

With this in mind, I can see myself perhaps finding the square root of a cube group element by representing it's corners and edges as matrices and diagonalizing them to find the square roots. Has anybody fully implemented or developed a method for doing such a thing? Is there even a use for finding such roots of cube group elements?

- Consider representing just the position of the corners in an 8x8 permutation matrix. Then, doing an R turn will result in a permutation matrix with a negative determinant. If such a matrix, let's call it A, is the square root of this permutation matrix, then A would need a complex determinant in order for det(A)^2 = -1. Thus, such a square root cannot exist.

With this in mind, I can see myself perhaps finding the square root of a cube group element by representing it's corners and edges as matrices and diagonalizing them to find the square roots. Has anybody fully implemented or developed a method for doing such a thing? Is there even a use for finding such roots of cube group elements?