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jstone Inner circle Someday I'll have 1473 Posts |
Hey fellow cubers...
Those who can solve the 4x4x4 are familiar with the "parity" problem that occasionally occurs when solving the cube. I know the algorithms to solve the parity, but I'm trying to understand what causes it in the first place. Does anyone know, specifically, what is happening that creates the parity? Thanks in advance. |
Philemon Vanderbeck Inner circle Seattle, WA 4694 Posts |
I suspect it has something to do with the 2-color edge "cubies." Since there are two identical ones for each edge, in one configuration, the cube is solvable, and if they're switched around, then the cube becomes unsolvable. Also, probably as long as there's an even number of "switched" pairs, then the cube remains solvable (hence, the "parity").
Of course, I'm just guessing here...
Professor Philemon Vanderbeck
That Creepy Magician "I use my sixth sense to create the illusion of possessing the other five." |
Scott Cram Inner circle 2678 Posts |
That's basically it, Philemon.
The same parity trick was taken advantage of by Sam Loyd when he put out a 15 puzzle with the 15 and 14 switched, and then offered money to anyone who could demonstrate a solution that ended up with the puzzle in 1-15 order. Because there was only one (an odd number) swap, the puzzle was impossible to solve. It also helped when he put a similar puzzle in which the pieces, when received, said: RATE YOUR MIND PLA and you had to solve it so that it said RATE YOUR MIND PAL You can try it here. At first, this seems just as impossible as the original puzzle, because there seems to be only a single swap. However, this one can be solved, if you just realize the "AHA!" I shamelessly stole the basic idea behind this puzzle for my VOTE THIS TUESDAY puzzle I published last year. Playing around with the simpler 15 puzzle version (learn how to solve it here) will help you understand the parity problem on the Cube better. |
stanalger Special user St. Louis, MO 998 Posts |
Quote:
On 2009-05-03 15:23, Scott Cram wrote: Impossible to solve? Not the problem as posed by Loyd. Loyd wrote: "It is safe to say that the thousand-dollar prize would have to be divided into the smallest coin of the realm to be portioned among all who have worked out solutions of some sort." [My source for the above information: Jerry Slocum's article "Sam Loyd's Most Successful Hoax" in Homage to a Pied Puzzler, edited by Ed Pegg, Jr., Alan Schoen, Tom Rodgers. The article concludes with a Martin Gardner quote: "Did Sam Loyd, America's greatest puzzle maker, invent the notorious 14-15 sliding block puzzle? He claimed he did, but the claim was a total lie. Loyd had nothing whatsoever to do with either the puzzle or its popularity."] |
Scott Cram Inner circle 2678 Posts |
I meant impossible to solve in the standard way, which it is.
There are lateral thinking approaches, of course. You can get them in 1-15 order if you don't mind all the numbers being sideways, for one. The solving of the puzzle sideways effectively acts as the 2nd switch, and thus satisfies the parity. I did it that way for a guy who claimed there was no way to get it in the 1-15 order whatsoever. |
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