Rubik's Cube/The Cube
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Ray Calvin Baker 19:31, 18 August 2011 (UTC)
| HOW TO FIND YOUR VERY OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE | | (Preliminary April 20, 2007 version) | | by Mr. Ray Calvin Baker | | FREE Public Domain Educational Material . | | | | Chapter Zero - - - - - - - - - - The Cube | | | | DO NOT PEEL ANY OF THE COLORED LABELS OFF YOUR CUBE! (I call this "the chemist's method, | | because it involves the breaking of octillions of weak molecular bonds. This is not meant | | to disparage chemists at all -- one of the first books on the Cube that I read was written | | by a chemist, and he described elegant geometrical and mathematical moves such as "Rubik's | | Maneuver".) | | | | There are other, less damaging ways to disassemble a Cube if you really want to see what's | | inside it (or if you want to cheat.) -- see Chapter Two, "Using Pictures, Diagrams, | | Notation, and Abbreviations", for "the Physicist's method of disassembling the Cube. Of | | course, if you can follow directions from a mathematician, you won't need to damage or take | | apart your Cube at all! | | | | From the outside, your Cube looks like it is made of 27 smaller cubes. (There's a picture | | in Chapter Two, "Using Pictures, Diagrams, Notation, and Abbreviations".) Each of these | | smaller cubes is called a cubie. A cube is a shape having eight corners, six flat square | | faces, and twelve edges. A Cube (upper case) is Rubik's Cube (or a cheap imitation of it). | | | | When it came from the factory, each face of your Cube had nine colored labels, all the same | | color -- six faces, six different colors. It may not seem likely to you now, but your Cube | | (unless it's physically damaged) can be restored to this condition. | | | | Your Cube consists of one central core, which holds six axles. Each axle supports one face | | cubie, which is free to rotate unless blocked by other components of your Cube. Each face | | cubie has a colored label -- six faces, six different colors. Each of the twelve edge | | cubies is clamped between a pair of face cubies. Each edge cubie has two different colored | | labels. Each of eight sets of three edge cubies clamps a corner cubie to the rest of the | | Cube. Each corner cubie has three different colored labels. (There are diagrammatic | | pictures in Chapter Two, "Using Pictures, Diagrams, Notation, and Abbreviations".) | | | | Even though each of the eight corner cubies and twelve edge cubies is a separate piece, | | they are all clamped together into a secure and stable assembly -- the Cube. And each face | | of the Cube can be rotated. Of course, that rearranges the colored labels. | | | | YOUR MISSION, should you choose to accept it, IS TO RESTORE YOUR CUBE TO ITS FACTORY | | CONDITION! | | | | How will you be able to tell where each cubie belongs when part of your Cube is scrambled? | | Look at the center square on each side. This piece is firmly attached to the core of the | | Cube, and is only free to rotate, The central square of each side of the Cube is your | | "fixed landmark" around which you must arrange all of the other cubies. (Once we start | | arranging edge cubies, we will temporarily use other landmarks, but we will always restore | | the positions of the center squares.) | | | | I suggest that you skim through this paper to see how the parts are organized, and to | | convince yourself that there really are ways to solve the Cube. Then, you can pick up a | | scrambled Cube to try to follow the exact steps you will need to solve it. Don't be | | discouraged if your first few attempts seem only to scramble the Cube more! Start again | | from the beginning if you have to. You will become more familiar with your Cube, and you | | will, with practice, become more confident that you can master each necessary operation. | | | # Just how many ways are there to arrange the cubies of the Cube? Let's calculate! But be # # aware that most calculators cannot hold numbers this large! (Some calculators may use # # scientific notation to express large numbers, but they almost always truncate the # # significant figures of the numbers, which is an approximation, not an exact number.) # # # # There are 8 factorial (usually written "8!") ways to arrange the eight corner cubies. # # This is 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 = 8 * 7 * 6 * 5 * 4 * 3 * # # 2 * 1 = 40,320. (I have used lots of calculators and computer languages which use "*" as # # the sign for "multiply" or "times".) # # . # # Then, There are ( (three to the eighth power) divided by three ) ways to orient the eight # # corner cubies. This is often written (on a computer) as "(3 ^ 8) / 3", which is # # (3 * 3 * 3 * 3 * 3 * 3 * 3 * 3) / 3 = 6,561 / 3 = 2,187. (We need to divide by three # # here, because it is not possible to rotate just one corner cubie. Whenever one corner cubie # # is rotated clockwise, another has to rotate counterclockwise, and vice versa -- unless, of # # course, you physically disassemble your Cube.) # # # # Next, there are 12! / 2 ways to arrange the edge cubies. This is (12 * 11 * 10 * 9 * 8 * 7 # # * 6 * 5 * 4 * 3 * 2 * 1) / 2 = 479,001,600 / 2 = 239,500,800. ( We need to divide by two # # here, because it is not possible to interchange exactly two edge cubies, unless you # # physically disassemble the Cube.) # # # # Lastly, there are ( ( 2 ^ 12 ) / 2 ) = ( 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 ) # # / 2 = 4,096 / 2 = 2,048 ways to orient the twelve edge cubies. (It is not possible to # # "flip" just one edge cubie, unless you disassemble... ) # # # # Multiply these four numbers together, and you get 40,320 * 2,187 * 239,500,800 * 2,048 # # = 43,252,003,274,489,856,000, which is exactly the number of ways the cubies of Rubik's # # Cube can be arranged. This is a twenty digit number -- forty-three quintillion, two # # hundred fifty-two quadrillion, three trillion, two hundred seventy-four billion, four # # hundred eighty-nine million, eight hundred fifty-six thousand. (OK, one of these ways is # # actually the unscrambled, pristine Cube, so there are really only # # 43,252,003,274,489,855,999 different scrambled Cubes, plus exactly ONE unscrambled Cube.) # # # # Each of those four factors of 43,252,003,274,489.056,000 corresponds to one of our major # # goals, which are discussed in the following chapters. # # # # By the way, if you do dissassemble your Cube, there are twelve distinctively different ways # # to put it back together. Twelve = 3 * 2 * 2, and these factors correspond to those numbers # # by which we divided in the previous calculations. Of these twelve ways, only one will allow # # you to solve the Cube. There will always (until you disassemble again) be something wrong # # with the other eleven ways of re-assembly -- an edge cubie will be flipped (disoriented), # # or two edge cubies will be interchanged, or a corner cubie will be improperly oriented -- # # you get the idea. (The twelve ways to re-assemble the Cube are sometimes called "orbits".) # | | *---------------------------------------------------------------------------------------------*
Ray Calvin Baker 15:01, 29 October 2011 (UTC)