# Does the Rubik's Cube in this painting have a solved state?

This is an image of a Rubik's Cube I found in the Men's Toilets during the first day of a big Scrabble tournament.

This position is impossible for the standard Rubik's Cube (White/Red/Blue opposite Yellow/Orange/Green respectively) for any number of reasons.

Is this position legal for at least one non-standard Rubik's Cube? (i.e. with different permutation of White/Red/Blue/Yellow/Orange/Green)?

• Don't be too serious in toilets ... (BTW I think it's totally possible, with Green/White, Red/Blue, Yellow/Orange as opposites.) Commented Mar 6, 2021 at 11:15
• Oh, I didn't know there is a standard of the ordering of the colors. Learn something new everyday! Commented Mar 6, 2021 at 11:26
• @StigHemmer No, that is not the case. Most (if not all) cubes by any manufacturer have the same color scheme. Or at least the companies don't care what Rubik's brand thinks. Commented Mar 9, 2021 at 4:47
• Remember when you could buy a set of stickers to cover up the ones on your unsolved cube? Commented Mar 9, 2021 at 16:51
• @StigHemmer I have an original genuine Rubik Cube from back in 1980 or so (I used to be quite good at it) and it has blue opposite white.
– SiHa
Commented Mar 9, 2021 at 18:17

## 2 Answers

Oo, this is a good one. Let's do some analysis:

We can see the centres of three sides, and the relative positions of the centres cannot be changed, so we know that when/if this cube is solved, the blue side will be adjacent to green and orange.

We can also see a blue-white edge piece, and a blue-yellow edge piece as well. This means that as long as the cube has a solved state,

the red side must be the side opposite blue.

Then, we can take note of the following facts about solved Rubik's cubes in general:

1. There are always exactly two corner pieces between any two adjacent colours
2. Out of these pieces, one has the two colours next to each other in a clockwise order, the other piece has them in anticlockwise order

It follows from these two facts that if we

1. see two sides of a corner piece, and
2. we know where those two colours are on the cube,

then we can uniquely place that corner piece on the solved cube.

Doing so, and remembering what we learned about red above, we notice that

both of the marked pieces belong in the same place, the top right corner of the orange side.

Because of this,

this cube cannot possibly have a solved state.

• Graffiti and a cheater! Commented Mar 6, 2021 at 23:56
• How annoying. :-( Commented Mar 8, 2021 at 14:29

I'm new here, and I might be wrong, but...

We can see the centers of three adjacent sides, and we can see a corner where three other adjacent sides meet, which happen to be the three we can't see directly. If we turned the cube around, we'd see red, white, and yellow in the same configuration.

So we can identify all the sides: White is opposite to blue, red is opposite to green, and yellow is opposite to orange.

Anything that contradicts this is proof that the cube is impossible—like the edge piece where white borders blue.

Can it really be that easy?

• The white-yellow-red corner can be rotated, though. So in theory, blue could be opposite to white, yellow or red. You've just proved that white cannot be opposite to blue, but it's not enough to prove the impossibility to solve. Commented Mar 7, 2021 at 8:02
• Ah, I knew there had to be a catch. Don't mind me, then. Commented Mar 7, 2021 at 8:03