Author |
Topic: Rubik's Cube without seeing (Read 5840 times) |
|
codpro880
Junior Member
Teachers strike, students hurt. English...
Gender: 
Posts: 89
|
 |
Rubik's Cube without seeing
« on: Jan 3rd, 2009, 6:35pm » |
 Quote  Modify
|
Is it possible to solve a rubik's cube without looking at it?
By not looking at it I mean without seeing it at all, hence you're blindfolded BEFORE you look at it (people can easily solve them blindfolded if they look at them first).
If it's not possible for a 3x3, could it be possible for a 2x2?
It it is possible, how?
If it's not, why?
|
|
 IP Logged |
You miss 100% of the shots you never take.
|
|
|
SMQ
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 2084
|
|
Re: Rubik's Cube without seeing
« Reply #1 on: Jan 3rd, 2009, 7:53pm » |
Quote Modify
|
With the addition of someone to tell you when you've solved it, I'm sure it's possible, but it's probably not practical. That is, I'm sure there is a sequence of moves which will cycle through all possible permutations of the cube, such that at some point in the sequence it must be solved. However, the number of permutations for the 3-cube is high enough that no human could cycle through all of them in a lifetime... I'm not sure about the 2-cube.
Without someone to tell you when you've solved it, no, it's not possible. Given two distinct starting configurations, performing the same sequence of moves on each will always give distinct ending configurations, so there is no "magic" sequence of moves, however long, which will end with the cube solved regardless of the starting configuration.
--SMQ
|
|
IP Logged |
--SMQ
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
|
|
Re: Rubik's Cube without seeing
« Reply #2 on: Jan 3rd, 2009, 8:09pm » |
Quote Modify
|
If the cube is solved on any given move, there's a unique possibility for the starting configuration. Since there are over 43 quintillion possible starting points, it is possible only in theory. One way is to make a list of all possible sequences of at most 22 moves. Then apply the first sequence, undo it, then apply the second, undo it, etc. Eventually the cube will be solved. But I'm sure there's a better way.
For a 2x2 cube there are "only" 3674160 possible configurations so at one move a second you're looking at a minimum of over 42 days.
Reversing things, you are trying to find the shortest sequence of moves such that if you make these moves simultaneously from all 24 solved positions you'll reach every position; I don't think you can identify rotations because the edge labels will be different. So it's similar to the metric traveling salesman problem for the Cayley graph. It's conjectured that Cayley graphs are Hamiltonian, but it's still an open problem whether this is true for the 3x3 Rubik's cube (source).
You might find the recently proved road coloring theorem interesting, even though it doesn't really apply here.
|
| « Last Edit: Jan 4th, 2009, 2:54pm by Eigenray » |
IP Logged |
|
|
|
Hippo
Uberpuzzler
Gender:
Posts: 919
|
|
Re: Rubik's Cube without seeing
« Reply #3 on: Jan 4th, 2009, 2:53am » |
Quote Modify
|
OK, without feedback you cannot be sure to end with solved cube.
If you can define several feedback types, it may became feasible.
(Feedback of type k faces are colored well, or edges/corners on k-faces are colored well will be very helpful).
Even the feedback a face is colored well is sufficient for 3 cube.
... You can find which one it is by careful turning edges on remaining faces. ...
|
|
IP Logged |
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7527
|
|
Re: Rubik's Cube without seeing
« Reply #4 on: Jan 4th, 2009, 8:31am » |
Quote Modify
|
on Jan 3rd, 2009, 6:35pm, codpro880 wrote:| By not looking at it I mean without seeing it at all, hence you're blindfolded BEFORE you look at it (people can easily solve them blindfolded if they look at them first). |
|
Being blindfolded before you look at it looks a bit contradictory.
And while some people can solve it blindfolded after looking at it, I wouldn't say people in general can do it easily.
OK, just nitpicking.
Well, I second SMQ that without any feedback, it is impossible to solve it. That is, if nobody even tells you whether it is solved or not, you cannot pick up a scrambled cube and return it solved. Unless you are really lucky.
@Eigenray: I don't see why you should consider 24 starting or ending positions. Considering that the centers don't move, the problem is to bringing the remaining 20 pieces to the right position. And for that there is only one solved position.
I wonder if there are "braille" cubes for the blind.
|
| « Last Edit: Jan 4th, 2009, 8:32am by Grimbal » |
IP Logged |
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
|
|
Re: Rubik's Cube without seeing
« Reply #5 on: Jan 4th, 2009, 2:48pm » |
Quote Modify
|
on Jan 4th, 2009, 8:31am, Grimbal wrote:| @Eigenray: I don't see why you should consider 24 starting or ending positions. Considering that the centers don't move, the problem is to bringing the remaining 20 pieces to the right position. And for that there is only one solved position. |
|
What I meant was that the graph has 24 components, and you don't know which one you're in. Say X' denotes rotating the cube X about some axis. If turning the top face clockwise takes you from X to Y, to get from X' to Y' you have to turn a different face. So the path you travel will be different depending on the initial rotation: the generators are permuted. But you're right that it doesn't matter, because it is still equivalent to finding a sequence of generators and inverses such that 1, x1, x2x1, x3x2x1, ... contains every group element, so it is just a metric TSP on the Cayley graph.
|
| « Last Edit: Jan 4th, 2009, 2:56pm by Eigenray » |
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print