Reading about people who solve the Rubik’s cube quickly got me wondering. If you remove the stickers and replace them without turning the cube, is it possible to arrange the stickers such that it becomes impossible to solve the cube?
Yes. All you have to do is remove and replace one corner piece in the wrong configuration and the cube will be impossible to solve.
Of course. Just put three stickers of the same color on one of the corner pieces.
Yes, you could make an impossible to solve cube. Start with a solved cube. Take one of the side pieces that only has two stickers and swap the stickers. You can’t flip that piece over without making some other piece out of place.
Sure, if you remove the stickers and replace them at random you can make an unsolvable pattern. Suppose you put two stickers of the same color on a side piece. They cannot possibly both face the same side.
Same if you take it apart and flip a side piece around or rotate a corner piece.
One kind of impossible is where you create an invalid piece, such as making one piece have all the same color stickers. That piece is invalid, so it could never yield a correct cube.
Another kind of impossible is where you twist only one piece. When solving a cube, you can’t make a change to just one piece. At least one other piece will also change. So if you pop out a piece, twist it, and replace it, you won’t be able to solve the cube. When you get that piece back to the correct orientation, you will have at least one other piece which has twisted.
By the way, how many different classes of non-invalid cube configurations are there?
Answering my own question - this article Counting the Permutations of the Rubik’s Cube, Scott Vaughen, claims that there are 12 such classes - any random rebuild of a cube from valid pieces has a 1/12 chance of being solvable.
I tried to solve a cube a teacher had in her classroom once, and ended up with one side piece turned around. I told her that meant that at some point someone had taken in apart and put it back together, or else moved the stickers. She refused to believe me. Made me crazy. She thought that because I knew how to solve the cube, I must be able to “fix” it without disassembling it or peeling off the stickers, and refused to let me do that to it.
As I understand it, Erno Rubik designed the cube to teach math students a complex set concept involving “moieties.” A properly finished cube represents one moiety, and no matter how you scramble it it’s still in the same moiety. If you physically change one pair of face colors, the cube represents a different moiety altogether, and cannot be “solved” back into the first moiety’s condition. (Someone who actually knows math at this level will certainly come along and fix my sloppy explanation…)
Then some student said, “Hey, this is fun to play with” or however you say it in Hungarian, and the rest is history.
(Proud to own a Cube from the first shipment to reach the US… the toy store manager had never heard of them and the owner had to speak up and say, “Yeah, those are in a case on the loading dock…”)
During my very brief Rubik’s Cube obsession a while ago, I had an odd experience. At a certain point, I stopped seeing a scrambled cube as different from a solved cube. It’s the same thing, just in different states of… I dunno, twistiness. It was like one of those odd mental leaps when you suddenly see the hidden penis is the painting of a fish, and then you can’t unsee it.
It’s hard to explain. It left me a bit dizzy. And, yeah, I know, you’ll have some of what I’m smoking.
Not sure if this has anything to do with what you’re talking about, but it feels like it might.
No, there’s some logic in that. But I’m curious: Did you also see a malassembled cube as the “same thing”, or was that different? How about a completely disassembled pile of cubelets?
No, a malassamled cube is a different thing. A pile of cubelets is just a pile of cubelets.
But I started seeing a scrambled cube as already solved, in a way. The physical twisting into place is just paperwork. The cube is just as solved when scrambled as when solved. Does that make any sense? I mean, in a logical way. Or, maybe logical isn’t the right word, as it was an intuitive kind of realization. I haven’t really applied any logic to it.
Very obvious - in a solved Rubik’s cube, each colour has one colour that is “opposite” face; there are no adjacent pieces (Quick Google, if the picture is right - green vs blue, Red vs orange, yellow vs white) Therefore any configuration where those are touching edges or corners is obviously because someone played with the colour stickers.
You got the colors right for an official Rubik’s brand licensed cube. Many of the knockoffs used different arrangements of the colors in an attempt to defeat the copyright on the original design.
The same sort of questions being discussed here also apply to other puzzles much older and much simpler than Rubik’s Cube.
Consider the good old “15 puzzle”
By sliding the pieces around, and you arrange the numbers in any permutation?
No. Of all the possible permutations, you can only get 50% of them. If you remove the pieces from the box and re-insert them in a random order, there is a 50% chance that you will be able to arrange them in numerical order.
If by chance you can’t do that, then the other half of all possible permutations are possible instead. (IIRC, this would include the mirror images of all the other permutations that you could have gotten.)
I have one of these with the numbers shown in normal number form and also using the Hebrew letters. (Nobody uses Hebrew letters for numbers, at least not in modern times, except for some kinds of decorative purposes, like we use Roman numerals.) The puzzle can be arranged in normal left-to-right numerical order. But if you try to arrange them in Hebrew-style right-to-left numerical order, you can’t.
Another unsolvable change is to take any color from a non-center square and swap its place with a center square of a different color. When you do that you’ll have two center squares of the same color. Being as all six sides have to have a center square of a different color, you can’t solve it. And the center squares cannot be moved, so you’ll never be able to fix the problem.
Since OP’s question has been answered, may I ask a different question? Roughly what percentage of Rubik’s cube owners can solve their cube? Are there gazillions of cubes out there that will never have the pleasure of being restored to their pristine state? Is there a website that allow an ordinary person to “solve” the cube without thought?
One reason I’m asking is that I’ve wondered whether a website titled
Restore your Rubik’s Cube to its “solved” state
would get a lot of hits? There are downloadable solvers on the 'Net, but a website with a step-by-step solution for any initial input configuration would be a fun and challenging project.
Sure, http://rubiksolve.com/ or Online Rubik's Cube Solver App
Like most things online, all the cool shit’s been done already =/
There are several web sites with step-by-step instructions. It’s not that hard if you practice the patterns a little. I used to do it while watching TV. Now I can usually solve one in under two minutes by muscle memory. The world champs can do it in seconds.
Yes, there are websites that will allow an ordinary person to do that. As I mentioned in the other cube thread, once you know a series of steps for solving your cube, doing so involves literally zero thought.
What percentage of cube owners can solve their cube? 100%, if they look up a YouTube tutorial.
Creating an solution from scratch? Probably quite difficult indeed. Executing a solution, if you know an algorithm (which in a cube context is just a fancy word for “series of steps”)? Beyond easy.
I think this has to do with what I tried to get at earlier, about the cube already being “solved” in a scrambled state. Solving it when you know an algorithm is less like problem solving and more like rearranging furniture. All the relationships between the pieces are there no matter how scrambled the cube is, you’re just pushing things around a bit.
The algorithm I learned (not a super-quick one, but I can do a cube in, I guess, a couple of minutes) will solve* any scramble using the same steps in the same order* (well, except when I get lucky, and I’m able to skip steps because of how the chips fall). There is no more or less scrambled, in a sense, there’s just the cube, and some ways of moving things around.
For a super-quick solve, I suppose you want to pick the best moves from a variety of methods, and be able to choose the optimal ones on the fly. But it’s the same thing in principle.