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 modificationsA real fully operational 2*2*4 cube with uniform cubiesAlready 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 CubeWell, 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.