Monday, December 17, 2012

Cubing without even trying to - Part 3 or, No, I don't get the picture.

I am so glad that they have not made a Jackson Pollock picture cube, but perhaps it would have been easier than the Minnie/Daisy picture cube that I encountered a couple of weeks ago. I remember as a child that there was some puzzle billed as "The World's Hardest Jigsaw Puzzle" and it had Pollock's "Convergence" as the subject.

To review the problem that I had with Minnie and Daisy, I need to review a bit of the basic mechanics of the 3x3x3 Rubik's cube. The Rubik's cube, for the sake of our discussion, has twenty-one pieces. There are 12 edge pieces with two stickers each, 8 corner pieces with three stickers each, and the center spindle mechanism with six stickers.

Let's say that we have a particularly difficult picture cube, but you recognize one of the pictures enough to know which nine of the 54 stickers go on one of the faces. Now, if you know what stickers go on one certain face, and can move them all there, then you have one center, four edges, and four corner pieces all on the correct face. If for some reason they aren't in the right place relative to each other, there are only a few ways it can be wrong, and it can be a simple matter to fix if you know what to look for. Taking the location of one corner and the center piece as a starting point, there are 3!*4! ways the remainder of the pieces can be placed, and on a picture cube there can be 4 different orientations of the center piece, so lets say that there are 576 possible permutations of the pieces on the one face. If you can correctly move just one more piece into place, you cut that number to at least 192, and with each successive piece you reduce the number of possibilities further and further. Even if you had to do it completely by trial and error to see what the side pictures are, we're not talking about so many different cases that you couldn't get it done in a few hours. The great part about this is that once that's worked out, you now have one third of the picture for the four adjacent sides put together also. (For those of you that have been away from a math class too long, the exclamation point represents the operation called 'factorial'. n!, or n factorial, is the product of the integers 1,2,...,n-1,n. For example, 3! is 3*2*1=6, and 4! is 4*3*2*1=24.)

Now, if you can't discern which group of nine stickers comprise any face of the cube, the challenge is orders of magnitude more difficult. If you select a particular corner to start with, the only stickers you have eliminated as possible other pieces for that face are the other two stickers on the same corner piece. Considering only corner pieces, you have 1330 possible sets of corners to try to go with the one you start with. Factor in the edge pieces and you have 10,626 possible groups of four edges to try out, not to mention having to try all six centers. Multiplying all that out means that just to get a single face on a hard-to-decipher cube means you would have to try almost 85 million possibilities if you took one corner piece as a starting point. This is far too many to do by trial and error. If you could check one configuration each second, it would take more than 2-1/2 years to check them all.

These previous two scenarios also assume that the cube has not been tampered with. If the stickers have been moved around, then your best bet would be to just consider any possible grouping of 9 stickers out of 54 to see if they formed a picture. Now you're at 531 million or so possibilities to check, just to get one face of the cube figured out.

So, in case I haven't made this clear enough, let me summarize. If you can't figure out any of the pictures on a Rubik's picture cube from a scrambled state, you probably don't have enough free time to sort it out by accident.

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