Message #45

From: mahdeltaphi <>
Subject: Re: [MC4D] Orientations of the centre cubes …
Date: Sat, 13 Sep 2003 10:09:44 -0000

>i don’t know if you included this but i suspect there may be a 3rd
>of ambiguity found in puzzles with more than 3 cubies along an edge.
>example when there is more than one "middle" cubie along an edge they
>can all be correctly oriented yet placed in interchangeable

Certainly, as the cube gets bigger, there are more orientation
problems. The fact that the 3x3x3x3 cube is uniformly
coloured makes it sensible to talk about visible and invisible
orientation issues, but other classifications could be

If we consider the 4x4x4 cube, it turns out the the 2-face cubies
cannot have invisible orientation problems. If they
are both correctly positioned as to colour alone, then they are
either both correctly oriented or else they are both
incorrectly oriented in a visible manner. The way in which the 4x4x4
cube was constructed was to have a central
sphere, with grooves along the octants. The 1-face cubies slid along
those grooves, carrying the 2- and 3-face cubies
with them (just about - the cube tended to explode if roughly
handled!). The fact that the 2- and 3-face cubies had
to slide around a central sphere meant that they had curved backs,
and the nature of those curves made it impossible
for the 2-face cubies to exhibit invisible orientation problems.
There were, however, invisible orientation problems
for the 1-face cubies. The four such cubies for each 4x4 face of the
cube could be permuted. Once a cubie was in a
particular position, however, its orientation was fixed (one corner
of the cubie would always point to the centre of
the 4x4 face). It was reasonably simple, then, to mark the 1-face
cubies in such a manner to force a unique solution.
Although the arguments given above for these limitations to the
possible orientations are mechanical in nature, the
device’s mechanism did permit all possible cube rotations, and so I
would regard these limitations as theoretically
justifiable as well.

If we considered the 5x5x5 cube, the nine 1-face cubies on each 5x5
face would split into 3 groups - the four
corners, the four edges, and the centre. The first two groups could
have their elements permuted, and the central
element could be rotated. If we consider the 3 2-face cubies on each
edge, I would expect that the central one could
be flipped, and possibly the other two could be swapped (although I
suspect that similar considerations to those for
the 2-face cubies for the 4x4x4 cube would prevent this).