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 Post subject: A question about parity?Posted: Thu Jan 04, 2001 7:45 am
My question concerns the Rubik's Cube and the Square-1 puzzle. I believe I have read that edge pieces must be exchanged in two pairs of two edges on the Rubik's Cube. That is, it is not possible to exchange exactly two edges while leaving the entire rest of the cube intact. However, I have a Square-1 puzzle that has exactly two edges exchanged and the remainder of the puzzle pieces are in their correct position and orientation. Can someone give me an intuitive (non-algebraic) explanation of why this is impossible on the Rubik's Cube, yet possible on the Square-1 puzzle?

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 Post subject: Square-1 parityPosted: Thu Jan 04, 2001 8:47 pm

Joined: Sun Dec 19, 1999 3:02 am
: My question concerns the Rubik's Cube and the Square-1 puzzle. I believe I have read that edge pieces must be exchanged in two pairs of two edges on the Rubik's Cube. That is, it is not possible to exchange exactly two edges while leaving the entire rest of the cube intact. However, I have a Square-1 puzzle that has exactly two edges exchanged and the remainder of the puzzle pieces are in their correct position and orientation. Can someone give me an intuitive (non-algebraic) explanation of why this is impossible on the Rubik's Cube, yet possible on the Square-1 puzzle?

Sure, no sweat, but in order to make it intuitive, I'll have to bring a little bit of math into it, I think. Suppose you have a puzzle with nothing but corners on one side. Let's number them in order: 1, 2, 3, 4, 5, 6. Now let's try making a few swaps. If we swap corner 1 and corner 2, the order becomes 2, 1, 3, 4, 5, 6. Swap 1 and 3, and we get 2, 3, 1, 4, 5, 6. Swapping corner 1 with each corner in turn, we get 2, 3, 4, 1, 5, 6, then 2, 3, 4, 5, 1, 6, and finally 2, 3, 4, 5, 6, 1, which gives us exactly 5 corner swaps. However, EXACTLY the same thing could be done much easier by simply rotating the all corners side counterclockwise by the space of 1 corner, which is where the odd parity comes in. If you have a pattern which gets you to nothing but corners on one side, then perform the pattern, rotate the all corners side either way by an odd number of corners, and then reverse your pattern, then lo and behold, the parity problem will be fixed! Good luck! L8r.

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 Post subject: More on parity.Posted: Fri Jan 05, 2001 3:33 am
Thank you very much for explaining to me how to solve the parity problem on the Square-1. I am not sure I have a pattern to obtain 6 corners on one side of the puzzle, but I will figure one out and then try out your method.

From your explanation, I gather that Square-1 is an odd parity puzzle while Rubik's cube is an even parity puzzle. I do not understand why this is so. In my naive view, both puzzles look similar in the sense that they have four corners and four edges on top and bottom when in the pristine state. I realize that it is not possible to obtain six corners on one face of a Rubik's cube, but I am unclear as to how this determines parity. In fact, I do not understand how to determine the parity of a puzzle at all. And further, I am not really sure why it is that the parity of a puzzle determines whether or not edges (and/or corners?) can be swapped in single pairs as opposed to pairs of pairs.

Is there a reference that explains parity and how it relates to atomic operations on puzzles?

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 Post subject: re: More on parity.Posted: Sat Jan 06, 2001 3:33 am

Joined: Sun Dec 19, 1999 3:02 am
: Thank you very much for explaining to me how to solve the parity problem on the Square-1.

You're welcome.

: From your explanation, I gather that Square-1 is an odd parity puzzle while Rubik's cube is an even parity puzzle. I do not understand why this is so. In my naive view, both puzzles look similar in the sense that they have four corners and four edges on top and bottom when in the pristine state.

They sure do look awfully similar, don't they? I'm sure I'm not the only one who got really cocky about that with my first puzzle...

: I realize that it is not possible to obtain six corners on one face of a Rubik's cube, but I am unclear as to how this determines parity. In fact, I do not understand how to determine the parity of a puzzle at all. And further, I am not really sure why it is that the parity of a puzzle determines whether or not edges (and/or corners?) can be swapped in single pairs as opposed to pairs of pairs.

: Is there a reference that explains parity and how it relates to atomic operations on puzzles?

: Thank you for your time.

Jaap Scherphuis has a section on his page (http://www.org2.com/jaap/puzzles/) relating to puzzle mathematics, but I think I can simplify it a bit. Let's take a standard Rubik's Cube with 4 edges and 4 corners on one side, with corners 1, 2, 3, 4, and edges a, b, c, d. The standard layout would be 1a2b3c4d. If we rotate this side, we would end up with, say, 4d1a2b3c. Effectively what's been done is swap corner 4 with corners 3, 2, and 1 in sequence (1, 2, 4, 3, then 1, 4, 2, 3, then 4, 1, 2, 3) and swaped edge d with edge c, b, and a in sequence (a, b, d, c, then a, d, b, c, then d, a, b, c), giving us 3 edge swaps and 3 corner swaps (ie. both are odd parity). If we make another quarter turn, then we'll have 6 corner swaps and 6 edge swaps (ie. even parity for both corners and edges). As we continue to make quarter turns, we get 9 corner swaps and 9 edge swaps, then 12 corners and 12 edges, then 15 and 15, etc. No matter how well the puzzle is mixed, the corner and edge swaps MUST BOTH be odd, or MUST BOTH be even. Unless you take the cube apart, there's no way to get around it. Therefore, if either the corners or edges look solved (ie. zero parity, which is even), then by definition the other MUST have an even parity as well, hence the need for a 3 cycle minimum (ie. the same corner is swapped with 2 others, or the same edge is swapped with 2 others). L8r.

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