Thread: "Virtual vs Physical 2^4 Correspondence"

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Hi, Michael,

Nice one! I like your idea of focusing on the missing I face moves as
an attractive way of reaching all of the virtual 2^4 states. The RIL
moveset is a good candidate to become one of our primary ways of
interacting with the phys 2^4 puzzle.

Melinda and I had been using the two kinds of "double restack" as our
only way of mixing the R and L faces with the other faces (packaging
each with a sequence of cleanup moves to produce FOro and FUro for ease
of understanding; the double restacks are the first two moves of each).
However, those double restacks are full-puzzle rotations that affect all
16 pieces, and look different than regular twists. So, it's lovely to
have a simple 8-piece twist that moves (half of the) R and L stickers
onto other faces.

My 4th video from my message of 6/10/2017 quite relevant to this
discussion, if anybody wants to revisit it. I demonstrate the I face
moves at 2:24: https://www.youtube.com/watch?v=Dsu-5TTDNbM&t=144s

In the video I say that the Iy and Iz moves (which I refer to as
"illegal" IU and IF quarter turns, but I like how you've given them
distinct names) are "failing badly", but all I really meant was, they
twist the pieces of the I face in ways that I didn't understand, and
that do not correspond to a nice 1-click move on the virtual 2^4.
They only "fail" if you were insisting on only using moves that have
1-click equivalents in MC4D. I'm actually strongly in favor of
loosening that up to include a few attractive phys 2^4 moves, as long as
they have macro equivalents in MC4D.

The same behavior (applying a lot of twists that we must either
understand or untangle) is true of the double restack, actually. If we
compare the double restacks to a full-puzzle-reorienting sequence using
I face turns, such as "Iy RU2ro Iy", they produce VERY similar results.

(By the way, Michael pointed out to me via email that the 4-cycle
"quarter puzzle twists" or "subface twists" that I mention in my 5th and
6th videos of 6/10/2017, and was hoping to add to the move set, actually
violate permutation parity of the 4-color pieces. I was
thinking of the 2^3 where pair swaps are OK. 4-cycles of pieces in the
2^4 need to be done strictly in pairs, if at all. So, when you're
executing I face turns on the puzzle, don't make a quarter turn to one
of the the end caps by mistake or you'll be in a bad permutation parity
state.)

Michael gives these macro equivalents that he found:

> Iy (physical) == IU' RO2 IF RO2 IU RO2 IF' RO2 IU (virtual)
>
> IU (virtual) == Iy' Rx2 Iz Rx2 Iy Rx2 Iz' Rx2 Iy (physical)
>
> [I think these are right, someone please double-check :)]
>

Yep, they work! Thanks for doing that crucial step, of showing a macro
for each of the "interesting" I face moves in the "other" puzzle. (The
ones for Iz and IF will of course be identical to Iy and IU modulo a
full puzzle x rotation.) With these two sequences, we can know that
we're reaching the identical set of puzzle states in both puzzles, even
if some solutions in one moveset are of a different length than in the
other.

> *Summary*: You can simulate 2^4 moves on the physical cube without too
> much trouble, but if you want a more comfortable puzzle that is not
> quite equivalent to the 2^4, but very similar and with the same level
> of difficulty, this is a nice way to do it. The allowed moves are to
> do any rotation to the 2x2x2 cube formed by either the left two rows,
> middle two rows, or right two rows, leaving the rest of the puzzle
> alone. That's it! Scramble and try to solve.
>

To clarify (and please correct me if I'm wrong), the RIL physical 2^4
moveset reaches exactly the same set of states as the complete virtual
2^4 puzzle in MC4D. When Michael says "not quite equivalent", he is
referring to how the moves Iy, Iy', Iz, and Iz' do not have 1-click
equivalents in MC4D, only macro equivalents. And of course the RIL
moveset is 3/8ths the size of the MC4D moveset, so solutions will tend
to be somewhat longer. But, you can fully scramble the puzzle, and
solve all MC4D scrambles, using only RIL.

Good stuff!

Cheers
Marc


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Hi, Michael,



Nice one!   I like your idea of focusing on the missing I face moves
as an attractive way of reaching all of the virtual 2^4 states.  The
RIL moveset is a good candidate to become one of our primary ways of
interacting with the phys 2^4 puzzle.



Melinda and I had been using the two kinds of "double restack" as
our only way of mixing the R and L faces with the other faces
(packaging each with a sequence of cleanup moves to produce FOro and
FUro for ease of understanding; the double restacks are the first
two moves of each).  However, those double restacks are full-puzzle
rotations that affect all 16 pieces, and look different than regular
twists.   So, it's lovely to have a simple 8-piece twist that moves
(half of the) R and L stickers onto other faces.



My 4th video from my message of 6/10/2017 quite relevant to this
discussion, if anybody wants to revisit it.  I demonstrate the I
face moves at 2:24:  
https://www.youtube.com/watch?v=Dsu-5TTDNbM&t=144s



In the video I say that the Iy and Iz moves (which I refer to as
"illegal" IU and IF quarter turns, but I like how you've given them
distinct names) are "failing badly", but all I really meant was,
they twist the pieces of the I face in ways that I didn't
understand, and that do not correspond to a nice 1-click move on the
virtual 2^4.    They only "fail" if you were insisting on only using
moves that have 1-click equivalents in MC4D.  I'm actually strongly
in favor of loosening that up to include a few attractive phys 2^4
moves, as long as they have macro equivalents in MC4D.



The same behavior (applying a lot of twists that we must either
understand or untangle) is true of the double restack, actually.   
If we compare the double restacks to a full-puzzle-reorienting
sequence using I face turns, such as "Iy RU2ro Iy", they produce
VERY similar results.



(By the way, Michael pointed out to me via email that the 4-cycle
"quarter puzzle twists" or "subface twists" that I mention in my 5th
and 6th videos of 6/10/2017, and was hoping to add to the move set,
actually violate permutation parity of the 4-color pieces. 
<Blush>  I was thinking of the 2^3 where pair swaps are OK. 
4-cycles of pieces in the 2^4 need to be done strictly in pairs, if
at all.   So, when you're executing I face turns on the puzzle,
don't make a quarter turn to one of the the end caps by mistake or
you'll be in a bad permutation parity state.)



Michael gives these macro equivalents that he found:

Iy (physical) == IU' RO2 IF RO2 IU RO2 IF'
RO2 IU (virtual)

IU (virtual) == Iy' Rx2 Iz Rx2 Iy Rx2 Iz' Rx2 Iy
(physical)

[I think these are right, someone please double-check :)]

*Summary*: You can simulate 2^4 moves on the physical
cube without too much trouble,  but if you want a more
comfortable puzzle that is not quite equivalent to the
2^4, but very similar and with the same level of
difficulty, this is a nice way to do it. The allowed moves
are to do any rotation to the 2x2x2 cube formed by either
the left two rows, middle two rows, or right two rows,
leaving the rest of the puzzle alone. That's it! Scramble
and try to solve.