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I'm learning/teaching myself the megaminx (3x3), based on my experience of cube solving (ranging 1x1 to 8x8, I have no problems with them, but I'm not fast).

Mostly it's pretty intuitive, I could get to last layer, with the edges orientated, then use one of my 3x3 algorithms to permute the corners, then R D R' D' to rotate corners in place.

Then I'm stuck. On the 3x3 I can permute 3 edges with V2 T V T2 V' T V2 (where V is the vertical middle slice rotating backwards), but that can't work on the megaminx because there's no middle slice.

All algorithms I've seen need up the position of the corners. Are there any algorithms that can permute the edges without messing up the corners? Or do I need to change my strategy?

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Most Megaminx solutions do the last layer edges first, and then the corners. The reason is that the corners are easier to move individually, as a corner is the intersection of three turnable layers. On a cube the same is true for edges (the intersection of two face layers and a slice) but not on a megaminx. If you do edges first on a Megaminx, then you can ignore corners and treat edges as the intersection of two face layers, which is quick and easy like a Pyraminx.

It is still quite possible to move Megaminx edges individually while keeping corners intact, it just tends to take more moves, and involve more faces than corners do.

I'm writing this up without a Megaminx available, so I am improvising this, and that means it probably uses more moves than necessary, but should be fairly intuitive. Here is my labelling of faces:

Megaminx face labels

I am assuming that the U face is the unsolved face in which you wish to permute edges. First we need a move sequence that takes out any edge from the U layer, but leaves the rest of the U layer intact. Let's take out the edge at the Ur location:

  • Do r' r' F F. This moves the Ur edge and its adjacent corners down to the DF location.
  • Do R L' D L R'. This separates the corners from the edge, replaces the edge piece by another, and rejoins the corners.
  • Do F' F' r r. This moves the edge and corners back up to the Ur location.

Let's call this move sequence S. Try out S and its inverse S' a few times to familiarize yourself with it. Note that S' is almost the same as S except that the D face moves in the other direction. This move sequence S can be called a monoswap.

Using this monoswap, any permutation of the U edges becomes fairly easy. If you wish to do a 3-cycle of edges A->B->C->A, then:

  • Hold the Megaminx so that edges A,B,C, are in the U face with A at the Ur location.
  • Do S to remove edge A.
  • Turn the U face until edge B is at the Ur location.
  • Do S' to replace edge B by edge A.
  • Turn the U face until edge C is at the Ur location.
  • Do S to replace edge C by edge B.
  • Turn the U face back to its original position.
  • Do S' to put edge C in place at the Ur location.

You can construct a double swap of edges in a similar way, though you could use two edge 3-cycles for that too.

You say you have already oriented the edges, but you could try to construct a monoflip, a ssequence of moves that flips a single edge in the U layer while messing up the rest of the Megaminx. Combining that with its inverse and U turns you can then flip any pair of edges.

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  • $\begingroup$ Thanks, that works! Even better, you clearly explained why and how it works. I trivially orient the U edges when I solve the last adjacent face (save for its U edge) it maps onto my block-building and (for me) intuitive cube solving. $\endgroup$
    – baralong
    Jul 7, 2023 at 14:34
  • $\begingroup$ On my megaminx I've just solved it (thanks for that) then I tried my corner permutation and orientation algorithms, they leave the edges untouched. Is there a simpler edge swapping algorithm, that leaves them oriented, but allows Courtney changes? $\endgroup$
    – baralong
    Jul 7, 2023 at 14:41
  • $\begingroup$ @baralong If you don't care about the last layer corners, a simple edge 3-cycle is R U R' U R U' U' R'. It basically uses a monoswap sequence of the single move R with turns of the U face in between. The only thing is that it does affect one non-U corner, so if you are more free with the U turns you will end up with the edges solved and 6 unsolved corners. $\endgroup$ Jul 7, 2023 at 15:10
  • $\begingroup$ @baralong My preferred method is exactly that. Solve everything except for the last layer and an an adjacent corner-edge slot. Use that corner-edge slot while solving the last six edges (orient, then permute). Then solve the last 6 corners (permute, and then orient). $\endgroup$ Jul 7, 2023 at 15:21

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