Sunday, January 11, 2015

What is the opposite of solved?

My holiday cubing was noteworthy this year. I spent some time working on solving large cubes, although I haven't managed to migrate to a modern solution yet. It was nice to knock three minutes off of my 5x5x5 solve time, although it's not really that noteworthy since I'm just down to 10 minutes from 13. I couldn't really do much with my 4x4x4 time, since I'm still using a Rubik's Revenge from the 80's and the stickers slide around a lot and I'm worried about losing one. Both of those were just warmups for the VCube 7x7x7 that I got this Christmas, but my only solve time for that was nearly three-quarters of an hour, and I'm nervous about the stickers on that as well as they're very small and I've managed to rough one of them up already. I'm not quite ready for anything bigger until I refine my solution, since nobody wants to sit around for hours while I solve a cube. (At least I'm not working on a 17x17x17.)

As a result, the 3x3x3 still has to be the go-to cube for going out in public and talking about cubing. It's still the most mechanically robust design, and it's what people are familiar with anyway. So, that's what I'm always carrying around with me. Last Friday night at a small gathering of friends, I was asked about something unusual - something quite the opposite of what's usually done with the cube. I was asked if it was possible to get all six colors on each of the six sides. I was pretty sure that it was, but I didn't have a ready answer because I didn't seem to be in the right frame of mind. (I had fumbled a question about the meaning of the Apple logo despite having just listened to an episode of The Nerdist with Benedict Cumberbatch the week before.) I knew that there should be several possible positions that satisfied these conditions, but it worried me a little because in my head I started with the most extreme case - throwing stickers on a blank cube and then wondering if it meets conditions of solvability afterwards.

If you take a cube apart and reassemble it at random, you have a one in twelve chance of having a solvable cube. If you start from a blank cube and apply stickers at random, you have a much lower chance of having a solvable cube. If you put the six centers on first, assigning random stickers, you have less than a 1 in 48 chance of having a solvable cube since all six have to be different. It just gets worse from there. Clearly, this was the wrong approach for the problem. So, as I walked back home, I imagined what you could do with conventional patterns, hoping that I could figure out something that wasn't too bad, but nothing came to mind. I could guarantee three or four easily, but once I put more one place, I would always have a side that came up short.

When I got home, I did a quick internet search, thinking that this was already a solved problem and I was just going to reinvent the wheel. While I found a couple people asking the question rather easily, I did not find the answer there. It was hard to find people that even understood the question, since I don't always hold out a lot of hope for Yahoo Answers on questions of this type. But, in the minute or two that I looked, I also didn't find anything that overtly stated that it was an impossible position, so I decided that I would stop checking the internet and leave it as a problem for Saturday

Often when presented with a problem like this, I find myself working on it even when I'm not working on it. As I put my head down to go to sleep, I have this overwhelming thought - "Superflip!" Now, this is not the entire solution, but it puts you a lot closer without much difficulty. The position referred to as the superflip is the state of the Rubik's cube where all of the pieces are in their correct places, the corners are all correctly oriented, but none of the edges are correctly oriented.

So, with the superflip, you automatically have 5 different colors on each side - the corners and the center are the regular color of the face, and the four edges are the colors of the four adjacent faces. The only thing left to do are things that manipulate centers or corners that mix colors from opposite faces. If you move centers around, you can either move six centers to adjacent faces, which we don't need, or you can move four centers to opposite faces, which gets us 2/3 of the way there.

The cube on the left is the superflip cube from the first picture with the centers moved, the cube on the right is a cube with only the same four centers manipulated.

So, I am left with only two sides that need a piece from the opposite side, and no available tool to move centers around. So, what can I use to manipulate some corner pieces without disturbing the edges? The first thought I had was one of the zigzag patterns.

So, once you do that, then the finished cube ends up like this:

So, now we have all six different colors on all six of the sides. If you need the cheat code to get here, try using B' D2 L2 B L' R' D B F' L' F2 R' B2 R' D' U' F2 R2.

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