Message #3781

From: Marc Ringuette <ringuette@solarmirror.com>
Subject: Physical 2^4 puzzle: the ROIL moveset
Date: Wed, 09 Aug 2017 13:57:07 -0700

Hi, physical 2^4 puzzlers,

Extending Michael Gottlieb’s RIL idea from last week, I present to you
… the ROIL moveset.

I think ROIL will be a very nice and useful way of manipulating the
puzzle. Also, I’ve found a straightforward version of FOro, our key
full-puzzle rotation, that fits right into the ROIL scheme. Somebody
will need to fill in a bit of theoretical work to systematize all of
this a bit better, but I think it’s a winner.

I’ve made five videos, the first three of which are less than 5 minutes
total and give the high points of ROIL. I’d recommend having a look
and skimming or skipping the rest of the explanatory text below.

12 ROIL moveset intro 2m14s https://youtu.be/oYeh5fK8LCI
13 ROIL version of FOro 1m10s https://youtu.be/3gvrda5fMto
14 ROIL scrambling 1m17s https://youtu.be/ymnvMzypUp8
15 ROIL strict subset vs macro equivalents 3m37s
https://youtu.be/Iu7MKzD4yq0
16 ROIL meaning of junctions out positions 2m54s
https://youtu.be/xN_vn-6RnV8

For those who don’t wish to watch the videos, the main idea is: starting
at the usual home position for the puzzle, shift the 4-piece right end
cap all the way to the left, and then take apart the puzzle into its new
left and right subcubes. You are now holding all pieces of the In face
in your right hand, and the Out face in your left hand. Perform any 3D
rotation on either or both of the I and O faces, trying not to freak out
about the fact that you are looking at the puzzle in an inside-out
"junctions out" projection. Put the subcubes back together, and shift
the left end cap to the right, to return to the normal "corners out"
projection. Continue as normal. Any I and O face turns you have just
done will either be one-click MC4D moves, or macros of around length 9
(in the cases of the I and O face moves, such as Ix and Oz, that make an
odd number of 90 degree turns around the x or the z axis).

Michael already gave us the key two macro sequences last week:

> Iy (physical) == IU’ RO2 IF RO2 IU RO2 IF’ RO2 IU (virtual)
>
> IU (virtual) == Iy’ Rx2 Iz Rx2 Iy Rx2 Iz’ Rx2 Iy (physical)
>

With these, and some (admittedly fairly major) bookkeeping work but no
new research, we should be able to complete a proof that the set of ROIL
states is identical to the set of virtual 2^4 states, by providing
macros in each puzzle for all moves in the "other" puzzle, so that each
puzzle can "follow along" with any moves made on the other puzzle.

Or, if you prefer to solve the physical 2^4 puzzle using a strict MC4D
subset – only moves that have one-click equivalents in MC4D – you can
add only the new FOro physical "macro" sequence, as a unitary move, to
the common-subset moves that are identical and length-1 in both puzzles.

   FOro == Iy Oy' RO2 BO2 UO2 RO2

You’ll note that this FOro sequence begins with a pair of opposite I and
O moves, a rotation of sorts, followed by four cleanup moves.

If you’re concerned about my slab moves UO2, DO2, FO2, BO2, a set of
strict ROIL substitutions are:
UO2 == Rz’ Lz Iy2 Rz Lz’
DO2 == Rz Lz’ Iy2 Rz’ Lz
FO2 == Ry Ly’ Iz2 Ry’ Ly
BO2 == Ry’ Ly Iz2 Ry Ly’
and of course RO2 == Rx2. Sorry about the still-unsettled mix of
virtual and physical notations.

Cheers
Marc