Last week i designed a new type of Rubik’s puzzle: The mechanics are identical with those of a Rubik’s cube, but the solution state is defined by a continues line that runs and twists over the surface of the cube instead of equally colored sides. The evil twist is that you dont know a priori which piece needs to go where. Which means you are not only solving a rubik’s cube but also solving a puzzle.

The solved state is a continuous black loop that runs all over the cube.

The video below gives a good idea of what this puzzle is all about.

Of course if you memorize the solved state, the difficulty is no more then that of an ordinary rubik’s cube. However, the solution state is a very complex, twisted path and thus not easy to memorize to start with. Further some of the pieces look very similar but are oriented in different directions that are not exchangeable.

So far all of my Rubik’s enthusiast friends (some with <2 min solving time on ordinary rubik’s cubes) have failed to solve this cube. The challenge is out. The best attempt at the time of writing was a solve to 2 layers and 3 corners but the remainder of pieces did not fit into place – a dead end.

Here is a picture of the cube in it’s scrambled state and a link to a video.So how did I arrive at this ?

I was thinking about NP-complete vs NP-Hard problems. In simplified terms the former is a class of problems where a solution is relatively easy to identify as such (i.e. in polynomial time)  but hard to find. The latter is a set of problems, where even when you stumble over a solution its not clear that you have one. This is certainly the case in global energy optimization problems such as Protein Folding.

Anyway, thinking about this i realized that the difficulty of an ordinary Rubik’s cube is in the restricted mode of movements rather then in the puzzle solving in the sense that when looking at a scrambled cube you already know exactly where each piece needs to go in the end (resembling vaguely the NP-complete problem). This lead me to conceive of a puzzle state in which the final position of the pieces is far from obvious (resembling alittle more the NP-Hard problem), while the piece movements are still highly restricted. I realize the analogy to computational complexity is not perfect but it served as a general guide to how to make a rubik’s cube much harder. Note that this cube has just as many states as an ordinary cube (519,024,039,293,878,272,000 unique states) times the fact that the center facet’s rotation is now relevant and can be changed adding a multiplication factor or around 25-20 or so. (i dont know exactly since i dont know the coupling of the center facet rotations).

If you would like a copy and try the challange, email me, paypal me 20$ and ill make a copy for you. Please indicate if you want it in a solved or scrambled state. Note that the true challenge is to obtain the scrambled cube and solve it from there! Be the first ever to solve it! If enough people email me i will try and get this produced in bigger batches and the price would go down for everyone, which would be awesome! Right now i’m building these by hand.

Note that i am not a mathematician and thus dont fully understand the group theory around cube states – i would be really interested in talking to some cube theory experts about the properties of this new puzzle !

Mike

Edit: A quick search found some other Maze-based cubes, but considerably less complex ones, since they dont allow bridges or underpasses as well as some other non-rubik’s related maze-like puzzles:

http://lohe.gmxhome.de/maze_cube.htm

http://cubesmith.com/mazes.htm

http://www.youtube.com/watch?v=abYO0Jm37Wo  (<– this guy even used the same style!)

Nice to see other people are thinking along the same lines (no pun intended)