Thread: "Calculate number of permutation of restricted cube"

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You don't need to visualize it but I want to, so that I can compare its
structure with that of MC2D and any other small, symmetric twisty
puzzles. We know some general things about these graphs such as their
valences and diameters, but I'm not sure we know a whole lot more.
-Melinda

On 6/30/2016 11:04 PM, phamthihoa4444@gmail.com [4D_Cubing] wrote:
>
>
> Why should I visualize it? What I need to prove now is why only 1344
> positions are reachable. But is 1344 too large to visualize?
>
> Just like how permutation of 2^4 = (15!/2) * (12^14 * 4) is
> calculated, hold one corner piece in its place and rotate the rest.
>
> I tried to solve the puzzle by MPUlt (the structure of puzzle file is
> really difficult to understand, and I have not understand many parts
> yet) and can solve it intuitively, then I think number of permutation
> is correct.
>
> Computer brute force given:
> + Any 7 cubies in same parity cannot be moved without affecting other
> pieces. Thus 3, 4, 5, 6 and 7-cycle are all impossible.
> + Any 4 cubies in same parity cannot be moved if 4 other cubies in
> that parity is fixed and the other parity can be moved.
> (let a piece be even if all edge-adjacent pieces of it are odd and
> vice versa)


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">


You don't need to visualize it but I want to, so that I can compare
its structure with that of MC2D and any other small, symmetric
twisty puzzles. We know some general things about these graphs such
as their valences and diameters, but I'm not sure we know a whole
lot more.

-Melinda

Why should I visualize it? What I need to prove now is why
only 1344 positions are reachable. But is 1344 too large to
visualize?

Just like how permutation of 2^4 =3D (15!/2) * (12^14 * 4) is
calculated,=C2=A0 hold one corner piece in its place and rotate the
rest.

I tried to solve the puzzle by MPUlt (the structure of puzzle
file is really difficult to understand, and I have not
understand many parts yet) and can solve it intuitively, then I
think number of permutation is correct.

Computer brute force given:

=C2=A0+ Any 7 cubies in same parity cannot be moved without
affecting other pieces. Thus 3, 4, 5, 6 and 7-cycle are all
impossible.

=C2=A0+ Any 4 cubies in same parity cannot be moved if 4 other
cubies in that parity is fixed and the other parity can be
moved.

(let a piece be even if all edge-adjacent pieces of it are
odd and vice versa)