I have put this up for Geert. It show his unique and clever method for building a 2x2x4 cuboid and then, a 2x2x6... read on....

Geert Hellings' 2x2x4 and 2x2x6 cuboid modifications



A real fully operational 2*2*4 cube with uniform cubies



Already some time ago 2*2*4 cubes were proposed and made by a kind of bandaging of a 4*4*4 cube, resulting in non-uniform "cubies"of size 1*2*2.

Wayne Johnson recently also presented a possible internal mechanism for a 2*2*4 with cubies having uniform sizes for all sides.

I have made such a fully functional puzzle in a slightly different way.



As a basis I have used a regular 4*4*4 cube.

As a next step, 8 (2*2*2)-cubies can be added to the 8 centre pieces of two opposite faces of a 4*4*4.

Evidently this does not result in a functionally working 2*2*4 puzzle since rotational movements of the 4 centre pieces will be blocked within the plane of the 12 surrounding edge and corner pieces.





To overcome this, I have first rounded and truncated the centre pieces in such a way that a set of 4 centre pieces can rotate freely within the plane of the surrounding 12 pieces.

This requires both a truncation of the top parts of the centre pieces ("rounding" them) as well as truncation of the lower parts avoiding that the (8) edge pieces still block the rotational movements of the (4) centre pieces.



Adding now the 8 (2*2*2)-cubies to the 8 truncated centre pieces will result in a real 2*4*4 cube with cubies of size "2".

A first fully working prototype has been made.





Rotating the pieces of this 2*2*4 prototype results in some nice shape variations, see the pictures below.



Evidently, the same principle can be extended to two or three dimensions by adding 16 or 24 (2*2*2)-cubies to a 4*4*4 puzzle with adapted centre pieces.

Adding 16 pieces creates the shape of a cross (see below for a "schematic" side-view).

Adding 24 pieces creates the shape of the so-called "Hexcross"-puzzle, however, now with the possibility of rotating and exchanging all added parts (the original "Hexcross"-puzzle is a 2*2*2 with attached, non-movable pieces).




The 8, 16 or 24 pieces added in this way all have a "double size" (2*2*2) compared to the cubies of the regular 4*4*4.

Adding 32 non-uniform "cubies" of dimensions 1*2*2 in 8 stacks of 4 "cubies" another "cross"-shaped puzzle (of height "4") can be formed. A "schematic" top-view of this

4-layered puzzle will again look like the cross-shaped figure below.



Moreover, the same principle can be used to create 6-layered puzzles, which I will illustrate next...









The first fully operational 6-layered Cube





Well, to be honest, it is not a full 6*6*6 cube in all directions, but it certainly is a fully functioning, operational 6-layered puzzle that allows full interchangeability of (corresponding) parts.



Using the same principle as for the "2*2*4" cube with uniform (2*2*2)-"cubies" (see separate message on this website), I have added 8 non-uniform 1*2*2 "cubies" to the 8 centre pieces of two opposite planes of a regular 4*4*4 cube. Prior to this the centre pieces have been truncated in the same way as for the 2*2*4 cube, allowing rotation of 2 sets of these 4 centre pieces within the plane of their 12 surrounding pieces (8 edge pieces and 4 corner pieces).





The result is a fully operational 2*2*6 cube.

A first fully operational prototype has been built, see figures below.

To my knowledge it is the first operational "cube" with dimension 6!!

Clearly a 2*2*5 variation can easily be made in the same way.



Non-trivial "cube"-type puzzles with 6 layers (or even more layers) have been made before by adding non-movable pieces or in the case of "siamese" cubes. However, in those cases, parts from e.g. the top half of the puzzle can not be exchanged with parts of the lower half of the puzzle.

In contrary, the puzzle presented here is fully 6-layer operational in the sense that all "cubies" can be interchanged with corresponding "cubies" at both sides of the puzzle.






In the case of the 2*2*6 puzzle, this principle is applied in one dimension.

However, the same principle can also be applied in 2 or 3 dimensions.

Applying it to 3 dimensions requires all 24 centre pieces to be truncated.

Next, 24 cubies of unit size "1" can be added to these centre pieces.

A mock-up of such a puzzle is presented in the last figure below.

An actual puzzle of this type is in preparation but has not yet been made, although the principle is identical to the 2*2*6 described above.

However, the truncation of the centre pieces probably requires to be more precise in order to keep sufficient rigidity to the puzzle.





Please note that the 24 added pieces of this puzzle can all be rotated and can all be exchanged with one another.

The puzzle basically can be regarded as a 6*6*6 puzzle in 3 dimensions, although with "truncated" (eliminated) edge parts.



Geert Hellings.

great work geert!!

This is definitely the first of its kind, but dont forget daniels magic attractor invention. this enables construction of a working cuboid of any dimension. and he has made a 6x6x6 so i believe he should be credited with the first functional 6 layered cube.

Yeah, it's kind of like crediting Larry Nichols with creating the cube. He actually invented the magic attractor thing as well, but it was the mechanism that Rubik designed that credits him with making the cube.

I'd definately say Geert is the first with 6 layers of a mechanical type. I'd like to see how well Daniel's invention works in the hands eventually.

I hope you guys dont get me wrong with this! im definitely not taking any credit away from anyone here. in fact i am, as always, in awe of the creative ability of the members of this forum, and perhaps even more at someone who has made a 'true' mechanical 6 layered puzzle. i do see the difference, and in a sense something like the magic attractor can be likened to someone designing a computer program with 50+ layers. its a different thing altogether. i guess the main thing is that there is certainly a point somewhere in the distance which we will never be able to get to - and the people who get very close are the really clever ones...

oh well, its not like me to ramble on this much

in case the message wasnt clear enough earlier, BRAVO!!!!!

that using magic attractor theoretically only makes the puzzle, but practically it doesn't score much when it come to playing with it.
The Magic Attractor slides around too much with a bit too muh force here & there, making it a very meticulous to be a truly playable Twisty puzzle, mor so on a higher order cube. I can imagine making realignments & readjustments every one to two major turns. Speedcubing is also unimaginable on a magic attractor...hehe
I still consider the 6x6 cube unmade yet at this point of time, maybe the Japanese Puzzle Reconstruction Studio might be able to do it, like what they did to the Master Pyraminx
That is a practical Twisty puzzle & one personally acknowledged one.
Maybe u right though, Daniel's Magic Attractor idea is really brilliant, but just not there yet unfortunately.

for the 6 layered mechanical twisty puzzle
Bravo!!
Does she post here by the way? Would like to see all the creative inventions/mods collection that these puzzle makers have amassed thru the years.

I have studied and studied these puzzles, and I can't figure them out. I can't figure out how the 2x2x2 cubies are added in a way that the puzzle still works. Is the mechanism a 4x4x4 in the end? Geert, do you have some photos of the internals?

I suppose you are takling about the 2x2x4 puzzle.
Start with a 4x4x4. This will become the centre 2x2x2 part of the finished puzzle, so the pieces on the sides will eventually be glued together. The centre tiles of the U and D faces are rounded so that they are like a circular platform of 4 parts, which can turn independently of the U/D layers. On each of these tiles, glue a single solid cube the size of a 2x2x2. This new top /bottom layer can then turn independently.

This 2x2x4 is huge - the size of two 4x4x4's stacked together.

Jaap

I'm starting to see it. So the middle pieces of the top and bottom layer COLLECTIVELY form a circle? They then can exchange with each other.

Also are the other middles cubes glued to the edges and then matching size cube "flats" attached to them?

Would the newer "mini" 4x4x4's mech be suitable for this modification?

Dear Carter, Jaap, Jeremiah,

Thanks for the comments.
Basically Jaap is right in his explanation.
However, I need to make one addition.
Just only rounding the 4 middle pieces of the U and D plane will not be sufficient.
In that case the side pieces (2 on each side) may still block the movements of the 4 centre pieces.
You can see this by looking at a separate centre piece and a separate edge piece.
They normally fit nicely "inside" each other, but after then turning e.g. the centre pieces the edge pieces will still block the movement.
The solution to that is to also remove some material from underneath the flat plane of the centre pieces (alternatively all edge pieces can slightly be adapted, but that is more work and will probably affect the integrity of the puzzle more).
The above is probably difficult to explain without pictures or photographs.
So, the coming weekend I will make some pictures and ask Wayne to add these to the article.

> So the middle pieces of the top and bottom layer COLLECTIVELY form a circle?

Yes, each of the 8 top/bottom face centres is a quarter circle.

> Also are the other middles cubes glued to the edges and then matching size cube "flats" attached to them?

Yes, apart from the top/bottom centres, the rest of the 4x4x4 cube is glued/fused together to form a 2x2x2. You therefore have a big 2x2x2 with turnable centres on the top/bottom for the extra layers to be attached to.

You can of course do the same thing to the centres of some of the other faces as well to create other puzzles.

> Would the newer "mini" 4x4x4's mech be suitable for this modification?

I don't think so. Its centre tiles can probably not turn independently. Its mechanism is based on a simplified 5x5x5.

This is a BIG puzzle!

Photos or drawings would help a lot. I am having some trouble visualizing the necessary modifications required, even with Jaap's help. Geert, what do you do for a living? Do you normally design complex 3D interlocking objects, or is this just a hobby?