# Puzzle on a Rubik's Cube

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.

It is

possible.

Proof:

One way to do it is
R2 U2 R2 U2 R2 F2 R2 U2 F2 U2 R2 F2

Another way to see it is that you can do two adjacent swaps:
U2 F2 U2 F2 U2 F2
and
U2 R2 U2 R2 U2 R2
This fixes the U layer pieces, but also does two swaps in the middle layer, leaving a 3-cycle. This 3-cycle can be solved with:
U2 F2 U2 R2 U2 F2 U2 R2

• Wow! Thanks so much. I got the two adjacent edge-swaps but thought the 3-cycle was unsolvable since 3 was odd... Jun 23 '21 at 23:29