Interdimensional Cubes

As a thought experiment consider the case of the familar 4x4x4 cube with a 2x2x2 cube embedded inside it, instead of the usual mechanism. I'll call this the "Interdimensional 4x4x4 cube" for lack of a better name. Now clearly if we turn the slices of the 4x4x4 cube it would have an effect on the internal 2x2x2 cube. Now moving the slice adjacent to the U face and moving the slice adjacent to the R face this would be the equivalent of turning the internal 2x2x2's U face and R face.

My question is: Is it possible to reach all the positions of the internal 2x2x2 without having any constraints on the 4x4x4 cube? How many positions are there?

Clearly it is possible to manipulate the internal 2x2x2 cube without touching the 4x4x4 cubes' corners, but what about the centre pieces and edge pieces of the 4x4x4?

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Read the bit about 'eviscerat

Read the bit about 'evisceration' in Singmaster's Cubic Circular:

There are many well known move sequences on the 3x3x3 that affect the corners of a cube without affecting the edges or even the orientation of the face centres. This sequence applied to a 4x4x4 will have the same effect on its corners without disturbing the centres or edges (or invisible centre). Any such sequence applied to the inner slices of the 4x4x4 cube however, will affect only the invisible inner cube in that way.

There is a parity constraint. Every slice move on the 4x4x4 cube is an odd permutation on the invisible inner cube, and an odd permutation on the edges. Therefore any odd permutation on the inner cube will be an odd permutation on the edges, and so will have to disturb some edges.

For each particular position of the outside layers, the inner cube therefore has 8!/2 * 3^7 = 44089920 possible positions. This puzzle has 44089920 times as many positions as the normal 4x4x4 cube.

Jaap's Puzzle Page:

Evisceration, interesting concept

I remember seeing this idea in the late 80's but never used it.

I've tried to eviscerate some processes and was able to move a ring of 4 edges adjacent to the F face. An eviscerated process that would rotate the U centre and F centre on a standard 3x3x3 cube would move the corresponding ring of edges adjacent to U and F on the 4x4x4. Thus a clockwise rotation of the U centre would end up causing a clockwise rotation of the 4 edge ring.

I'll like to hack up some software to help to see the invisible interior 2x2x2 while manipulating the 4x4x4. Maybe it could have partly translucent colours or just small squares of the 2x2x2 appearing over the big squares of the 4x4x4.

More thoughts

I see. Since we are talking about half of the positions of the interior 2x2x2 (8!/2 * 3^7) inside the 4x4x4 there is no division by 24.

So 88,179,840 / 2 = 44,089,920

So the "interdimensional" 4x4x4 would have:

44,089,920 * 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000


326318176648849198250599213408124182588293120000000000 or
3.263 x 10^53