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For two Rubik cubes to be identical, what is the minimum requirements? for example are 2 faces and their position sufficient? e.g. 2 side by side face, or two opposing faces of two cubes being same would be sufficient for two cubes to be identical?

What about a square from not the middle of all faces being specified with their relative positions, for example would specifying bottom left hand side of each face be sufficient? e.g. on Top Red , on Bottom Blue, on left face Green, ... etc, would that be easy to see whether two cubes are identical.

Is there a name for minimum set of requirements that makes two structures isomorphic/identical to each other?

jimjim
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    What do you mean by "isomorphic"? That you can transform one cube to be identical to the other? – Eric Wofsey Jul 10 '16 at 04:59
  • @EricWofsey I don't understand what is unclear about meaning of isomorphism. Maybe I should say identical ? – jimjim Jul 10 '16 at 05:15
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    Why on earth would 2 faces make the cubes identical?... oh, never mind. Um, identical up to what purpose? Indentical can mean whatever you want it to mean. – fleablood Jul 10 '16 at 05:31
  • @fleablood two identical rubik cubes will have colours on their sides matching each other, there would be nothing to differentiate them by colours positioned on their faces. – jimjim Jul 10 '16 at 05:48
  • @fleablood give two examples of defining identical for two rubik cubes please – jimjim Jul 10 '16 at 05:53
  • If you know the definition of identical (for your purpose) what is your question? – fleablood Jul 10 '16 at 05:56
  • Of course two faces aren't sufficient. You need all six faces all nine pieces per face. Although if you have all but one piece, and you got the layout by usual legal moves, then the last piece must be in place. However it's possible to have all but two pieces in place. So minimum requirement is all but one piece identical. – fleablood Jul 10 '16 at 06:01
  • @fleablood minimum sufficient conditions that make two cubes identical – jimjim Jul 10 '16 at 06:02
  • @fleablood are you sure? Don't you think 5 faces bring identical will make the 6th identical as well? – jimjim Jul 10 '16 at 06:03
  • @fleablood are you familiar with triangles? What are the minimum condition that make two triangles congruent or identical – jimjim Jul 10 '16 at 06:06
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    Don't think of in terms of faces. Think in terms of pieces. There are 20 mobile pieces. If 19 of them are in place properly orientated the 20th will be too. But if only 18 are in place that isn't sufficient. – fleablood Jul 10 '16 at 06:24
  • @fleablood I guess that settles the question – jimjim Jul 10 '16 at 06:27
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    @Arjang: No, not even knowing 5 faces is enough; see this answer. – hmakholm left over Monica Jul 10 '16 at 12:15
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    @fleablood: Knowing the positions of entire pieces is a different problem -- if we're looking at faces, we can get partial information about some of the pieces, which is not possible in your formulation. And to get full information about a piece requires looking at two or three faces of the entire cube. – hmakholm left over Monica Jul 10 '16 at 12:17
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    @fleablood: And even so, if the 18 pieces you have full information about are 11 edge pieces and 7 corner pieces, then you can deduce both the location and the orientation of the two remaining ones; assuming that the cube is in a reachable-from-solved state. – hmakholm left over Monica Jul 10 '16 at 12:23
  • Where in that question does it say five faces is not enough? Yes, it's different problem but the question was phrased minimum requirements. Not the minimum information about faces. Piece information is more efficient. I did not know that about the 18 pieces although I should have. – fleablood Jul 10 '16 at 15:13
  • Oh I see. Yes the cube has to be unsolved. But knowing the invisible top face has two pieces adjacent to yellow you can not tell which of the two visible yellow pieces are which. – fleablood Jul 10 '16 at 15:21
  • Note, nobody ever considers knowing whether the center piece maintains its orientation as a criterion. – fleablood Jul 10 '16 at 15:22

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Just considering cubie permutations (ignoring their individual rotations), the corner and edge cubies together have exactly $\frac{8!12!}2$ valid configurations so you'd have to identify 7 out of the 8 corners, 10/12 edges. and two adjacent centers (to check the total orientation) to uniquely identify the permutation of all the cubies. Then you'd have to examine the rotations of 7 out of the 8 corners and 11 out of the 12 edges to uniquely identify all the cubie rotations.

NovaDenizen
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