User:Arjun sunel/Solvability of a 3x3x3 Rubik's Cube State?: Difference between revisions

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rosettacode>Hajo

Revision as of 04:44, 13 April 2013 (view source) rosettacode>Arjun sunel (Created page with "==Introduction== This article is about the solvability of a 3x3x3 Rubik's Cube. One thing might come in your mind that what is motive behind studying so much about a Rubik's C...")		Latest revision as of 23:31, 21 October 2014 (view source) rosettacode>Hajo (→‎Edge Parity Test: Typos)
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Line 54: ==Solvability Understanding Using A Solved Cube== ===Case 1: Twist a corner=== Suppose, you are holding a solved Cube. Now, pick a corner and twist it clockwise ~~(or anticlockwise) by 120 degree (by removing it out and rotate by 120 degree and put it again in the same place)~~ (or anticlockwise) by 120 degree (by removing it out and rotate by 120 degree and put it again in the same place) The cube state so obtained is unsolvable because it is not possible to solve the twisted corner without disturbing the positions of already solved cubies using any legal moves. Line 61 ⟶ 62: File:flip_edge.png\|Flip an edge. File:Swap_edges.png\|Swap just two edges. File:Swap_corners.png\|Swap just two ~~edges~~corners. </gallery> Line 205 ⟶ 206: ===Edge Parity Test=== Correctly-oriented edge: AAn edge in any position is said to be correctly-oriented if by using rotations of only four faces LEFT, RIGHT, TOP and BOTTOM, we can get into the correct position. ~~If afer applying such rotations, we get the edge flipped, then the edge is not correctly-oriented.~~ If after applying such rotations, we get the edge flipped, then the edge is not correctly-oriented. It should be noted that rotation of the above four faces ~~don~~does not change the orientation of the edges. But, we have two more faces. They are FRONT and BACK. ~~Rotation of these two faces by 90 degrees, flips all the four edges on that face.~~ Rotation of these two faces by 90 degrees, flips all the four edges on that face. Let, Line 225 ⟶ 228: Since, LEFT, RIGHT, TOP and BOTTOM flips zero edges while rotation of FRONT and BACK by 90 degrees flips 4 edges. So, any legal move will change the edge-orientation value only by an even number which is divisible by 2. Hence, ana legal move will change the Rubik's Cube state from one solvable state to another solvable state and from one unsolvable state to another unsolvable state. ====How to compute edge-orientation?==== Line 231 ⟶ 234: a) Pick an edge and initialize n=0. b) Try to bring the edge in the position where it is ought to be in a solved Cube by using only four faces LEFT, RIGHT, TOP and BOTTOM and also ~~note~~take notes of the moves you make ~~on a paper~~. c) When the edge is in the correct position, check whether the edge is correctly-oriented or not. If the edge is not correctly-oriented, then increment the value of n by 1, else move to the next step. d) DoUsing ~~the~~your notes, reverse ~~moves as noted in~~ the ~~paper~~moves. After applying the algorithm, check whether all the 20 edges have been checked. If yes, then go to step e) else go to step a). e) Check whether n=total number of flips is divisible by 2 or not. If yes, then the Cube state passes the Edge- Parity -Test, otherwise it fails and we can say that this ~~Cubic~~ Cube-State is not solvable. ==Rubik's Cube Solvability:== Line 302 ⟶ 305: Please don't disclose this trick and keep it with yourself. The trick is that we can remove the cubies apart and recombine in such a way that the cube is solved. But, these are all illegal moves!!!! ==References== [[Category:Rubik's Cube]] [[Category:Combination puzzles]] [[Category:Puzzles]] [[Category:Multi-dimensional geometry]]