# partly obscured Rubik's cube

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

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

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