# The 4x4x4 Rubik’s Cube Puzzle – Intuitive Solve Part 2

This puzzle video is in regards to the 4x4x4 Rubik’s dice. I have been doing solves of a bunch of puzzles, in an effort to lastly resolve the 4x4x6 CrazyBad Fisher Dice. Sooner or later…

Rubik’s playlist (hopefully this works!): https://www.youtube.com/watch?v=vK9XjwmseSk&record=PLcpX2qo49TGbCvpVZ7sW2YfimfDIDQ53T

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Comments

I am looking forward to your next video in the series. My 2x2x4 hasn't arrived yet so I have abstained from watching that video but will do so as soon as I figure it out for myself. BTW what do you plan on doing next – 4x4x4 fisher or 3x3x5.

Just to clarify, you use "B" as in "bottom" but in Rubik's terms you should use "D" as in "down" and use "B" for "back".

can you explain, how did you come up with this algs?

I’m surprised you got the little r and l notation correct

It's possible given any position to figure out what parity state you have and what type of moves you need to do to place it into the goal state if any exist not just for this but almost all twisty puzzles. You can do this by subtracting 1 from the cycle count for the pieces in question. If you have a 4 cycle of corners like on this puzzle with a face turn the minimum number of swaps that need to occur for this to happen is n-1 or 3. That is an odd number hence the change in parity with a face turn.

If you calculate all the parity changes for all pieces given all possible minimum moves you can get a very good idea of how your sequence needs look.

face turns: Corner parity changes, center parity changes, edge parity does not change. we add the parities of the 2 separate 4 cycles together because at least one moves exists on the puzzle that can change that piece orbit(a slice move)

Slice moves(layer under a face): Edge parity changes, center parity does not change for the same reason as edges above, and corner parity does not change.

Given that we can already deduce that the number of face turns that need to occur to should be even and the number of slice turns should be odd. After your fix(the slice turn) you used commutators to reposition the other pieces. They by definition can't change the parity of a position as the number of moves is always even for any move type(the inverse of every move exists and the parity changes cancel)

As a speedsolver, I can safely say that there are two distinctive parities: OLL, in which one 3×3 edge needs to flip (by swapping the two wedges that compose it) and pll, in which two 3×3 edges need to swap. Using your algorithm and doing each wedge individualy, you avoided PLL parity completely, and I imagine it could also be solved using the centers, almost like a void cube (which hasd this same parity). What you got was some variation of what I call OLL parity, as two individual wedges need to switch places (which using my method would flip a 3×3 edge's orienation). Just for the purpose of experimenting, I would suggest you try to use a OLL parity alg from the internet to reach the parity state you were in, and it should work.

BTW, The way you overcame that parity by yourself was simply ingeneous!

Your notation is not self consistent. You refer to the left and right inner slices with the same letter as the outer slices but lowercase, whereas the up and down inner slices you refer to using the letter for a 3×3 horizontal slice in upper and lowercase…

3x3x5 Fisher cuboid would be good to know how to solve, because it helps understand how the fisher mod applies to cuboids.

btw, this is a much nicer cube than the 3×3 you used!

Just like.. slightly straining the brain on that whole vid. Very well explained, just challenging to follow!

*Undoes pronounced 'Un-Duz'

10:49 YES! OLL parity is turning layers and PLL parity is switching pieces! As a cuber, I have just been told this and learned very little about the actual theory of why it happens. It's really interesting to see you actually figure it out.

Rotating four pieces involves three swaps, that's an odd permutation. Outer layers have four sets of four pieces, so rotating an outer layer once involves 12 swaps, an even permutation. Center layers have three sets of four pieces, so turning a center layer involves nine swaps, an odd permutation. That's why you had to move a center layer and resolve the centers to solve the cube. On a 6x6x6, rotating and resolving an outer layer would actually work, because they consist of nine sets of four pieces. Mathologer has good videos on the parity of permutations.

Edit: Parity errors are as nasty on a 6x6x6 as on a 4x4x4. Just checked.

Hey FLEB, after you solve the 4x4x6 CrazyBad Fisher Cube you should solve the mirror blocks (bump).

It is way easier but very fun.

I Feel like the 3x3x5 might be easier, so that's a relief I guess.

Having only two edges switched is an odd permutation. Turning a center layer a quarter turn cycles 4 edges, and that's also an odd permutation. Thus when you have both it's a even permutation, and you can attack it with your even permutation algorithms.

I struggle myself solving parities on the 4x4x4, 5x5x5 and 6x6x6 intuitively. I can get everything else just fine. I might be able to find something for myself after this video, though i'm not sure i completely get it yet. It sure makes it more interesting.

When I solved the 4x4x4 cube years ago, I used the approach of putting the six center faces together first, and then all the middle edge pairs together second. Then I tried to solve using 3x3x3 alg's. I have a 50-50 chance of success (parity). I found a parity switching algorithm that was about 5 moves (which I've long since forgotten), that didn't destroy too much of my original work. As I recall, all 6 center faces were restored, but I had a few edge pairs to put back together. Now solving using the 3x3x3 method worked fine.

I think you suggested this approach at the beginning of your part 1 video. You might want to go back and try that, just for kicks.

Hey you're a puzzle designer wright? Could you maybe make a video tutorial on how to design your own puzzles. Would be great.

Hey Fleb, I got this sliding puzzle I'm having trouble with. Is there a place I could send you a photo of it?

On the topic of parity. The 4x4x4 is really just a 5x5x5 without the center slices. It helps to consider what's missing from view when solving these cases.

For example, on the 2x2x2, you can have corner parity if the edges aren't properly positioned, even though there aren't any edges to show. By just performing an edge swap, you solve the corner parity. I hope this helps.

It is annoying with those constant cut-outs in the video. I can't watch this.

Are you going to learn a more traditional reduction technique after this series is done?

Yay more