Earlier today I heard a passing mention of a Rubik’s Cube and my mind drifted to cube-solving robots, and I started thinking about how many views would be needed to be sure of the configuration of every side, given a complete understanding of the physical structure of the cube. For instance, looking at just one side would tell you the color of the center square opposite it, since those are in a fixed relationship. Looking additionally at an adjacent side would tell you the color of the center square for every side. But what if you know 5 sides? You will then know the center square for the final side. You will also (I think?) know the color for the 4 remaining corner pieces. But will you always know the color of each of the 4 remaining middle pieces based on the known configuration of stickers on the pieces and how they can move, or is there more than one possible arrangement?
There may be more than one possible arrangement. E.g, start with a solved cube and flip all four middle pieces on the top layer. If you can’t see the top layer, then you won’t be able to tell the difference if you further mix around the middle pieces on the top layer.
Experimentally speaking, 3 sides seems to be enough, since that’s what the cube-solving app I had used. Give it a photo or photos with three visible sides, and it would produce the sequence of moves to solve the entire cube. (I think all three sides had to share a corner, but I don’t recall for sure.)
Try it with a cube prepared as I have suggested. What sequence of moves does it produce?
Actually, there is no need to bother trying, since if you start like I said and flip a couple of adjacent edge pieces on one layer, looking at the cube from the opposite corner it is still indistinguishable from a solved cube, so there is no way that app could work.
Can you actually DO that, though, using only valid moves and not pulling the cube apart? Swap two of them and leave everything else exactly untouched? There are numerous potential configurations that cannot actually happen using real moves.
Because while it was a couple years ago, and I didn’t do exhaustive clinical trials, the app DID work for the cases I tried.
I must be misunderstanding you. If you can see all 5 sides except the top, and then you flip a middle piece on the top layer, it changes a piece on a side as well. So you would see that one of the 5 visible sides is no longer solved.
Googlebing shows a reddit and a youtube video claiming to be able to create a cube that’s solved on three shared-corner sides but scrambled on the other three – along with tons of comments from people claiming it didn’t work for them, and others claiming this can only be done by disassembly or sticker movement.
I don’t own a working cube any more, (or enough interest in the outcome) so I can’t try the proposed move sequences, but it’s possible there’s a case that can’t be solved from just three sides. (Which the app from a few years back would not be able to solve, presumedly). However, I’m convinced that given any four sides, the state of the cube can be completely determined.
Get your hands on a cube. Let’s start with the following configurations:
Cube A: RBLFUF’UL’B’R’F’U’FU’
Cube B: F’U’L’UB’U’U’BLUFU’RU’U’R’
Cube C: BUL’BLB’U’B’R’F’U’FUR
I claim that Cube A has three unscrambled faces sharing a corner, so it you look at it with that corner right in front of you, you can’t tell it isn’t a solved cube.
Next, I claim that Cubes B and C have five sides the same. But not the sixth!
ETA it is impossible to swap two cubelets, or to flip a single edge. But you can flip two edges (without swapping their positions); see Cube A.
To clarify: R means turn the right face 90º clockwise, and R’ means 90º counterclockwise, and likewise for L meaning left. I’m guessing that F means front, but I don’t know if B is back or bottom, and no idea what U is.
sorry:
F = front
B = back (opposite the front)
U = top (up)
All turns are 90º clockwise unless indicated with a ', in which case they are anti-clockwise.
If anyone wants to see these but doesn’t have a cube handy, check out Rubik’s Cube Solver. You can enter the above sequences of moves into the applet there and see them executed (though you’ll have to put a space between each move or the applet will get confused.)
Reported, because the post by puzzleSolver changed the URL in the quoted post