Message #3478

From: carl.n.hoff@gmail.com
Subject: How to make a better 3x3x3x3
Date: Sat, 23 Jul 2016 12:30:02 -0700

Better here is very subjective. I just really like playing with Super http://twistypedia.oskarvandeventer.nl/index.php/Super-(puzzle_name) and Multi http://twistypedia.oskarvandeventer.nl/index.php/Multi-(puzzle_name) puzzles and the current 3x3x3x3 is neither.

Let’s take a quick look at the current 3x3x3x3.
http://wwwmwww.com/3x3x3x3/3x3x3x3.gif http://wwwmwww.com/3x3x3x3/3x3x3x3.gif
There is an excellent video that explains it very well here that I’d highly recommend.
https://youtu.be/yhPH1369OWc https://youtu.be/yhPH1369OWc
So here I will just cover some of the basics.

This puzzle exists in 4D space. In addition to the x, y, and z directions we also have the w direction.

The positive X direction is what we call Right and I’ll color stickers on that face/cell Red.
The negative X direction is what we call Left and I’ll color stickers on that face/cell Orange.

The positive Y direction is what we call Forward and I’ll color the stickers on that face/cell Blue.
The negative Y direction is what we call Backward and I’ll color the stickers on that face/cell Green.

The positive Z direction is what we call Up and I’ll color the stickers on that face/cell Yellow.
The negative Z direction is what we call Down and I’ll color the stickers on that face/cell White.

The positive W direction is called the Ana direction and I’ll color the stickers on that face/cell Magenta.
The negative W direction is called the Kata direction and I’ll color the stickers on that face/cell Cyan.

The cell with the White stickers is missing from this image and is the 8th cell. It can be placed below Down, above Up, to the right of Right, to the left of Left, in front of Front, or behind the Back. In truth it is actually in all these locations at the same time.

So for a 3x3x3x3 we would expect to find 81 different hypercubies. In this post http://www.twistypuzzles.com/forum/viewtopic.php?p=351831#p351831 Brandon discussed them all. But suffice it to say we have:

8 pieces which correspond to cell centers and they are only marked with a single sticker.
24 pieces which correspond to cell faces and they are marked with 2 stickers.
32 pieces which correspond to cell edges and they are marked with 3 stickers.
16 pieces which correspond to cell corners and they are marked with 4 stickers.

This comes to 80 pieces and we were expecting 81. So we are missing one. So this is NOT a Multi puzzle. And where is the missing piece?

Let’s check to see if its a Super puzzle. In a Super 3x3x3x3 each piece would need a fixed position and orientation in the solved state.

Each cell center can reach 24 possible orientations. With just a single sticker we cannot distinguish any of these orientations. So that alone means this is not a Super puzzle. But let’s also look at the cell faces. These contain just 2 stickers which are from neighboring cells. These 2 stickers contain enough orientation information to reduce the ambiguity in the possible orientations by a factor of 6 but that still leaves 4 possible orientations that it could be in and still appear solved. So the cell faces also keep this from being a Super 3x3x3x3.

Both the cell edges and the cell corners already contain enough information to give them a unique position and orientation in the solved state.

Can we do better? Can we make a Super Multi 3x3x3x3?

Well before we answer that let’s see what we need to do to make a Super Multi 3x3x3.

Here is your normal 3x3x3.
http://wwwmwww.com/3x3x3x3/Rubiks.png http://wwwmwww.com/3x3x3x3/Rubiks.png
And let’s look at an exploded view.
http://wwwmwww.com/3x3x3x3/RubiksExp.png http://wwwmwww.com/3x3x3x3/RubiksExp.png

This view allows us to see all 3x3x3 or 27 cubies of the 3x3x3. As was the case with the 3x3x3x3 there is one that is missing any stickers. This is the core cubie and keeps the normal 3x3x3 from being a Multi puzzle. Also the face cubies are only marked with a single sticker and this does not provide enough information to distinguish between its 4 possible orientations. So the normal 3x3x3 is also not a Super puzzle.

But we do know a way to make a Super Multi (aka Real) 3x3x3. We call it the Circle Cube. So let’s examine how it pulled that off.
http://wwwmwww.com/3x3x3x3/CircleCube.gif http://wwwmwww.com/3x3x3x3/CircleCube.gif
Now let’s look at an exploded view of this puzzle.
http://wwwmwww.com/3x3x3x3/CircleCubeExp.png http://wwwmwww.com/3x3x3x3/CircleCubeExp.png
Here we see how all the stickers map to the 27 cubies. Let’s look at them in turn.

Here is the core cubie. It now has stickers on ALL 6 faces.
http://wwwmwww.com/3x3x3x3/CircleCubeCore.png http://wwwmwww.com/3x3x3x3/CircleCubeCore.png

Here is a face cubie. This now has stickers on 4 of its 6 faces and now contains the information needed for solving its orientation.
http://wwwmwww.com/3x3x3x3/CircleCubeFace.png http://wwwmwww.com/3x3x3x3/CircleCubeFace.png

Here is the edge cubie. It now has stickers on 4 of its 6 faces as well.
http://wwwmwww.com/3x3x3x3/CircleCubeEdge.png http://wwwmwww.com/3x3x3x3/CircleCubeEdge.png

And here is the corner cubie. It has maintained the 3 stickers that it originally had.
http://wwwmwww.com/3x3x3x3/CircleCubeCorner.png http://wwwmwww.com/3x3x3x3/CircleCubeCorner.png

So… we know its possible to make a Super Multi 3x3x3x3. Can we do the same thing to the 3x3x3x3? If so how?

Well the answer is yes… and I’ll show you how. Instead of adding circular cuts to the faces, we need to add spherical cuts to the cells. And this is what the Super Multi 3x3x3x3 (or Sphere HyperCube) looks like.
http://wwwmwww.com/3x3x3x3/Sphere3x3x3x3.gif http://wwwmwww.com/3x3x3x3/Sphere3x3x3x3.gif
Just as the circle is fixed in the Circle Cube while the rest of the layer rotates about it, the sphere is fixed while the rest of the cell is rotated about it.

And just as you can interpret the circles on the faces of a Circle Cube as windows that allow you the see the layer below, the same is true for these spheres on the cells of the 3x3x3x3. We can now look though these windows and see many of the previously unstickered cubes« of the 81 hyper-cubies that make up the 3x3x3x3.

So now let’s look at the individual piece types.

Here is the Core of the 3x3x3x3. It is totally unseen in the standard picture but it now has all 8 cubes stickered. The White sticker is in the 8th cell off screen.
http://wwwmwww.com/3x3x3x3/Sphere3x3x3x3Core.png http://wwwmwww.com/3x3x3x3/Sphere3x3x3x3Core.png

Here is a Cell Center. It now has 6 of the 8 possible stickers and now has a unique position and orientation in the solved state. Just was with the Face Centers on the Circle Cube, the Cell Centers do not have a sticker in the cell they represent. And the other missing sticker is from the opposite cell.
http://wwwmwww.com/3x3x3x3/Sphere3x3x3x3CellCenter.png http://wwwmwww.com/3x3x3x3/Sphere3x3x3x3CellCenter.png

Here is the Cell Face. It too now has 6 of the 8 possible stickers and has a unique position and orientation in the solved state.
http://wwwmwww.com/3x3x3x3/Sphere3x3x3x3CellFace.png http://wwwmwww.com/3x3x3x3/Sphere3x3x3x3CellFace.png

Here is a Cell Edge. It goes from having 3 stickers to having 5 of its 8 possible stickers.
http://wwwmwww.com/3x3x3x3/Sphere3x3x3x3CellEdge.png http://wwwmwww.com/3x3x3x3/Sphere3x3x3x3CellEdge.png

And finally here is the Cell Corner. It gains no new stickers but it already had enough to give it a unique position and orientation in the solved state.
http://wwwmwww.com/3x3x3x3/Sphere3x3x3x3CellCorner.png http://wwwmwww.com/3x3x3x3/Sphere3x3x3x3CellCorner.png

So we have now made the Super Multi (aka Real) 3x3x3x3.

You may say that is an awful lot of work to expose a core that can never be scrambled anyways. And yes… that is true for the 3x3x3x3. But this method has implications that go FAR beyond the 3x3x3x3 itself. We can now make a Sphere 4x4x4x4 and not only would it allow you to solve the 2x2x2x2 on the inside but it would also be able to give the 2-sticker face pieces a unique position as well. So you can make a Super Multi 4x4x4x4. I believe you can make a Double Sphere 5x5x5x5 which would map to the Super Multi 5x5x5x5 and if so that could in turn be used to make the Complex 3x3x3x3. And you can even take this in different directions. If we allow some of the spheres to rotate with their cell but not all we will then have the Crazy Plus 3x3x3x3’s that we can play with. This enables all sorts of possibilities…

Now to see if I can figure out how to post this to the MC4D Yahoo Group…

Enjoy,
Carl

« I was tempted to say "cells" here but cell is already being used for the 8 faces/cells of the 3x3x3x3. I didn’t think "faces" was appropriate as that implies 2D squares and the faces of a hyper-cube are cubes. So I have used the term "cubes" but "the previously unstickered cubes of the hyper-cubies" sounds very odd to me. Is there a better term? In 3D I would say sides of the cubies. Is "sides" a better term to use here as well?