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Math Help - Rubik's Cube Riddle

  1. #1
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    Rubik's Cube Riddle

    Anyone who has ever played with a rubik's cube knows that it is possible to only complete 1 of the 6 sides.

    Anyone who has ever played with a rubik's cube knows that it is NOT possible to only complete x of the 6 sides.

    find x

    (I am proud of this one I made it up last night!)
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    Grand Panjandrum
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    Quote Originally Posted by MathGuru
    Anyone who has ever played with a rubik's cube knows that it is possible to only complete 1 of the 6 sides.

    Anyone who has ever played with a rubik's cube knows that it is NOT possible to only complete x of the 6 sides.

    find x

    (I am proud of this one I made it up last night!)
    The only way that I ever solved a Rubik's cube is by levering the
    thing apart, and reassembling it in a solved configuration

    (better still reassembling it in a configuration which cannot be solved)

    RonL
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    MHF Contributor Quick's Avatar
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    Quote Originally Posted by MathGuru
    Anyone who has ever played with a rubik's cube knows that it is possible to only complete 1 of the 6 sides.

    Anyone who has ever played with a rubik's cube knows that it is NOT possible to only complete x of the 6 sides.

    find x
    I personnally have spent a lot of wasted time on my Rubik's Cube. And through the many minutes/hours spent on it I had discovered that two adjacent sides control all the other ones. If two adjacent sides are correct, than the whole cube is correct.

    Therefore, it is possible to get two opposite sides completed without completing the cube, but a third completed side would automatically be adjacent to both opposite sides, so that side could not be completed unless the cube is.

    So,  3 \leq x \leq 5

    on a side note, I think there are 1.69267*10^13 different combinations of a Rubik's Cube.
    Last edited by Quick; June 10th 2006 at 06:42 PM.
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    I can solve a Rubik in less approximately 5 minutes.

    I believe the solution algorithm was obtained thought group theory. Because it is a puzzle my intuition tells me it has something to do with premutational groups. I was always interested to see how mathematicians derived an algorithm.
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    Quote Originally Posted by Quick
    I personnally have spent a lot of wasted time on my Rubik's Cube. And through the many minutes/hours spent on it I had discovered that two adjacent sides control all the other ones. If two adjacent sides are correct, than the whole cube is correct.

    Therefore, it is possible to get two opposite sides completed without completing the cube, but a third completed side would automatically be adjacent to both opposite sides, so that side could not be completed unless the cube is.

    So,  3 \leq x \leq 5

    on a side note, I think there are 1.69267*10^13 different combinations of a Rubik's Cube.
    I have solved my cube for 3 sides, so this is possible.

    Note: The Rubik's Revenge (4 squares to a side) CAN be solved for 4 sides.

    And though I loath to suggest that I compete with ThePerfectHacker in anything , I can solve the original Rubik's cube in less than 2 and a half minutes.

    BTW: If you can solve a Rubik's cube and a Revenge, you can also solve the Professor's cube (5 squares) and any other that they might make with a higher number of cubes. (The guy who sold me the Professor's cube would be mortified to know that I've had it out of the box. )

    -Dan
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    Quote Originally Posted by ThePerfectHacker
    I can solve a Rubik in less approximately 5 minutes.

    I believe the solution algorithm was obtained thought group theory. Because it is a puzzle my intuition tells me it has something to do with premutational groups. I was always interested to see how mathematicians derived an algorithm.
    I know of two ways to argue a solution to the Rubik's cube. The first is cheating: figure out how to undo what the person did to mix it up. Obviously if you do it backward, it will be solved. (This is not as trivial as it sounds...there are many physical systems that you can't do this with.) Another one is to do what the person did to mix it up over and over and over and over... I forget what theorem of group theory this refers to, but if you perform a series of moves on the cube and repeat this series long enough, the cube will eventually (several life-times of the Universe, no doubt!) come back to a solved state. I have verified this for a few simple patterns.

    Obviously, since we have to know what the person did to mess up the cube in the first place, neither of these methods of proof for the existence of a solution are very satisfying!

    -Dan
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    Grand Panjandrum
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    Quote Originally Posted by CaptainBlack
    The only way that I ever solved a Rubik's cube is by levering the
    thing apart, and reassembling it in a solved configuration

    (better still reassembling it in a configuration which cannot be solved)

    RonL
    Which as those who know me will tell you that is my approach
    to most puzzles/problems.

    (Don't tell the puzzle editors of the Sunday Times or NewScientist
    or they may ask for prizes back)

    RonL
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    Site Founder MathGuru's Avatar
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    Quote Originally Posted by Quick

    Therefore, it is possible to get two opposite sides completed without completing the cube, but a third completed side would automatically be adjacent to both opposite sides, so that side could not be completed unless the cube is.

    So,  3 \leq x \leq 5
    This is not correct. you can see a jave applet with a rubik's cube that has 3 sides completed here.

    No one has recognized the obvious answer yet. . .
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    well you obviously can't only complete five sides, as then the only piece that would have a chance of being incorrect on the 6th side would be the center square, and these don't even move. if one square is in the wrong place another must also be. hence if 5 sides are complete then the 6th must also be.

    i don't think four sides are possible either, though.. since if you rotate one corner, you cause 3 sides to be incorrect, and you then must rotate another corner to do this. also if you switch one edge piece with another edge piece you effect three sides. but not sure on this one.
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    MHF Contributor Quick's Avatar
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    Quote Originally Posted by MathGuru
    This is not correct. you can see a jave applet with a rubik's cube that has 3 sides completed here.
    Where does it show three sides completed? I couldn't see any that didn't have all the other sides completed as well.
    Last edited by Quick; June 13th 2006 at 05:13 PM.
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    Quote Originally Posted by Quick
    Where does it show three sides completed? I couldn't see any that didn't have all the other sides completed as well.

    Are you sure? I have never been able to get three sides, and believe me I've tried.

    Picture attached, this is two views of the same cube. It has to be possible because you can play the video that will solve this cube from this configuration.

    Aradesh is right about 5 not being possible because 5 means all 6 are correct. I am not sure about 4 . . .
    Attached Thumbnails Attached Thumbnails Rubik's Cube Riddle-rubiks-3-sides-complete.jpg  
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  12. #12
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by MathGuru
    Picture attached, this is two views of the same cube. It has to be possible because you can play the video that will solve this cube from this configuration.

    Aradesh is right about 5 not being possible because 5 means all 6 are correct. I am not sure about 4 . . .
    Darn!
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  13. #13
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    By the way Quick, I love your signature. . . maybe it's time to add an avatar!
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    Quote Originally Posted by MathGuru
    Picture attached, this is two views of the same cube. It has to be possible because you can play the video that will solve this cube from this configuration.

    Aradesh is right about 5 not being possible because 5 means all 6 are correct. I am not sure about 4 . . .
    The only time I have ever solved the cube for 4 sides is when one of the pieces popped out and I put it back in the wrong way. I don't have a proof, but 4 sides are impossible. (That is to say I couldn't solve the cube from the starting position of when it was in a state of 4 sides solved, so the initial state is "unnatural" in some sense.)

    -Dan
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    Is it a good riddle?

    As far as the riddle is concerned . . .

    with allowing popping out pcs and rearranging them only 4 sides is possible. so 5 sides is the answer to the riddle.

    Is it possible to only complete 4 sides of a natural rubik's cube? I do not know.

    Is it a good riddle? It is the first one I ever came up with.
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