First there was the cube - and then.....

Surely this cannot be real!!!

This is the 3x5x7 cuboid - it was designed by the truly amazing Gregoire Pfennig and is available (not cheap) from Puzzle Maestro’s Shapeways store. At present I think it is probably the toughest cuboid available in the world although I’m sure not for how long - it is my absolute favourite twisty puzzle just now and has stimulated me to write a more general post about cuboids.

As you have probably realised over the last year, I have become progressively more addicted to twisty puzzles. I embarked on this odyssey with some trepidation - I emailed a few of the more vocal twisty puzzlers and asked them “what was their secret?” and “how do you go about a approaching a new puzzle?”. Now Rox was her usual mad self and told me that the puzzzles "talk to her" and she just listened to them and they would tell her how to solve them! Well, if I waited for that to happen, I’d never manage to solve anything - I may (or may not) hear voices but couldn’t possibly say in public for fear of being carried off by the men in white coats. Crazy Bad Cuber and SuperAnonioVivaldi were very helpful but couldn’t really tell me what they do. But they basically said that they had learned basic algorithms and added some more complex algorithms and over time and after experimentation they had learned what other effects they had on the puzzle. After that, solving came fairly naturally with trial and experimentation. All of these “Gods” of the twisty puzzle world were very encouraging, trying to entice me into their world. I was still very skeptical about it but decided to give it a go and buy a “few” twisty puzzles!

Well 85 puzzles later I can categorically say that they were right! I have actually reached a point where I can pick up a new puzzle and after some investigation, I can often (although not always) work out an approach to solving it using just existing algorithms (sometimes used in unusual ways) and am now even starting to understand the process of making commutators (techniques for cycling pieces by combining short sequences). I am far away from the genius of at least a dozen of the guys on the Twisty Puzzle forum but I feel I have reached a point at which I can even consider myself able to teach - at the moment I’m sort of assisting a young man in Australia via Facebook messenger - who'd have believed it?

Enough rambling!!! What are you going to review?

The main focus of this topic is one that a few correspondents have requested that I write about - when you have learned the basic cubes (3x3 & 4x4) then the options I’ve discussed before open up to you. In that article I chose an eclectic mixture of shape mods, crazy cubes and alternative shapes but ultimately my real favourites have been the cuboids of which I now have about 15. A little while ago I was amazed to see that SuperAntonioVivaldi had produced a summary of the different families of cuboid based on approach to solving. I think that this wonderful treatise needs to be shared so I asked his permission (and received it) to publish a summary of it on my blog for all general puzzlers to read. Thanks so much! If you want to watch his original video then visit his YouTube classification video here.

All the cuboids can be traced back in their ancestry to the standard (3x3 & 4x4) Rubik cubes and require systems based on them with some modifications to solve. This huge family provides some wonderful solving experiences requiring some thought as well as a moderate knowledge of algorithms. Many are very cheaply available and I’ll provide links where possible.

3x3x2	3x3x4
Ayi's 5x5x4	Ayi's 4x4x5

The Domino cuboids are most puzzlers’ first experience of a "cube that’s not cube shaped any more" - I remember getting mine and being totally bamboozled by the inability to turn the sides 90 degrees! How on earth can you do that? Well let me tell you that they are the simplest to do and are therefore a great starting point in your odyssey.

They take the generalised form:

N x (N + O) x (N + O) or N x N x (N + O) where O is any odd number

Examples are the 2x2x3, 3x3x2, 3x3x4, 4x4x5, 5x5x4, 5x5x6, 6x6x7 and 7x7x6 (not all of these have been mass produced but are available on Shapeways). They are cuboids that do not shapeshift in any direction and only layers parallel to the top and bottom faces can turn by 90º.  When scrambled they are not particularly fearsome:

This is as bad as they get!

The key with the domino solve strategy is initially to reconstitute the centres using the standard methods used in the 4x4 (and bigger) cube, then to reduce the small edge pieces into giant combined edges and then to use techniques to corner switch between layers, then corner swap within layers, and finally adjacent or opposite edge swaps to solve. This is done in layer pairs. This group are the simplest of cuboids to solve and their are no parity problems (Phew!). Once you mastered these then you are ready for others because they are what you ultimately reduce all cuboids to in order to solve.

Rubik's Tower

TomZ' 4x4x6

So far I can see you yawning through your computer screens and mobile phones - they don’t sound very exiting (or even difficult) and you are right until you get to the big ones but they are the basis for others. The Shapeshifting cuboids are most people’s next step and are wonderfully confusing to first play with and they really boggle the brains of your non-puzzling friends! They have the general form:

N x N x (N + E) where E is any even number

These include the 2x2x4, 3x3x5, 4x4x6, and 5x5x7. Because they are the same order (even or odd) in each direction they can be turned by 90º on any face and when the lengths of the sides don’t match they obviously shape-shift in every direction.

4x4x6 scrambled - shape-shifted in every direction

The solve strategies for these are a multi-phase approach - firstly, the aim is to solve the lowest common denominator cube (e.g. in the 4x4x6 you first solve a 4x4 cube) to return the cuboid back to it’s proper shape - Yep! you use a cube solve strategy to return a shape-shifted cuboid to cuboid form! Next the aim is to complete the center placement giving you effectively a very oddly proportioned Domino cuboid! Finally, using domino strategy, the aim is to fill in the corners of the other layers until solved. This one also does not display any parities other than the standard edge flip that may be required in the even order cube section. It is great fun for the multiple types of approach required - cube and domino strategy.

2x3x4 by MF8

3x4x5 by TomZ

Now things start to get really interesting!
Next up are the Brick cuboids - these are of the general form:

N x (N + O) x (N + E)   or   N x (N + O) x (N + O + 2) where O is odd & E is even

Including the 1x2x3, 2x3x4, 3x4x5, 4x5x6, 3x5x6, 2x4x5, 4x6x7, or the 2x3x5, 3x4x6, 3x6x7. Only 2 of these have been mass produced but they are fantastically challenging  Basically, you have two even and one odd layer, or two odd and one even layer in the cuboid which means that it does shape-shift, but only in 2 of the 3 axes. This has the advantage that it looks horrendous to your friends and also has a particuarly interesting solution strategy.

3x4x5 scrambled - less horrific than the Shape-shifters?

The strategy here is different again! The aim is to return it to the cuboid form first but this time instead of cube algorithms you have to use Domino techniques with the edge pieces as your guide! Confused yet? You will be! After returning to cuboid shape the aim is to manipulate the cubies around the puzzle using ONLY algorithms that utilise 180º turns! Have you ever tried to scramble and solve a Rubik cube or 4x4 using ONLY 180º turns? No? Well you’d better start practicing right now! It’s tough but is a really fun challenge. This is quite a complicated solve as the cuboid configuration may require quite a number of “pre-algorithm” set ups and deconstructions i.e. you may need to take it out of cuboid shape temporarily before it is finished. PLUS, for the first time you will come the dreaded parity problem.

What is cube parity? I hear you ask across the intertubes (voices again!) - a cube parity occurs when a reduction of one puzzle to another (simpler) one produces a conformation that is not actually possible in that second one. This means that the reduction needs to be undone and then redone again without that odd positioning. So for the brick cuboids you can end up in a position with 3 pieces out of place, each a 180º degree rotation away from each other and apparently no easy way to cycle them. Undoing this is also a nice fun way to end the solution!

*Please note that if you do buy the MF8 2x3x4 then you buy the version with the pieces to make it fully functional (unfortunately it has an internal defect which causes it to lock up) and watch this video to learn how to fix it.

Not more! I hear you cry from around the world! (Damn voices!) Oh yes one final fantastic group!

2x4x4 from Hunter Palshook

3x5x5 from Hunter Palshook

The final group is the Floppy cuboids. I have no idea who named them but based on the original 4x4x2 and its’ general shape I can see where it came from.
This fantastic bunch are of the form:

N x (N + E) x (N + E) where E is any even number

Included here are the 2x4x4, 3x5x5, 3x7x7, 5x7x7 and also if the Es are each different even numbers the puzzle can be asymmetric like the 2x4x6 (recently announced on the Twisty puzzles forum) and my favourite, the 3x5x7. These do look very similar to the Domino group but are classified differently because of the presence of a “floppy parity”. Due to the entirely even or odd order of the puzzle they wildly shape-shift and in doing so “bandage” parts such that they cannot be easily returned to the cuboid shape. At present there are no “normal” mass produced versions of these. Fairly recently Witeden produced a non-proportional 2x4x4 which is quite fun (although not strictly fully functional). If you want one then they presently need to be bought from puzzle modders - Hunter from TP made me a wonderful pair - 2x4x4 and 3x5x5 - if you’d like one then I’m sure he will oblige if you ask (contact me via my contact page or pm him via TP.

 3x5x5 scrambled - it's truly an amazing puzzle to solve!

For these, the solution consists of starting with normal cube strategy as before for the lowest layered side, again we require center and edge placement to free up the end layer bandaging, and then use algorithms and common sense to reduce it back to the flat cuboid form. This alone is a major feat but you are still not finished! This hard work is then followed by a Domino strategy to finish the solve. It doesn’t sound particularly hard BUT this is amongst the most difficult of solves due to the “floppy parity” that occurs extremely frequently i.e. often several of the edge pieces are mixed up during the reduction and not easily undone. This mandates the use of special parity algorithms to undo them - these algorithms are not new or difficult (in fact you have used it before on a 4x4 cube) but here they need to be strung into a series of 3 of them with intervening setup moves.

3x5x7 scrambled - Is this solveable? Oh YES!

My latest cuboid was the 3x5x7 pictured at the top of the post. It shares features of both the Floppy cuboids and the Bricks including the parities from each. If you do become hooked and work your way through this hierarchy yourself then this is the ULTIMATE cuboid - it is magnificently difficult and well worth the rather high cost. I am aware of at least one prominent TPer who seriously struggled with this puzzle - considering the huge extent of his genius, this shows just how challenging this “King of the cuboids” really is! Buy one if you can - you will not regret it!

Have I tempted you? I hope so! If I have then Contact me here to let me know of your experiences. These are a fantastic series of challenges but require only a few new ideas to solve! Certainly one of each deserves a place on the shelves of every puzzler. Try it, you’ll love it!

Here is a list of links to help you purchase some for yourselves:

Domino cuboids:
2x2x3 US, HK
3x3x2 UK, US, HK
3x3x4 UK, HK
4x4x5 US, HK
5x5x4 US, HK

Shapeshifting cuboids:
2x2x4 US, HK
4x4x6 US, HK

Brick Cuboids:
2x3x4 UK, HK
3x4x5 UK, US, HK

Floppy cuboids are not mass produced yet apart from:
Non proportional 2x4x4 HK