The puzzle is as follows:
Consider a solved Rubik's cube. Apply the move sequence M2 U M2 U2 M2 U M2 (Commonly known as the H-permutation in the cubing community). Is it possible to solve the cube using only double-moves (when you do a move, you must repeat that move)?
I suspect the answer is no because the best double moves can do is swap two edges but that doesn't preserve its adjacent edges -- consider R2 U2 R2 U2 R2 U2 for example.