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How many possible combinations are there in a 3x3x3 rubics
cube?
In other words, if you wanted to solve the rubics cube by
trying different combinations, how many might it take you
(worst case senerio)?
How many for a 4x4x4 cube?
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Answer / guest
There are 4.3252 * 10^19 possible combinations for 3x3x3
Rubics and 7.4012 * 10^45 possible combinations for 4x4x4
Rubics.
Let's consider 3x3x3 Rubics first.
There are 8 corner cubes, which can be arranged in 8! ways.
Each of these 8 cubes can be turned in 3 different
directions, so there are 3^8 orientations altogether. But if
you get all but one of the corner cube into chosen positions
and orientations, only one of 3 orientations of the final
corner cube is possible. Thus, total ways corner cubes can
be placed = (8!) * (3^8)/8 = (8!) * (3^7)
Similarly, 12 edge cubes can be arranged in 12! ways.
Each of these 12 cubes can be turned in 2 different
directions, so there are 2^12 orientations altogether. But
if you get all but one of the edge cube into chosen
positions and orientations, only one of 2 orientations of
the final edge cube is possible. Thus, total ways edge cubes
can be placed = (12!) * (2^12)/2 = (12!) * (2^11)
Here, we have essentially pulled the cubes apart and stuck
cubes back in place wherever we please. In reality, we can
only move cubes around by turning the faces of the cubes. It
turns out that you can't turn the faces in such a way as to
switch the positions of two cubes while returning all the
others to their original positions. Thus if you get all but
two cubes in place, there is only one attainable choice for
them (not 2!). Hence, we must divide by 2.
Total different possible combinations are
= [(8!) * (3^7)] * [(12!) * (2^11)] / 2
= (8!) * (3^7) * (12!) * (2^10)
= 4.3252 * 10^19
Similarly, for 4x4x4 Rubics total different possible
combinations are
= [(8!) * (3^7)] * [(24!)] * [(24!) / (4!^6)] / 24
= 7.4011968 * 10^45
Note that there are 24 edge cubes, which you can not turn in
2 orientations (hence no 2^24 / 2). Also, there are 4 center
cubes per face i.e. (24!) / (4!^6). You can switch 2 cubes
without affecting the rest of the combination as 4*4*4 has
even dimensions (hence no division by 2). But pattern on one
side is rotated in 4 directions over 6 faces, hence divide
by 24.
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