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(If you are not interested in the backstory, you can skip right to the picture.)

At Most Complicated Illegal Partial State of Rubik’s Cube, user @happystar asks a very interesting question about cubers' favourite pastime: instead of finding sloppily created pictures of cubes, and then proving that the depicted cube cannot possibly be solved, what happens if you intentionally colour in a cube in a way that's illegal in the subtlest way possible?

Here's my go at the task:
(Cross-posted as a separate question, because the original question is a "request for puzzles")

enter image description here

Here are the rules:

  • This cube might have a non-standard colour pattern. White being next to yellow is not an error in itself.
  • Apart from the colour pattern seen in the picture, nothing else is known of the cube
  • Yet, this cube cannot possibly be solved so that each side only has one colour

and most importantly,

  • It is possible to rigorously prove it.

So now your task, should you choose to accept it, is to find a reason why any cube that looks like this cannot ever be solved.

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3 Answers 3

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We can see

all four yellow edge stickers and also all four orange stickers. But there is no orange-yellow pair.

Therefore

Yellow must be opposite Orange.

But the centre pieces tell us that

Yellow cannot be opposite Orange,

Which is a contradiction. Therefore this cube cannot be solved.

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    $\begingroup$ Very nice! This argument uses only 10 stickers. $\endgroup$
    – Jaap Scherphuis
    Commented Mar 22, 2021 at 20:16
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    $\begingroup$ Aargh! Turns out it's really difficult to not create unintentional proofs :-) $\endgroup$
    – Bass
    Commented Mar 22, 2021 at 20:20
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    $\begingroup$ @Bass I'd say it's related to how the "God's Number" for Rubik's cubes is 20 - even the most complicated of shuffles can be solved relatively simply if you happen to know how. What you're trying to do is similar to trying to find one of those rare shuffles that require 20 moves. $\endgroup$
    – Rob Watts
    Commented Mar 22, 2021 at 20:45
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    $\begingroup$ @RobWatts I'm afraid you may be correct. Having tried to kill this proof for a while now, it seems the existence of this proof is an unavoidable consequence of being able to do the intended proof. Not giving up quite yet though, I'll give it another go in the morning, and there'll of course be a bounty coming this way too, whether I manage to salvage the original idea or not. That should teach me never to post a puzzle after only two hours of quality control :-) $\endgroup$
    – Bass
    Commented Mar 22, 2021 at 21:02
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    $\begingroup$ Hmm. Couldn't be saved. The original idea was to show that all the green and blue edge pieces just barely won't fit on the hidden edges, but any construction that allows that deduction will, by some interesting duality-of-proofs property of the cube, also cause this proof to work. $\endgroup$
    – Bass
    Commented Mar 24, 2021 at 12:39
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Here's one way to show the cube cannot be solved:

Suppose the bottom face is white. Then the corner piece with yellow and white visible is in the right place, and is the fourth blue corner. That means the corner piece with green and white belongs in the hidden corner and the right-side hidden face is green. That leaves the left-side hidden face to be red, but that would put the corner piece with red and green visible on it also in the hidden corner. This means that the bottom face cannot be white.

Because we know that white is next to yellow, we now know that the right-side hidden face must be white. With red next to orange, red must be the left-side hidden face and green must be the bottom face. However we now have two blocks that both need to be in the lower-right visible corner - the corner with yellow and white on it and the corner with green and white on it.

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    $\begingroup$ If this solution works (I'm afraid it might), I'll have to fire my QA department (into the sun), because this is not the intended solution :-) $\endgroup$
    – Bass
    Commented Mar 22, 2021 at 19:38
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    $\begingroup$ Yeah, cannot find any holes in that logic. Damn. The intended solution goes a bit deeper than this, so I'm tempted to drop you a big fat bug bounty, and turn the blue corner at the very top into an orange one. (I think that would just barely break your beautiful chain of logic, because the blue corners aren't fixed anymore.) Would this be an acceptable arrangement for you? $\endgroup$
    – Bass
    Commented Mar 22, 2021 at 19:58
  • $\begingroup$ @Bass I'd be fine with that, but you could also just ask it as another question. $\endgroup$
    – Rob Watts
    Commented Mar 22, 2021 at 20:00
  • $\begingroup$ @Bass using the scoring from the question that inspired this, I think this solution considers 16 stickers. How does that compare to the solution you had in mind? $\endgroup$
    – Rob Watts
    Commented Mar 22, 2021 at 20:10
  • $\begingroup$ the intended solution has ~24, and another bug has just been found :-) $\endgroup$
    – Bass
    Commented Mar 22, 2021 at 20:19
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The color layout of the cube is in such a way that the final four corners can not be solved, much like if you try to do a pattern by changing each faces bordering colors like making yellow have a white border and white with a yellow when they are opposite and doing that for every face you can not actually solve because when the try the last algorithm either 2 or none of the corners will be correct and while using the traditional algorithm you need to get only one corner in the correct place because it rotates the other three so with two in the correct spot you can never have all four go to the correct spot. The question also clearly states "explain why it's unsolvable" meaning there is no solvable answer. This is why I believe it is unsolvable. No matter what face is what color on the faces you can't see. The last corners will never be correct, so you'll have a completed cube aside from 2 corners.

Take for example a normal cube layout. If you have white opposite to yellow, red opposite to orange, and blue opposite to green. When trying to make each face have an opposite color border like yellow with a white center and white with a yellow center. When trying to aline the cube in such a way the last four corners will not line up. Say you use the og beginner method algorithms to solve and start with white making your cross and such but trying to keep in mind the pattern you are trying to create. When you get to the last algorithm of setting the corners in place before you turn them the correct way after making the cross on the yellow face. You need to do the algorithm with only one corner in the correct spot, meaning the right three colors on the right three faces, so the other three faces that are in the wrong spots will rotate untill they are. The issue is since you are trying to flip the cube essentially, your corner pieces end up not lining up because they are flipped. So you will always have two corers that are in the correct spot and two that are not. So you always end up just rotating but never getting all four.

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    $\begingroup$ The question says "This cube might have a non-standard colour pattern. White being next to yellow is not an error in itself." - so saying that the color pattern can't be standard isn't an answer $\endgroup$
    – bobble
    Commented Mar 23, 2021 at 23:31
  • $\begingroup$ It's saying that yellow next to white is not an error because there are two different types of cubes. The error is there is a red-orange piece. At least in my interpretation of the question with the rules. What I understand is it's telling you white next to yellow is not incorrect because the Japanese style cube does have white next to yellow so this would lead me to believe this is a Japanese style cube which still can't have red next to orange and since the question only address yellow to white and never states that it can be any color layout. The standard is western style $\endgroup$
    – Matthew Fleury
    Commented Mar 23, 2021 at 23:47
  • $\begingroup$ So the non-standard part is referring to the Japanese style cube. To me the only two reasons the cube would be unsolvable would be an actual error in the layout of the cube like I explained or with all cubes being known they could never actually line up, similar to when you would try to do a pattern like changing the color of each face. When you get to the end you can't fully solve because when doing one of the final algorithms to put the final faces' corner pieces in the correct spot, they won't go to the right place $\endgroup$
    – Matthew Fleury
    Commented Mar 23, 2021 at 23:53
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    $\begingroup$ I think "non-standard color pattern" includes all layouts not made into production. So the opposite of red can be any of white, blue, orange, green, or yellow. Maybe a clarification by OP is needed here though. $\endgroup$
    – Bubbler
    Commented Mar 23, 2021 at 23:57
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    $\begingroup$ The first part is incorrect for the context of this question, so you should definitely erase it - it's easy for someone to look at it, think that you've completely misunderstood the question, and downvote your answer without finishing reading it. As for the second part of your answer, it's not very clear what it is about the cube as shown that makes it so you won't be able to line up the last four corners. $\endgroup$
    – Rob Watts
    Commented Mar 24, 2021 at 15:52

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