Message #3345

Subject: 4^4 Parity solutions
Date: Sun, 24 Apr 2016 18:17:05 -0700

Hello all

My recent experience solving the 4^4 shows some interesting parity situations. You may or may not need the solutions for solving the flip parity that affects the block of 2 edge pieces on the 3D Rubiks Revenge cube, it depends on your solving method. I tried adapting the one that I use on the RR (link below) but couldn’t get it to work and that is due to the odd number of moves in the original sequence from 3D and this means that there is a lot of irreversible or difficult to undo scramble in the other layers of the cube. All of the techniques that I found on Youtube that address this parity of the 3d RR have an odd number of moves and so are not adaptable in my system so I had to find a new technique. I think this was good though as I used one of the techniques off of YT for the 3D RR so cheated on that one. There is another way though which involves re-scrambling the cube and taking advantage of the 50/50 chance of incidence of this parity to get a non-parity game and solve it that way. That may be fine for 3D but you wouldn’t want to do this on a 4^4 though not least due to the 1 in 4 chance of a non-parity of the 2 colour pieces.

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The log files provided show parity solving and there’s also some screenshots I include of these situations.

The first instance - 2 Colour Full-Block Parity.

If you have assembled all the 1, 2, and 3 colour pieces into their respective blocks thus reducing the cube down to a 3^4 and then proceed to solve as such you may come across this parity when you come to solving the last of the 2 colour blocks of 4 pieces and find one of them flipped. The sequence to resolve this takes up to 45 moves and can be done as soon as you recognize this or at the end of the game.

The second instance -2 Colour Half-Block Parity.

This parity crops up as you come to the end of the 2 colour piece assembly phase and is best resolved before
you move on to pairing the 3 colour pieces. It will take only 20 moves at this point, but leave it late in the game it will need 81 moves as the sequence affects a number of other pieces too and these would then need to be restored.

The third instance - 2 Pairs-3 colour position parity.

This is just like the one found on the RR. Both pairs of pieces have the same pairs of colours on the Top face, the other pairs of colours are flipped and the pairs of pieces are opposite to where they should be. The sequence to fix this is again adapted from 3D and is 52 moves long.

The fourth instance - 3 colour piece 3 cycle parity.

The methods I use which is to group all the 1, 2, and 3 colour pieces and then solving as a 3^4 with the LBL technique rules out this parity but it may occur with other solving methods. It is similar to the 3D version but this time it is a 3 cycle not a 2. This sequence is 82 moves long and uses the adaptation of the 3D version mentioned above, some other moves, and then the reverse of the first sequence or its mirror image from the back face then the reversal of the other moves and so get the 3 cycle parity.