Sunday 2 October 2022

It's Easy But Very Hard!

David Pitcher's 4 Corners Cube
Named this way because only 4 corners can be moved - notice that the top front corner is fixed
Back in May I bought a bunch of twisty puzzles that had been released in the previous year that I had been fascinated by and I felt the "need" to catch up (yes, I told Mrs S that I needed them!) I was particularly interested in the 4 Corners cube which was a design by the incredibly talented David Pitcher who I had been delighted to meet at the Paris IPP. I was amazed that he designs these incredibly complex puzzles and has them 3D printed and yet does not solve them. He understands geometry and yet cannot solve twisty puzzles. I, on the other hand, studied Geometry in some the Open University courses but never got to understand the sort of shapes that David deals with. Like him, I struggle to solve them as well.

The 4 corners cube was mass manufactured by Lan-lan and they did a superb job. This puzzle moves and works beautifully. My initial interest (beyond owning one of David's creations) was because I absolutely adore edge turning puzzles and the jumbling/shapeshifting that goes with them:

Edge turning and jumbling
This photo was taken after just 2 moves and you can see the shapeshifting and immediately see something that frightened the Bejeezus out of me! Firstly, it isn't really an edge's a sort of corner and edge turner together. This means that my feeble bwain is going to get very confused when it starts to move and, second, the split between edge and corner is aligned in such a way that it blocks subsequent moves - it bandages. Aaaargh! For this reason, I had it in my work bag with me for 4 months and did nothing more than fiddle. It was fearsome. 

I did realise that simple 3 cycles would be possible and this might be useful but after working this out, I could not see any way that this might be helpful. I put the bloody thing down again! Time to Think© but Allard wasn't going to be any help here.

I could see that using setup moves I could 3 cycle the centres into more or less wherever I wanted them to be but how was this going to help with all the other pieces? It wasn't going to be a very fun challenge if all I dared to do was scramble the centres. I took the puzzle with me up to Edinburgh with the intention of spending some decent time with it while not having to work but again it frightened me to death and I got side tracked by the fabulous Akaki Picnic basket puzzles.

Bring up bottom edges
Perform 3-cycle
Move edges back
After a bit more study, I realised that I could 3 cycle the edge pieces along with the corners as shown above. It just needed the correct setup moves again. I was making some major thought breakthroughs just by idly fiddling and trying just short sequences at a time.

Having found the above cycle, I had an epiphany - I am certain that you all will see it. Having done my cycle ending with that third picture it is clear that the top corner is now isolated and can be moved away. This means that I have found the first part of a commutator and a potential piece isolating sequence.

The second half of the commutator
Different corners but a corner 3 cycle found
Having worked this out and managed to perform it several times I was ready for a scramble! Except it is not an easy cube to scramble. It gets blocked up very quickly. I had to do setup moves to scramble edges and corners properly without getting blocked and then only at the end of a bunch of these could I just let lose and randomly turn. It got progressively harder to find sections that would actually turn properly or even at all. I (actually, David) had created a monster:

That was a very silly thing to do!
I seriously regretted what I had done and had serious doubts about being able to solve it. It has taken me most of this week to get to grips with this challenging puzzle. First of all, try to get the wing shapes (those that sit on either side of the centres) back into the correct orientation. This was fun but not easy - it was soooo blocked up that it was like unravelling a knotted piece of string to get it into something resembling cube shape. Once this was done then the 2 colour edges need to be placed into position and I thought this would be pretty straightforward until I found a scenario rather like this:

Here I have created it with just one edge but there may be several like this
Somehow some of the edges and corners have been positioned but flipped 180º. OMG! This is a nightmare for me! I know that it means that at least one of the centres has been invisibly flipped as well but trying to fix it is so so hard. I am sure that several of you brilliant twisty puzzlers out there are screaming at my website that it is an easy fix. But whole knowledge of how to do this puzzle is based on performing simple 3 cycles and using a pair of simple 3 cycles to move corners around. I literally have NOTHING more complex than a 4 move cycle (yes, it's a variant of my up, up, down, down system). 

I needed to fix the problem using my 4 move sequence and a whole bunch of setup moves and maybe do it several times to place things out, rotate them and place them back (often having to move them all the way around the cube in the process). I have performed this several times during multiple scrambles and have yet to actually understand a simple way to do it. I can manage it but only after multiple attempts and a LOT of swearing! Once the edges are all placed then the next step for me is centres (it could be the other way around):

Edges placed - the easy bit left to do
As I had mentioned above, placing the centres is really easy but can take a bit of setting up and then all that is left is to magically use my commutator to position the corners (easy peasy!)

What the hell is this?
All corners placed but 2 rotated in opposite directions! Aaaargh!
The first time I solved the puzzle, my corner commutator worked it's magic perfectly and I had my cube finished, let out a whoop and frightened a cat as well as annoyed the present wife! I was on a roll! Time to scramble again and see if I could master those blasted edge positioning moves (nope! still not managed it). I struggled yet again with those edges but finally got them done and expecting success, I moved my corners around boldly and sat back in disbelief. What was going on here? I had everything placed but 2 corners rotated - Nooooo! Fixing this took me several hours - as stated, I only have my 4 move 3 cycles - nothing more complex than this and I had to try and use them to take the corners out and rotate them back into place facing the right way. I have managed it a couple of times now and in the process found another variant where I all corners are in but 3 are rotated all the same direction. I can fix it but it takes me a long long time!

Should you buy this puzzle? Absolutely yes! It is a fabulous design and turns beautifully. I am sure that you will be able to solve it with much less trouble than me. The amazing thing is that it does not require anything fancy to solve it. It just needs a 4 move up, up, down, down cycle of 3 pieces used in imaginative ways. If you do it this way then it requires a lot of thought and planning but that is half the fun. If you do know of any better ways to fix the reversed edges then please let me know.

Thank you David for a fantastic challenge - I hope to get some more of your amazing puzzles soon (I am aware that there is a plus version which un-bandages the edges allowing for a much more complete scramble. I don't own this one yet but it is on my list of puzzles to buy. I need to wait for Mrs S to be away visiting the outlaws again before I order anything else! She keeps picking up sharp objects and looking at me with narrowed eyes!

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