Invisible <voi### [at] devnull> wrote:
> 4.33 10^19 Number of valid combinations for a Rubik's Cube.

  I think the correct term is permutations, rather than combinations.
(There *is* a difference between these two terms. They certainly are not
synonyms.)

  Btw, the classic Rubik's Cube has about 4.33*10^19 permutations, but if
you mark the pieces such that their orientation becomes discernible and
thus part of the correct solution (relevant only with the center pieces,
but the other pieces also have to be marked in order to distinguish the
proper orientation for the center pieces), the total number of permutations
increases to 8.9*10^22. In other words, there will be a bit over 2000 times
the amount of permutations, making it slightly more difficult to solve.

  The Rubik's Revenge has about 7.4*10^45 permutations, ie. about 10^26
times more than the Rubik's Cube. Curiously, though, it's only slightly
more difficult to solve. The solving difficulty is in no way related to
the increased number of permutations.

  (The main reason why Rubik's Revenge is only slightly more difficult to
solve than the Rubik's Cube is that the former can be partially solved to
a state where it becomes completely equivalent to the latter, after which
Rubik's Cube solving techniques can be used to solve it. Thus all the
additional techniques you need to solve Rubik's Revenge are the ones needed
to make it equivalent to the Rubik's Cube. Then just solve the it like the
Rubik's Cube.)

  (Ok, there's one slight difference: With the Rubik's Revenge it's possible
for a single "edge piece" (which in this case will be an edge piece pair) to
end up in the wrong orientation, which is impossible in the Rubik's Cube.
However, it can be turned around with a fixed sequence which you can
memorize.)

  Likewise larger versions of the cube exhibit the same phenomenon: The
amount of permutations grows exponentially, but solving them doesn't
become significantly more difficult. Only the amount of rotations needed
increases with the size of the cube, but the same principles can be applied
to all cubes from 5x5x5 up.

-- 
                                                          - Warp

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