So a $2 \times 2 \times 2$ cube has $8$ distinct pieces. With each of them having 3 colours(one on each of their exposed edges). Thus, the as echo piece has $3$ different orientations and there are $8$ pieces, won’t there be a total of $3^8=6561$ combinations?
I looked it up and found out there were around $3,674,160$ combinations. Where am I wrong ?
On
Pieces may be permuted too – this contributes an additional factor of $8!$ – but the orientation of seven corners determines the eighth, so we then divide by $3$. There is nothing to fix the orientation of the whole puzzle in space, so we divide by $24$ (the order of the rotational group of the cube $O$). This gives $3674160$ combinations.
On
Let $G$ be the group of the $2 \times 2 \times 2$ cube, generated by the moves R L U D F B. We want to calculate $G/24$, diving by $24$ because we consider two permutations equivalent up to rotation of the entire cube.
Permuting Cubies
Using the T-perm R U R' U' R' F R2 U' R' U' R U R' F' you can swap any two cubies without affecting their orientation. So we have 8 cubies that can be permuted freely. This gives $8!$ possible ways (since we have a full Symmetric Group on 8 points). So $|G|$ will be $8!$ times whatever twists we can apply.
Twisting Cubies
It is known that you cannot solve the cube after twisting just one cubie. So there are limits on what twists can be applied. Consider a group homomorphism that measures the $3$ possible twists of each corner.
$f : G \to \mathbb Z / 3 \mathbb Z$
In the default solved state of the cube the four upper cubies have white facing up and the four bottom ones have yellow facing down.
Define $f$ to sum up how many clockwise twists ($0$, $1$ or $2$) each cubie will take to get its yellow or white face to face the top if it's on the top layer or bottom if it's on the bottom layer.
Now we can analyze the generators:
They all come out as zero.
It is possible to twist any two cubies in opposite directions, for example R U2 R U' R' F R' F2 U' F U twists two adjacent cubies. This can be conjugated to allow you to apply twists to any two cubies without affecting anything else.
So we can perform any twists 7 of the cubies, the twist of the final cubie is determined.
So the final result is $|G|/24 = 8! \cdot 3^7 / 24 = 3674160$
Sources:
The pieces also move relative to one another - they don't just rotate in their places.
Then (as for the $3\times 3\times 3$ cube) there are constraints in the permutations and rotations - not all are achievable.
Also note that (because there are no central squares fixed in place to govern the colours of the sides), whatever state the cube is in, I can, for example, choose to put the same corner cube at top-front-left and then rotate the whole large cube to put that corner cube in a standard orientation.