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?

Is this your cube? http://www.redbubble.com/people/yolanda/works/6358896rubikscubeatmaroubra 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/6351863surfingfunatmaroubrabeach for the alternate view showing the corner. 





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 redblue edge cubelet, and the two corner redblue cubelets must not both have a redblueX 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 redX 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 

