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 Post subject: Has anyone made 2x2x4, 2x2x6, 3x3x5, etc?Posted: Thu May 23, 2002 1:17 am
The basic idea for 2x2x4 is to start with a 2x2x2 with a small core, and drill cylindrical holes in the centers of opposite faces (this results in all 8 octants being affected). These holes are then given a groove on the inside to catch lips. The outside pieces are solid cubies each with a quarter cylinder extention to fit into those cylindrical holes. Those extentions are given lips to engage the grooves.

This puzzle is a fully working 2x2x4, but unlike the 2x2x3 or 2x3x3 it retains the ability to handle 90 degree rotations. It will morph into unusual multi-cube shapes just like a glued 2x2x4, but whenever a face shares 4 cubie extensions, the extended face can rotate also!

This concept is extended to 2x2x6 in a straightforward manner by using concentric trunk extensions. The outer layer's trunk extensions must extend all the way to the base 2x2x2 in order for those pieces to stay firmly attached. Unfortunately, these extensions are visible when the puzzle is in a morphed shape, harming the aesthetics.

With multiple concentric trunk extensions, 2x2xn is possible for any even n. (For odd n, start with a 2x2x1.)

A similar idea will work for 3x3x5, although it will be much more challenging to build. The extensions of the face cubies is straightforward enough--add on a cubie with a swivel mount. The extensions of the corners and edges is the challenging part. Instead of a small solid cylindrical trunk, a hollow trunk with an inner radius greater than sqrt(1/2) and an outer radius less than 1.5 is needed. A smaller radius is preferable, with a depth of almost an entire cubie. The shape of the extensions will have to match the cutout portions in order to not interfere with the 3x3x3's rotation (the edge pieces narrow just under the surface to accomidate the corners; the corner pieces widen just under the surface). As with the 2x2x4 modification, a lip on the extensions will engage a groove to keep the pieces in place. It is important that the lip depth is sufficient to prevent the lip from being exposed in mid-rotation. If a lip is fully exposed, then its cubie may slip off of the puzzle in mid-rotation.

This puzzle can morph in shape just like a glued 3x3x5. However, if a face has extended cubies on all 8 of the edge and corner squares, those 8 can rotate in unison as a face. (This is possible whether or not the central cubie extension is present.) Also, the central cubie extensions can trivially rotate if not obstructed by neighboring edge extensions.

The principle of concentric trunks allows 3x3xn, but these would be extremely challenging to build and probably too delicate to be worth the effort.

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 Post subject: some remarksPosted: Thu May 23, 2002 1:31 am

Joined: Mon Dec 10, 2001 2:15 am
Location: Amsterdam, The Netherlands
a 2x2x4, 3x3x5 is impossible to make with same size cubies, This beacuse after a 90 degree turn the cubies will "hang" outside the cube.

So any N x N x N+2 will not work with same size cubies using a mechanical solution.

So the current bandages version of a 4x4x4 to make a 2x2x4 is probably the best and simplest solution.

Even so for a Bandage 5x5x5 to make a working 3x3x5.

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 Post subject: Of course they're possible!Posted: Thu May 23, 2002 6:10 am
I posted the necessary mechanism to prevent the outer cubies from falling out. The trunk extensions into the core 2x2x2 keep the outer cubies locked in place relative to the underlying cubie at all times unless aligned with 3 other extended cubies (in which case the 4 trunks form a single lipped cylinder which can freely rotate).

The 4 extended cubies are essentially like a group of 4 knob pieces in a Tricky Discy.

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 Post subject: cube extensionsPosted: Thu May 23, 2002 9:06 pm

Joined: Wed Mar 15, 2000 9:11 pm
Location: Delft, the Netherlands
The single layer extension is actually an old idea, and was patented; see US4593907 on Joshua Bell's patents page http://www.calormen.com/TwistyPuzzles/twisty.htm

I have not seen the idea before of extending to several layers using concentric 'trunks'. It seems to me however that not only will that be aesthetically less pleasing, it will also be very fragile unless you extend all faces in that way. This is because when it is mixed using quarter turns, the outermost cubie will be held only by a long thin trunk which is completely exposed and therefore easily broken. Unless of course you put a groove in the cubie it is adjacent to, so that it is properly attached to that also.

Jaap

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 Post subject: He's right it is posiblePosted: Fri May 24, 2002 6:10 am
It is possible actualy fairly simple. I do not think that there is anything original to a 2x2xN if N is an odd number greater than 3 or an even one greater than 4 because all of the extention piecse would act the same. The same hols true for an 3x3xN with N greater than 2 if even, or 5 if odd.

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 Post subject: Some more remarksPosted: Fri May 24, 2002 11:16 am
If you look at the corner moves of a extended 3x3x5. Any mechanisme with croves makes the corners act instable turnning it 20 degrees. In this situation a corner will only be suported by two faces and a twist off the corner and two other pieces is possible. This will make the cube instable.

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 Post subject: 2x2x6 comments...Posted: Fri May 24, 2002 9:06 pm
Well, there really isn't much point for a 2x2x6 from a pure intellectual standpoint anyway, since solving the outer extensions is essentially the same as the middle extensions.

However, the situation can be different with more complex shapes or irregularly cut extensions. For example, with Square 1 style extensions, neighboring faces cannot be extended without interfering with each other's rotation. Neighboring faces extended two levels can have the outermost layers rotate without interfering with each other.

Things can potentially be pretty interesting if some cubies are extended one level while others are extended two levels. The restrictions on rotation are interesting. If you have a face where all four cubies are extended one level but only some are extended two levels, then it's still possible to rotate the extended face but it's not possible to rotate the outer level independently.

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 Post subject: a 3x3x5 with a 5x5x5 kernelPosted: Sat May 25, 2002 1:12 am

Joined: Mon Dec 10, 2001 2:15 am
Location: Amsterdam, The Netherlands
I think if you use a 5x5x5 as kernel you can make a 3x3x5 with same size cubies.

The idea is just like the 2x2x3 that has a 3x3x3 kernel. The 5x5x5 kernel is bandaged to function as a 3x3x5. You can use 3x3x3 cubies. You must cut from 4 side half of every piece, and this leaves enough support to make a working 3x3x5 with same size cubies as a 3x3x3.

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