Sunday 22 October 2023

A Cube Without Edges

Solves With ONLY The Edge Piece Series!

The Crazy 2x2x2 trio
They all look identical but the numbers on the face describe different turning
I have had a bit of a splurge on some twisty puzzles recently and have been exploring them this last week or two with some success (to my great surprise). Here I discuss the Crazy 2x2x2 (0,0,0 version) and how it looks ferocious but solves with the simplest technique and a lot of thought.

I pretty much always have a standard 3x3 and 4x4 Rubik cube in my bag at all times. Partly because people finding out about me for the first time always ask whether I can solve a cube and then I have to show them when they don't believe it. I also have it to make sure that I don't forget my basic techniques and it does make a wonderful fidget toy.

One thing that a lot of non-puzzlers seem to think is that "there is a single magic algorithm that you can do over and over again" and the cube will miraculously get solved. I have to disabuse them of this idea but always tell them that it is possible to solve the entire thing using just one simple technique but it requires a lot of thought and planning. If I am to solve a cube quickly (ish - I have no interest in learning dozens of algorithms to discern and move quickly) then I solve layer by layer with a very basic system for the top layer which is obviously the hardest. Doing it this way, I average about a minute and I'm happy with that. Alternative techniques include the Rubik ultimate solution described by Philip Marshall and evangelised really well by Rline with his Twistypuzzling YouTube channel in which one solves all the edges first and then the corners. This technique relies on two simple algorithms - the 4 move edge piece series (EPS) and the 8 move corner piece series (CPS). It requires a bit of thought and understanding but is the simplest way to do it. I also use a block building solution, a corners first solution and for fun I do it using the EPS only.

YES! You can solve the Rubik cube with a single algorithm of just 4 moves!


I was quite surprised when I started to investigate this puzzle which, as a 2x2, has no edges and I realised that it does have edges in a way and that it absolutely requires to be solved using JUST the EPS!

The 0,0,0 designation means that 3 of the 6 faces are 0 faces i.e. turning those faces leaves the centre circles unturned (the red, green & white), whereas the other 3 have 1 faces and the circles are fixed to the outer parts. The result of this is that the blue/orange/yellow corner never separates from its' circle and you can only really scramble it or solve it by turning the 3 zero faces.

Does this sound confusing? It is a bit of a mind-bender but is not that tough if you have mastered the EPS completely.

 So what is the EPS? It is a simple 4 moves (up, up, down, down) or the other way around. If done just once then it does what it says on the tin - it swaps three edges amongst the front edge and the UR and UL edges. It muddles up the corners but in Marshall's method we don't care about the corners yet. That does seem very simple but the fun this happens when you do it more than once. On a solved cube if you do the EPS three times (D, D, U, U) then it leaves all the edges alone and moves the corners:

Having done the EPS 3 times, the top 2 diagonally opposite corners have swapped and the two from corners have swapped up and down. 

This can be used to move corners where you want them to be using just a 4 move sequence and some setup moves - it's ingenious and devious!

This all sounds great - you can move edges around easily (and orient them) with the 4 move sequence, you can move corners around using the same sequence multiple times but what about orienting those corners? Yes, you guessed it, that same sequence can be used to rotate the corners without moving them. It is destructive when done once but if done 3 times or done in the opposite direction immediately afterwards then the destruction is undone:

EPSx2 starting L - destructive
Turn U face anticlockwise
EPSx2 starting R - undoes it
So now you can see how it can be possible to solve every part of a Rubik cube using just that 4 move algorithm. It's not easy the first time that you do it but it is fun. How can this be applied to the crazy 2x2x2?

During my initial look at the crazy 2x2 I realised that the blue orange yellow corner was the source of the craziness - it was fixed to the central circle and I effectively could not turn the blue, orange or yellow faces since doing so would move that fixed corner around and would prevent any useful moves on new faces. I needed to orient the puzzle with that corner bottom? back and then solve it moving just the 2 front and the top faces. This is an odd way to look at a cube - I had to stop thinking of it as having a front, left and right face as I normally do. 

Having scrambled the crazy 2x2 it became clear that the circle centres are effectively edges. The white and yellow opposite colours are attached to each other. If I rotate the white face then the white circle pieces don't move. If I turn the red face then it moves the white and yellow circle pieces attached to that side but moves them together, the same is true when turning the green face. Therefore the circle pieces in a pair are effectively and edge but marked by the colours of the face at either end. It takes a while to get your head around this but once you do then start solving edges first:

Pseudo-edges all solved
Opposite sides (corner block visible)
Interestingly, it is possible to get the same sort of parities with this one. The first happens to me almost 50% of the time:
2 pseudo-edges to be swapped
This is a "parity of false equivocation". Each of the faces' edges have the same two colours on them for each side (i.e. there are 4 white/yellow edges and the same for each colour pair) it is possible to place them in any of 4 positions on each face and all that is required is to take one from a top edge and place it in the other top position. It is, again, just a matter of the simple EPS.

Once the circles (edges) are solved then the aim is to solve the rest of the puzzle by turning only those front three faces. It sounds awful but it really isn't too hard. Using the EPS only in multiples of three the corners can be moved around into position until they are all in place.

Again, every other solve there seems to be a parity - I am left with 2 corners that need to be swapped. If you know the Rubik cube then you know that swapping 2 pieces is impossible - there is ALWAYS another piece or pair of pieces that need to be swapped...even if you cannot see it. In this case, the parity is caused by the edges first solution to the cube. If you try to solve edges first on a 3x3 then it frequently turns out that all the edges are in place apart from 2. The reason for this is that when solving edges first, it is possible to have the top face turned 90º without realising it and the solution is to turn it that 90º and then re-solve those edges. It is very easy and the end result is all the circles (edges) solved.
Equivalent 2 top edges swapped
All edges solved
In the crazy 2x2 it is impossible to discern this scenario until we find 2 corners alone needing to swap. Unfortunately, the solution is to go back and resolve those edges/top circles again with the top face rotated 90º in either direction. Having done that, the next time that the corner positioning is done, all the corners will be placed correctly.

Corners ready for orientation
In the beginners method we would use the down face but cannot turn that here. We need to orient those corners using just movements of those 3 faces. Easy peasy! Remember what happens when the EPS is done twice? It rotates a corner and destroys a bunch of stuff. Turn the top face until the next corner is at the front and do the EPSx2 the opposite way and it undoes that destruction (see the pictures at the top of the post).

If the rotated corners are on the bottom face then rotate the cube about that fixed corner until the required corners are on the top face and do it all again. This is continued until all the corners are done.

Solved it! Effectively, you have solved a 3x3 with hidden edges and used just the EPS. It is great fun and a lovely, not too tough, challenge. I urge you all to go back to your standard 3x3 and learn the EPS only solution. It is not hard and just needs some thought and planning.

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