Sunday 21 December 2014

Curvy Copter 3 - A fab puzzle improved by commutator theory

Curvy Copter 3
Some considerable time ago after I decided that a general puzzler should also be able to do twisty puzzles, I branched out and discovered the hugely fun world of twisty puzzles beyond the standard cubes. I have written about the options open to puzzlers wanting to branch out here. One of the very first "non-standard" puzzles I played with and then raved about was TomZ's Curvy Copter. This fantastic edge turning puzzle rapidly became one of my favourites because it can more or less be solved intuitively and is a great introduction to other edge turners including the marvellous Gem series from MF8.

Now 3 years later MF8 have released a new puzzle which they have called the Curvy Copter 3. This was designed by them entirely independently and only after they saw the similarity to TomZ's earlier Curvy Copter and subsequent Curvy Copter 2, they decided to name it after his initial puzzle as a mark of respect. The first release was as shown in the picture above, made from coloured plastic with no stickers (I love these because they look pristine forever) and a black plastic with stickers is shortly to be available. I would consider one of these an ESSENTIAL addition to a twisty collection and if you want it then it can be bought from the usual Chinese/Hong Kong stores for example HKNowstore where I got mine.

So what makes this one so special? As you can see from the picture it retains many features of the parent puzzle - it is an edge turner but now the cuts are deeper than before and actually cross midline making the centre cubies and petals which get moved around. The edges, corners and pentagons remain as before. It also jumbles as the parent did (jumbling occurs when it is possible to partial turns to line up planes of the cuts and then swap pieces completely out of their normal orbits and even out of position.

Scrambled AND jumbled
So what does this add to the original that excites me and warrants a whole new blog post? The presence of the centres and petals is completely new and it requires a whole new approach to that part of the solution. No longer can it be solved algorithm free, you need a technique for moving particular pieces around without destroying other parts of the solved cube. Yes what you need is a dreaded........ COMMUTATOR!!!!!

Before you all run away screaming at the thought of a commutator, bear in mind that if you can solve a 3x3 using beginner technique or using the Ultimate solution then you are already using commutators without realising it - the corner piece series is a commutator (shock!) The fun thing for me here was that, presented with this totally new puzzle, I needed to try and find my own commutator and actually managed it with little difficulty. In fact this approach is used on much more complex puzzles. Probably my favourite twisty puzzles of all time (Eitan's star and the Master Curvy Copter) had similar problems which I solved by looking for a commutator. With very complex multi-part puzzles a common approach is to solve them, not by layers but by piece type and after a few types have been placed it becomes impossible to move the remaining ones about without ruining what has already been achieved and hence a commutator becomes necessary.

I probably cannot give you a definition of a commutator properly but the overall idea is to perform a sequence (preferably a simple one), then another sequence, then reverse them in order. If designed properly then the outcome will be a cycle of a particular piece type without upsetting the others. Many puzzlers use a fabulous piece of Java based emulation software called Gelatinbrain to try and find these commutators on a computer before playing with a puzzle - although the best twisty solver I know (Burgo from the Twisty Puzzle Forum) always does this on a puzzle itself rather than in software which I find astounding! With his encouragement and Konrad from TP I was actually able to find a couple of suitable commutators for the Curvy Copter 3.

To prove to you that you also can achieve such wonderful results I will show you how a basic commutator can be found on the CC3. Taking a non-scrambled cube if you perform the sequence UR, UL, UR, UL then you end up with 3 centre lines having been cycled (U -> L -> R ->U)

After this if you do a UB, UF, UB then the effect of this is to have replaced just one petal within those three lines

UB isolates the blue petal
UF brings it down then UB

After this then the next sequence is to undo the first set of moves so UR, UL, UR, UL becomes UL, UR, UL, UR

After the first sequence is undone
And finally we just undo that short second sequence - because there are 3 moves of 2 edges then it actually remains the same UB, UF, UB. This whole sequence can be written as:
[UR, UL, UR, UL], [UB, UF, UB], [UL, UR, UL, UR], [UB, UF, UB]
The final effect of this is simply to produce a 3 cycle of the petals from top-front to front-top to right-back with absolutely no effect on any other piece as you can see in the pic below. This can be used to move the petals effectively wherever you wish with the use of setup moves to position the required pieces to those correct positions

3 petals cycled around

This means the Curvy copter 3 is a revelation of a puzzle - partly solved by intuition and then requiring a nice easy commutator and lots of setups - so much fun and a really nice challenge without being too horrific. So go on all you general puzzlers out there - it's time to branch out! I would say that this is one of the best twisties released this year! Buy it as it is essential!


  1. Fantastic blog!

    Birthday on 10th jan, will be ordering myself one of these as a treat to myself!!

    1. You will love it Andy! Of course it will not be a tough challenge for a solver of your skills!
      Have a happy birthday on the 10th!

  2. Shame about the centre pieces that keep popping out. I can't messy around with this puzzle in my lap for fear of losing centres regularly it's very annoying. Great read though Kevin!

    1. Hi Bruce, thanks for the compliment - I'm glad you enjoyed the post.

      You can wedge the centre pieces in place - I took a small piece of paper and put that on the centre hole and pushed the cap onto it. This just tightens up the grip the cap has on the edge of the holding piece.

    2. I will try this... I wish this puzzle was a little larger. I am not sure why this puzzle is so small given the number of pieces and the degree to which faces have to be aligned to turn the edges. It would be much better if this puzzle was 4x4x4 sized I think.
      I am busy unlearning all previous twisty knowledge in favour of the ultimate solution. Early days but definitely a twisty faze I'm going through at present. Your Aussie mate's site and youtube videos have peaked my interest. I like the idea of adapting simple rules to achieve complex results. Mhuti

    3. Good luck with the Ultimate solution! I do use it sometimes for fun and it is a good technique to supplement by beginners method but I do find it too restrictive - even Rline has admitted that and occasionally added in the use of SUNE and antiSUNE. He has also used commutators for particularly tough puzzles.

  3. I had heard him say something similar on one of his videos. I realise it's not a one stop solution but I'm intrigued enough to take a look at adaptive algorithms such as this. I will look at SUNE/antiSUNE as well no doubt, do you have any suggestions/links in that regard? mhuti

  4. Petal 3-cycles can be made even shorter than 14 turns as shown here.

    Try: (UR UL UR FR BR FR)2 or [UR UL UR, FR BR FR]

    where [A, B] = A B A' B'

    THere are also other possibilities for making 12-turn commutators permuting 3 center petals.

    Feel free to explore :-)

    Per Kristen Fredlund

    1. Thank you for that. I am very poor at working out my own algorithms and commutators. Gradually I hope to get better.