Difference between revisions of "Parity"

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Parity is a case that may occur (in some form) on all current cubes larger than the 3x3x3 and many other different puzzles, such as the [[Square-1]].
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A parity is where two or more seperate pieces are either correctly orientated, but not correctly permuted or correctly permuted but not correctly orientated.
==Parity Cases==
==Flipped Edge parity==
On even cubes,([[4x4x4]] and [[6x6x6]]) the first type of Parity that may come up is the Flipped-edge parity. This is where two (or three) edge pieces (equivalent to one edge on the 3x3x3) are flipped. (This parity cannot occur on a 3x3x3 unless a piece is removed and replaced the wrong way around). This parity is easy to identify by immediately after the F2L instead of either a no edges, two edges, or the entire cross orientated there is either one or three edges orientated correctly.
This parity may also occur in the [[7x7x7]] V-cube and the 5x5x5 puzzle whilst trying to aquire the last two edges. On these cubes the Parity is identifiable by all edges being solved apart from one or two, both of which have all the correct pieces in the correct edge, but flipped the wrong way around. There is only one case of this on the 5x5x5 but three different cases on the 7x7x7, all of which are solved the same way. It is important to take note that on the 7x7x7 cube only the incorrectly flipped edge pieces should be included in the algorithm. This may require slicing the cube.
It is solved on the 4x4x4/6x6x6 with this algorithm, performed immediately after the Parity is identified.
''''(R*2 B2 U2) (L* U2 R'* U2) (R* U2 F2 R*) (F2 L'* B2) R*2''''
(For the 6x6x6 R*2 represents moving all three slices on the right side 180 degrees.)
Afterwards, the cube is solved normally.
==Switched Edge Parity==
This parity ONLY occurs on even cubes, (4x4x4 and 6x6x6). This parity is where the entire cube is solved besides two edges, which are not permuted correctly. This parity can be first easliy spotted by there being only two unpermuted edges before performing the last 2-look PLL algorithm.
It is solved on the 4x4x4 with this algorithm,
''''(r2 U2) (r2 U*2) (r2 U*2 U2)''''
On the 6x6x6 cube you can use the same algorithm, although here is a clearer version.
''''(3r2 R2 U2) (3r2 R2 3u2) (3r2 R2 3u2 U2)''''
Please note that 3r2 means that you turn all three slices on the right side 180 degrees. Similarly with the 3u2 parts.
==Switched Corner parity==
This parity can only occur, to my knowledge, with the Begginners method. After solving the centres and edges using the begginners method for solving the "reduced" 3x3x3 can come up with this parity. It is where two corners on the cube are incorrectly permuted. This parity is actually a variation of the Switched Edge parity. Because the beginners method requires you to permute the edges BEFORE the corners it causes the corners to be incorrect, so in theory this parity is no different from the previous one. Aside from that, here is the algorithm to fix the two different cases of this parity.
Diagonal corner switched.
''''U*2 L*2, U2, l2, U2, L*2, U*2, R, U', L, U2, R', U, R, L', U', L, U2, R', U, L', U''''
Adjacent corner switched.
''''U*2, L*2, U2, l2, U2, L*2, U*2 F', U', F, U, F, R', F2, U, F, U, F', U', F, R''''
It is important to note that these parity cases are completly avoidable using any technique that uses an OLL/PLL that orients the top layer before permuting the edges.
==Other Puzzles==
Parity can occur on many, many different puzzles. The [[Square-1]], for example can come up with a parity case that requires a large number of moves to solve.
[[Category:Puzzle Theory]]

Revision as of 01:21, 15 September 2009

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