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## Rubik’s nxnxn Cube Algorithms

Rubik’s nxnxn Cube Solution Algorithms

The Professor’s Cube is an extension of the Rubik’s Cube and the Rubik’s Revenge. It ismade of ﬁve rotating slices, from which it follows that the Professor’s Cube is composedby 98 cubies: 8 corner cubies (possessing 3 stickers each), 36 edge cubies (2 stickers)and 54 remaining center cubies (one sticker only). At ﬁrst glance the Professor’s Cubeturns out to share a remarkable feature with the Rubik’s Cube: one cubie in each face,namely the most central one, is ﬁxed. This represents a big diﬀerence with respect to theRubik’s Revenge and any cube with an even number of slices, where each center cubiecan be moved. Furthermore, the number of edge cubies is exactly the sum of the 24 edges(twelve pairs, in particular) of the Rubik’s Revenge and the 12 edges of the Rubik’s Cube:we will refer to the formers as coupled edges (indicated by black spots in Figure 1), whileto the latters as single edgesThe Professor’s Cube is an extension of the Rubik’s Cube and the Rubik’s Revenge. It ismade of ﬁve rotating slices, from which it follows that the Professor’s Cube is composedby 98 cubies: 8 corner cubies (possessing 3 stickers each), 36 edge cubies (2 stickers)and 54 remaining center cubies (one sticker only). At ﬁrst glance the Professor’s Cubeturns out to share a remarkable feature with the Rubik’s Cube: one cubie in each face,namely the most central one, is ﬁxed. This represents a big diﬀerence with respect to theRubik’s Revenge and any cube with an even number of slices, where each center cubiecan be moved. Furthermore, the number of edge cubies is exactly the sum of the 24 edges(twelve pairs, in particular) of the Rubik’s Revenge and the 12 edges of the Rubik’s Cube:we will refer to the formers as coupled edges , while to the latters as single edges.

3.1. The even n×n×nCube. The ‘even cube’ always 8 corner cubies (3 possibleorientations), while the number of edges and centers depends on n. It is not diﬃcultto check that the numbers of centers as well as that of edges is even. In particular, thenumber of center cubies isc= 6(n−2)2,(5)while the number of edges ise= 12(n−2).(6)In the special case of the Rubik’s Revenge (n= 4), we have c=e= 24. Notice thatin the limit case n= 2 we get a cube with no edges nor centers (only corners!), which isexactly the case of the 2 ×2×2, the so-called Pocket Cube.

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