Wednesday 20 June 2012

A Parity of False Equivocation!?

Fisher Cube
Fisher cube other side
There! I bet that title made you do a double take!!! This terminology was coined by my puzzle friend Jon - he is SuperantonioVivaldi on YouTube. Go and view his channel - he is absolutely amazing! If you think that you can't do twisty puzzles then let him teach you - EVERYONE can do twisty puzzles and Jon is one of the best teachers I have ever seen. He is also starting out as a puzzle maker/designer so if you really want to try something special then have a look at his Shapeways store (this is NOT for beginners however). Whilst I am advocating good teachers for twisty puzzles, please go and view Rline's site - this teaches many twisty puzzles using just intuition and 2 (yes JUST 2) very simple algorithms. It provides some fantastic techniques. I personally use a mixture of methods and just now am working my way through the Crazy 3x3 planet series and seem to find myself mostly using Rline's techniques because they really go back to absolute basics.

The Fisher cube was designed by the world famous puzzler, Tony Fisher - he has been designing cube modifications since 1981 when he unleashed the Siamese cubes on the world and has designed at least another 100 puzzles over the years.

I got the Fisher cube from Puzzle Master as part of my recent large order - they have a huge selection of Meffert's and other manufacturers' Twisty puzzles at decent prices (if you prefer it in black then Puzzle Master have it in that colour too). This can also be bought from all the usual specialist twisty outlets. It follows the standard Rubik's cube puzzle colouring and is a 5.7 cm cube also like the standard cube. Puzzle Master rate it as a level 9 in difficulty and this is correct if you are not a twisty person but if you can solve a regular cube then it is not so hard (apart from the alleged "parity of false equivocation")!

So, if you can do a 3x3 cube should you get one of these? Absolutely yes! This uses the same basic techniques but adds some very interesting twists to them, making it a rather interesting and really fun puzzle to solve without being ridiculously tough.

As you can see from the pictures, it is just a 3x3 cube with top middle and bottom layers as normal but the vertical cuts in the cube have been rotated by 45º. The faces, therefore do not turn at all - it is an edge turner in the vertical orientation. This makes for a very attractive cube but it also means that when it turns, odd things happen - it very rapidly starts to change shape until after a full scramble it becomes a very shape-shifted nightmare:

Doesn't look too cube-like anymore!
The challenge therefore is initially one of orientation - trying to work out which pieces are centres, edges and corners. This particular problem will not take you too long to overcome - it is not as horrific as the Mastermorphinx, for example. It also has the same issues mentioned in that review - the Fisher cube does have some Supercube properties. This means that some of the centres have more than one colour on them and hence orientation does matter - this rules out some of your usual algorithms which you will find rotate the centres (up until now you won't have realised it because on a normal cube the centres are rotationally indifferent). Having realised this - you blithely go ahead and solve it just as you would a standard 3x3 supercube.

Not so fast Batman!!! There are 2 approaches you can take! The first approach should be solving it as a normal cube with a flat first face - easy peasy. BUT you could make it interesting by solving a non-flat face first - just for the extra challenge:

Flat first face
Alternative first face
Each approach has its own little quirks so you should definitely do both!

These aren't the only issues with this puzzle - now we get on to SuperantonioVivaldi's lovely little term. By either approach you can solve the cube as you would any normal cube - I usually do it layer by layer. As with all these puzzles the last layer is where the difficulty lies because you have to move a lot around without mucking up all your previous hard work. The first thing I would usually do on the top layer is orient the edges correctly (i.e. have all the correct colour facing up). Normally there would be 0, 2 or 4 edges to orient. Look at the picture below - we have an impossible situation - only one edge the wrong way up!!

Impossible position???
Is it a parity? No! Parity really only occurs in even order cubes when we try to reduce them to an odd order to simplify the solve. This is a "parity of false equivocation" - the error is that the edge pieces of the centre row are just 1 colour. This has allowed us to place one the wrong way around and yet not be able to tell the difference. This is therefore, an easy problem to fix once you realise where the problem lies. It is always something to keep in mind with any puzzle where a piece that should have 2 colours on it only has 1.

When solving with the alternative start, an even more interesting parity seems to occur - the edges are orientated correctly but they are permuted incorrectly:

2 edges in place and 2 reversed
It should look like this!
In the upper left picture there are 2 of the edges out of place - again this is an impossible situation - ALL permuting algorithms of a 3x3 cube cause the swapping of a trio of pieces - therefore it is not possible to swap just the 2 edges required above. What is wrong here? This is much more subtle and much harder to fix. In this case, the issue is because this is a supercube but has not been stickered as a complete supercube. The yellow and white centres can be rotated the wrong way without you realising it! In this case just one of the centres is orientated wrong - a different "parity of false equivocation". If you know how to do a supercube then you should have a technique in your armoury to deal with this.

Now all of you are aware that I have become a complete twisty puzzle freak - I have 64 of the damn things in my study just now with a list of more to buy (just so I can annoy the present Mrs S even more)! I would say that ALL puzzlers should at least own and be able to solve a standard Rubik's cube - it is one of those "basic collection" puzzles. If you get into them then there are lots more to try.

I would suggest that the Fisher cube is another one that is worth owning early on in anyones collection because of the several extra interesting problems that it poses without being too horrific. So, all of you puzzlers out there - go and get one of these! It is great fun!


  1. Great post. Can't believe I'm the first person to comment on this. Couldn't agree more that the Fisher cube does not have true parity...despite a bunch of YouTube videos to the contrary. Cubers today are so focused on using algorithms they don't even look for the simple solution: edge on top is reversed? Bring it back down, then put it back up the right way. It works! Love your suggestion about solving the non-flat's like having 2 cubes for the price of 1. I'm a Fisher fanatic...I'm up to 18 varieties which I think is just about every type that is available...and I enjoy solving them just like you suggested. My favorite is the Arrow cube which actually gives you 4 different solves depending on which layer you start with. I think you can still find it at if you're interested.

    1. 18 different Fisher cubes? Wow! I don't really have very many and am always interested in more. I will bear your suggestion in mind when I go to buy some more twisty puzzles