Sunday 4 January 2015

An old puzzle teaches something new!

4x5x6 Cuboid
Shapeshifts on 2 faces
Yep you've seen that one before! It is TomZ's amazing 4x5x6 cuboid. I struggled this week to think of something to write about for you - I have published 2 extra posts over the holiday period and actually not done much puzzling. Then I remembered, I had actually worked on a pretty wonderful twisty and learned something rather special from it!

It had been an aim of mine for a very long time to get this puzzle and I even met and asked Tom Van der Zanden about it at the last IPP. When Shapeways effectively killed puzzle development with their "priceapocalypse", I more or less gave up on the idea! A lucky occurrence happened when a Facebook puzzle friend, Austin (yes he does actually live in Texas) had a spare one going at pre-rise prices. I jumped fast and it arrived in Sheffield in mid December. Yippee - Happy Xmas!!!

As another gratuitous twisty photo here it is scrambled! It is monstrous!!!

Fun fun fun!!! I know not al of you believe it but bear with me!
Why have I shown it again? Is there method in my madness? Well yes! This particular puzzle taught me a very hard earned lesson! When you think you truly understand a puzzle or puzzle type, another one can come up with something totally unexpected that kicks you in the butt and makes you actually go back to basics and THINK©! This one really hit me hard and took me several days to beat!

I have previously explained the classification of the cuboids and explained how each part of this particular classification is defined by the method used in solving them (they often have their own particular type of parity). One of my favourites has been the brick cuboids of the form:
N x (N + O) x (N + E)   or   N x (N + O1) x (N + O2) where O is odd & E is even
AI cube - build 2x2 blocks up
Examples include the mass-produced 2x3x4 and 3x4x5 made by mf8 and available widely for very little money. Unusually, this particular type of cuboid is not solved by initially solving the base cube (e.g. a 4x4x6 is first solved by resolving the central 4x4 cube within). The bricks are solved by initially solving a Domino cuboid to return it to brick shape and then using edge swapping algorithms to move edge pieces around in pairs. So far these sequences all can be a feature of other cuboids but the marker of this puzzle is the "brick parity" which, like all parities, occurs because the initial reduction technique creates blocks that could not occur if the puzzle had not been reduced in such a way. I dare say that using an entirely different approach - e.g. a block building approach similar to the AI cube then the brick parity is avoidable but half the fun of these is finding these parities and working a way around them! (Editors note: remind me to have a try at solving this using AI techniques)

The brick parity takes the form of a 3cycle of edge pieces - each at 180º from the other across a face. The parity algorithm ultimately involves taking it out of shape reduction and swapping pieces around before re-reducing. Yep! Us twisted twisty solvers love that kind of thing! In this scenario the algorithm is:
F2 U R2 U' L2 U R2 U' L2  F2 r2 F2 L2 U R2 U' L2 U R2 U'  F2 r2  (6x5) face at the front  
My first few solves were agood bit of fun and a couple of times I even came across the brick parity. It did take me a while to work out how to solve it again and I was very proud of myself when I actually managed it. BUT (there's always a but!) my 5th scramble seemed to be going well when I ended up with something new. It appeared that I had produced another brick parity on the second layer.

All done apart from 1 piece
Identical on the other side
Normally a parity on the second layer is not a problem as you can adjust an aspect of the algorithm to move pieces in lower layers. I quickly positioned the puzzle to fix it using the standard algorithm and....... Uh oh! It's appeared to be absolutely impossible!

It won't line up - can't do an R2 turn!
The normal algorithm as stated above requires a U R2 move at the start - as you can see after the U the layers don't line up to allow the rest of the required algorithm! This parity had occurred on the wrong face! I had a brick parity but a totally new one that I had never seen before or even heard of with the lower order cuboids! It took me quite a while to a) realise it and b) solve it!

I posted on FB about it after a few hours of being stuck and was given support and even pointed to a SuperAntonioVivaldi video on it but I wanted to work it out myself if at all possible! My first job was experimenting with the parity algorithm and adjusting the faces/layers I turned and the number of slices I used and I began to understand more about how it worked and how I could move pieces around. I then had another brainwave - I could not use the large face as my Front face (as I showed above) but maybe I could do something with it on the 4x5 face? A quick play about gave me this:

Moved the parity pieces around!
I had corrected the single pieces and moved the parity to the edge! Incidentally I had created a swapped centre piece. It is easy to swap centre pieces around giving me this:

Left with just edges swapped
The centres were easily fixed leaving just the edges BUT despite shrieking Eureka in the coffee room at work and startling the nurses at lunch, I was not quite there - the edge pieces that were off could still not be fixed using the brick parity but they were amenable to a re-attempt at pairing up using techniques learned from a layer by layer approach to the standard cube! The last layer can be manipulated using corner and edge manipulations!

All in all - this puzzle took me a week to completely fathom and reminds me why I just lurve the twisty puzzles - there is always something new to learn! This was a rather expensive puzzle and "just an extension" of those already available in mass produced version but it was worth the cost for the challenge of trying to work out how to beat it! This larger size brought something totally new to the table and I learned a lot!

Go on - jon us with the twisty puzzles - they are a great group of puzzles to play with and a great group of enthusiasts giving advice and assistance. If you want to know where to go beyond the initial basic cubes then start with my advice for beginners post.

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