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Rubik's Cube Comeback
aheath writes "Today's Boston Globe has an interesting article on the revived popularity of the Rubik's Cube. The article mentions that Winning Moves Games 'hopes to capitalize on the renaissance of the original Rubik's Cube in the next several months by rereleasing a version of the supersize 'Rubik's Revenge,' a 4-by-4 cube with 16 squares per side.' You can compare your best Rubik's Cube solution speed to the world champion's record. If the manual solution method doesn't appeal to you, you can always use the Lego Robotics automated method." I remember having a cube that had letters on each sticker instead of colors, so that the solved product spelled words across each face. That thing got me through a lot of childhood car rides.
I wrote a program in Qbasic to solve it. Is that cheating?
I used to do the cube in under a minute regularly. My best averages were around 50 seconds. Nowhere close to the world champion of course, but still more than enough to impress your friends
I used the corners first algorithm. Its not what the fastest cubers use, but its much easier to learn, because its more "natural". It also has a shorter average number of moves (under 60, IIRC). Though I say "easier", it takes several months!
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Among corner pieces, all positions are possible (reachable from a given configuration), but only one out of 3 orientations are possible.
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Among edge pieces, only half the positions are possible (diagonally opposite edge pieces can't be switched) and only half the orientations are possible.
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Among center pieces all positions are possible.
So that means only 1/12 of the total configurations are possible!So if you take the cube apart and put it back randomly, chances are you'll end up with an insoluble position :)
This also allows us to count the number of possible configurations.
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Corner positions: 8! (8 factorial)
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Corner orientations: 3^8
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Edge positions: 12!
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Edge orientations: 2^12
Total = 8! 3^8 12! 2^12 / 12 = 43252003274489856000 = 4.3 * 10^19Now a math teaser:
The set of operations (operation = sequence of moves) you can perform on a cube forms a group. Two group elements x, y are said to commute if xy = yx (in this case, 2 operations commute if it doesn't matter in which order you perform them). Question: Find an operation that commutes with every operation.
Rot13d spoiler follows.
Syvccvat nyy gur rqtr cvrprf.
In fact this is the only answer. If you know group theory, this is the only group element other than the identity which lies in the center of the cube group.
Those truly interested in the importance of the Rubik's cube should dig into the archives of Scientific American (if it's not online as a subscriber, try a library). There was a very good, in-depth (cover) story published as the popularity was beginning to wane. For the mathematically inclined, the Cube delves heavily into Group Theory and the article points this out. It also refers to "God's Algorithm", which means if you were to take a cube of any arrangement, hand it to a machine programmed with the algorthim, it would solve the problem with the same twists & turns with no decision making - just the same twists & turns. If you stop to ponder this - a universal algorithm with the same steps applied to any arrangement - this is pretty incredible! IIRC, the number of steps is on the order of 10^60 steps. Back to the mathematics. The arrangement of the tiles is important: at least if any of the pieces are pulled off of the framework and put back on (and this is not the same as peeling the colored paper off). When I was in college, there was a guy named Bert (seriously - his middle name could have been PointDexter) who bragged he could solve a cube in any arrangement in thirty seconds. Someone came to my room and asked if there was any way to shut this guy down. I grabbed a cube, took a corner cube off, gave it a 1/3 twist, and snapped it back into the framework. Scramble the puzzle a bit more, and down we went to visit Bert. Thirty seconds, forty seconds, fifty seconds. At three minutes, Bert exclaimed, "Someone's been tinkering with this cube!" (and he was pretty worked up by now) Obviously, this was pretty funny because his "any cube in thirty seconds" rep was gone and the fact he'd tried harder & harder as time progressed over the goal was gone. By 1982 or 1983, you could go to Target and buy Cubes for $0.50 (closeout). For those with a passing knowledge of the Cube wanting to delve into the history, one of the good places to go is eBay. There was a Saturday morning cartoon (copies sell on eBay), earrings, keychains, 4x4x4 cubes, 5x5x5 cubes, etc. I'm sure there are plenty of web pages and other online resources dedicated to the Cube as well (remember, there was no web when the Cube was hot).
The way I learned how to do it was this:
I would choose an arbitrary side to be the "top", and another side to be the "front", of the cube, and I invented a notation for describing different ways of turning the cube: ie, left side clockwise 90, bottom side 180, right side counterclockwise 90, and so on.
Then I would take an already solved cube and start making different turns with it, writing down each move I made as I went. After some number of moves (say 8 or so), I would analyze the entire cube and see which pieces had moved to which positions. I wrote that stuff all down, and then reversed the moves to start with a freshly solved cube again. I kept doing this for a while until I had enough moves that could shuffle the positions of small numbers of pieces, while leaving the rest of the pieces where they were. It took me a whole weekend to come up with enough useful moves to solve any cube, and the actual number of move combinations I ended up needed filled about 3 pages.
But using that method I was able to solve any cube, no matter how mixed up it got. 4 to 5 minutes, every time. I'm no speed demon, but hey... at least I can do it.
File under 'M' for 'Manic ranting'
Ahhh, the Rubik's Cube. Video games nowadays may entertain for hours and stimulate the mind, the cube did in the early 80's though. I was a freshman in HS in '82 and that and the next year were spent perfecting my solutions for it instead of study :) I averaged under a minute with a best of :24 once. I was so excited, the :24 time was when the standing world record was I think :24. This was just a week or so before the competition that was televised from Budapest where someone did it in :22.
I also scrambled it and worked it the exact same way over and over enough times that I could do it blindfolded or behind my back. It was like 109 total moves to solve and they were standard moves for my method, just memorized. Blew people's minds to see it done...
18 isn't rare. I've had a Celeron solving random positions for 154 days now (using God's algorithm); here is the solution length distribution (half-turn metric):
15 27
16 400
17 4343
18 11020
19 623
So 18 is by far the most common as well as being the median.