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I just came back from a beach which features a large Rubik's cube (2m high). The base of the cube is not visible and the top is not coloured. The four vertical sides are each divided $3\times 3$ into coloured squares as they should be. I was idly wondering: how can I tell if the patterns appearing on those four sides are actually possible for a real cube? Clearly there are necessary conditions like having at most 6 different colours and at most 9 squares of each colour, but is that sufficient? Can the legal configurations be characterized?

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Very nice question! But clearly the criterion you mention is not sufficient, since the role of center, edge and corner pieces are relevant. You can't have two centers of the same color; can't have more than four edge pieces or more than four corner pieces of the same color; can't have two sides of the same edge or corner piece the same color. –  Joel David Hamkins Oct 10 '12 at 3:30
    
Assuming all the constraints on having the right number of squares of the right type of the right colour, there is the question of whether the configuration is solvable. Alternatively, you could ask how many 'legal' colourings the obscured/uncoloured squares have, aside from the obvious choice of how to colour the two blank centre squares. –  David Roberts Oct 10 '12 at 3:44

3 Answers 3

Is this your cube? http://www.redbubble.com/people/yolanda/works/6358896-rubiks-cube-at-maroubra If so, it is not solvable. The dark blue (grey?) and white centred faces are opposite each other, but there is a corner with both colours.

Edit: the linked site's interface changed slightly, so now the two images are on different pages. See here http://www.redbubble.com/people/yolanda/works/6351863-surfing-fun-at-maroubra-beach for the alternate view showing the corner.

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Yes, that's it (though it looked a lot cleaner today than in the photo). Pity it isn't solvable; it would have been easy for them to use a real one as a model. –  Brendan McKay Oct 10 '12 at 6:32
    
New question: $\:$ What's the smallest number of face colors that could be $\hspace{1.8 in}$ changed to get a solvable cube? $\;\;$ –  Ricky Demer Oct 10 '12 at 7:55

You might be interested in this:

How to tell if a Rubik's cube is solvable

and Henning's answer.

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Note that this is different from taking apart and reassembling an actual cube, since the real cube is properly coloured in some configuration and has just those 12 different arrangements. The sculpture is more like a repainted cube, which may still be unsolvable even if it satisfies the corner/edge/permutation parity - just take a solved cube and repaint two of the small faces. So there should be more configurations to discard as impossible - like the one at Maroubra. –  Zack Wolske Oct 10 '12 at 6:33
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@Zack: It's easy to first check if you have the right pieces based on the center squares. Then you can apply the test in the link. –  Sam Hopkins Oct 10 '12 at 8:29

Having two opposite faces obscured leaves a lot of room for filling in. The primary criterion after making sure the number of facelets (squares) does not violate cardinality constraints is to check that adjacent visible faces of the same cubelet also do not violate cardinality and chirality constraints. To wit, one should not have more than one red-blue edge cubelet, and the two corner red-blue cubelets must not both have a red-blue-X orientation going clockwise when viewed from the appropriate angle. Beyond that, it gets harder depending on what choices are made to color in the missing squares. If you can partially unscramble the cube and there are two red-X edge cubes available, say, then if other constraints are filled, almost any parity conflict with edge cubes can be resolved by coloring in the appropriate colors. The coloring of the corner cubes is an independent problem; you might look at the literature for the 2x2x2 problem, which should be computationally tractable.

Gerhard "Not A Rubik Cubic Expert" Paseman, 2012.10.09

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Actually, there is a little more to worry about. The coloring of the edge cubes must be consistent with the coloring of the corner cubes, e.g. having rb, by, and yr edge cubes implies an rby corner cube, for example. So certain triples of edge cubes must also be considered. I suspect there will be no procedure that a human could use to test mentally a configuraion. A number of small tests can detect some invalid configurations. Gerhard "Not Looking That Mentally Tractable" Paseman, 2012.10.09 –  Gerhard Paseman Oct 10 '12 at 4:43

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