Message #3779

From: qqwref@yahoo.com
Subject: Virtual vs Physical 2^4 Correspondence
Date: Wed, 02 Aug 2017 02:27:05 +0000

Marc Ringuette sent me a physical 2^4 recently (thank you SO much!) and I’ve been playing around with it a bit, trying to get a good feel for how it relates to the virtual, theoretical version of the puzzle.

Here’s the notation I’ll use for 2^4 moves. I call the eight faces I (in), O (out), and the standard six 3D face names F, R, U, B, L, D. When discussing the virtual 2^4 cube, I’ll label moves as something like "FR", which means rotating the F(ront) face 90 degrees through an axis centered at the R(ight) face, clockwise as if you were looking from the R face. For physical 2^4 turns, I’ll instead use a face name followed by x, y, or z, which follow the speedcubing notation, of rotating the cube 90 degrees clockwise relative to the right, top, or front face respectively. So "Ry" means rotating the R(ight) cube clockwise relative to the top of it. A move followed by 2 (e.g. FR2) means doing it twice, and followed by ‘ (e.g. FR’) means doing it counterclockwise instead of clockwise.

First off, the rotation moves (which I think he calls FOro and FUro?) he’s shown are very long, and I was hoping to find a more efficient way to do them. But I actually felt like it might be more useful to just find a better way to do single moves of the I face. This will allow us to do arbitrary turns of R, I, and L, and since R and L turns together are actually cube rotations that can place any face in the I slot, this is enough to do any moves on the puzzle without the full rotations FOro and FUro.

Moves of R and L just involve arbitrary rotations of the right or left half of the physical puzzle; and you can do an IR move by just rotating the middle half of the physical puzzle around the long axis of the puzzle, that is, Ix. So, how do you do an IU or IF move? To do an IU move, I found out that you can take the leftmost and rightmost layers off the puzzle, and then apply the following moves to the middle half *as if it was a 2^3 puzzle*:
F2 U2 R2 F B
Note that for every move you’ll have to take half of the cube off, rotate it, and put it back on. After this, place the rightmost and leftmost layers back on. In this notation a letter means "rotate this half of the cube 90 degrees clockwise", and with a 2 after it, do the move twice, that is, 180 degrees. You can verify that this does indeed correspond to an IU move on the physical cube. Similarly, for an IF move, in the same fashion, apply the following 2^3 sequence:
U2 F2 L2 U D

That’s not that bad, right? Using those two sequences (plus IR) you can apply any moves to the I face. Combined with the R and L moves, that’s enough to play around with any 2^4 sequences on the real cube: if you want to do a move on one of the other 5 faces, use R and L moves as appropriate to rotate it into the I position, do your moves, and reverse the R and L moves.

But… that’s still a bit complex for casual play. How about just allowing arbitrary rotations of the right half (R), left half (L), and middle half (I) of the physical cube?

It turns out that this is very comparable to the above constrained 2^4 (where we only allowed R, I, and L rotations). In fact, every move on the 2^4 is possible with these moves, and vice versa. The equivalence of the R and L moves (e.g. Ry = RU) is already pretty clear. As for the I moves, it’s pretty clear that if we have two of the three we can generate any possible move, so we just have to be able to do the equivalent of Ix and Iy on the virtual cube, or the equivalent of IU and IR on the real cube. Here’s how:
Ix (physical) == IR (virtual)
Iy (physical) == IU’ RO2 IF RO2 IU RO2 IF’ RO2 IU (virtual)
IU (virtual) == Iy’ Rx2 Iz Rx2 Iy Rx2 Iz’ Rx2 Iy (real)
[I think these are right, someone please double-check :)]

If you have one of these physical cubes and want to try it out, I wrote a quick scrambler for it. Load up http://mzrg.com/qqtimer/index%20-%202222.htm and select 2x2x2x2 (under "specialty scrambles") from the dropdown. It will give you a 25-move scramble in the RIL/xyz notation.

*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.

Enjoy,
Michael Gottlieb