Under Construction

Many years ago, I was in London for a h2g2 Meet. I went to the puzzle and game shop that was in the street opposite the British Museum. (It has since closed down.) I met another h2g2 Researcher there. I think it was Solsbury Phil. I bought a 4x4x4 Rubik's Cube, and told him that I thought it would take me about a month to solve it. Years later, I still haven't fully solved the puzzle. I'm going on my holidays in a week or so; I'm going to bring the cube along and see can I actually solve it.

Eventually I gave in and looked at a published solution on the web, but I quickly realised that the person who claimed to have solved it didn't understand it as well as I did - he had a solution that would work only one time in twelve. Obviously he'd been lucky and had hit the solution, and then had thought that it would work every time.

So I still don't know a general solution and would really like to solve it myself. I solved the original cube in 19 days, an achievement I'm still proud of. I'm a bit older now (more than 40 years older), but I should still be able to use my brain.

The Rubik's Cube was one of the ubiquitous puzzle toys of the 1980s. Millions of them were sold and many people knew how to solve them. Even today there are still people who specialise in speed-solving the Cube, and new records are being set that are way faster than those set in the 1980s.

Rubik also produced a bigger cube which looked similar but was 4×4×4. That is, it had 4 cubes along each side, making 16 'cubies' (little cubes) on each face. This was called the Rubik's Revenge.

I'm working on a solution of this, and have made some progress. I'll record here methods that I have developed to solve this.

Notation

It is still possible to turn each of the faces, leaving the rest of the cube untouched. I call these 'face turns', and the existing notation of UDFBLR can be used to describe them. (See the entry on the Rubik's Cube for an explanation of these.)

Another way you can turn the cube is to split it down the middle, turning one half of the cube relative to the other. I call this a 'central split'. In effect, if you imagine the cube as four slices of 4x4 cubies each, you keep two slices stationary and turn the other two together. I'll use the notation of lowercase greek letters υ δ φ β λ ρ for these, corresponding to the equivalent face turns of U D F B L R. You may find this confusing if you're not familiar with Greek letters - I don't.

Other ways of turning the cube such as sliding one of the inner slices while keeping the outer slices fixed can be done by a combination of face turns and central splits, so there's no need to have any special notation for them.

Identifying the Faces

In a standard Rubik's cube, the centre of each face can turn, but it can't move from its location. The six centres stay put. This means that the colours of the centres are all located in the correct orientations relative to each other.

In the 4x4x4, on the other hand, any of the visible cubies can move:

• Any face cubie (which has only one face visible) can be moved to any of the 24 face cubie locations. (The corner of the cubie that is at the centre of the cube face will always be at the centre of a face.)

• Any edge cubie (with two faces visible) can be moved to any of the 24 edge cubie locations, but its orientation can not be changed - it is determined by its position. If you "flip" an edge, you change two cubies - their orientations flip, but they also swap positions with each other.

• Any corner cubie (with three faces visible) can be moved to any of the 8 corners in any of three orientations.

This means that you might sort out the central face cubies so that each face had four face cubies of the same colour, but might choose an arrangement of colours that doesn't allow the cube to be solved. For example, if the final solution has red on the oppositie side to white, then there's no point in putting the red face cubies on the side next to the side with the white face cubies.

The first task in solving the 4x4x4 cube is to identify which colour goes on each face. Pick one colour as the top. Find the four corner cubies that have this colour to identify what colours the fours sides should be, and in what arrangement. The sixth colour which has not appeared on any of these corner cubies will be the colour of the bottom face.

My cube has the red face opposite the purple face, the blue opposite the green and the yellow opposite the white. There are two possible ways of arranging these, but I can't remember which my cube uses.

Approach

There are presumably a number of possible approaches to solving the cube. The one I'm going to use is this:

• Fix up the face cubies on each face to be the correct colours.
• Fix up the edge cubies so that each edge cubie in a side by side pair is the same colour. There's no need at this stage to match these two-edge-cubie groups to the face colours.
• Treat the cube as if it were a 3x3x3 cube with a much thicker middle slice, and solve the cube by normal methods using only face turns. Obviously you must be able to solve a standard cube to do this.

The problem with this approach, which I have not yet solved, is that we usually (11 times out of 12? 3 times out of 4?) end up with situations which are not possible on a standard cube. For example, it is possible to end up with all the edge-cubie pairs in the correct positions, but with one flipped over while all the others are in the correct orientation. The equivalent could not happen on a standard cube so there is no sequence using only face turns that can solve the cube from this position.

If I could come up with a sequence of moves that flipped one pair of edge tiles without affecting anything else, I could solve for this problem.

Thinking about this a little more, I see that the corner cubies on this cube are exactly the same as on a standard 3x3x3 cube. So it is never possible to get them into a situation that is not possible on a standard cube, such as:

• One corner cubie rotated while others are all correct.
• Two corner cubies rotated by 1/3 in the same direction while others are all correct.

There appear to be two "impossible" situations, that is, situations which would be impossible on a standard cube:

• One edge flipped
• Two edges swapped

This suggests that the solving approach outlined above will give the correct solution one time in 4 and an "impossible" solution three times in four.

Fixing the Faces

This is reasonably straightforward and can be done using only a central slice ρ and ρ'.

Ignore edge and corner cubies completely until the faces are sorted. Gather face cubies together in groups of two of the same colour. Look at up, down, back and front faces. If any group crosses the central plane, turn that face through 90 degrees. Do a ρ move. Repeat until all the face cubies are in groups of 2.

Now use a similar move to unite them into groups of four.

Finally, swap the faces around until they are in the right relative positions. Use this:

[ρU² ρ'][ρ'F²ρ]

This should swap the front and up faces.

Getting the Edges into Groups of Two

I've been experimenting with this move:

[L²ρULU'ρ']

This swaps around the FU, FL and FD edges, so that one cubie from each goes into another edge. By careful positioning of the edges in advance of this move, you can sort out all the edges.

Solving the cube using 3x3x3 methods

By only doing face turns, we can sort the cube out without disturbing the four face cubies on each face, and without splitting up the 2-cubie edge groups.

Sort out the top three slices using simple techniques. Then turn cube over and sort the top face using these moves:

• [LFUF'U'L'] - position edges
• [LFUF'U'L'][R'F'U'FUR] - flip edges
• [L'URU'][LUR'U'] - position corners
• [LD'L'][R'D'R] U [R'DR][LDL'] U' - twist corners

Sorting it Out when you arrive at the impossible position

Don't have any ideas on this one yet. I think it might be possible to sort using a variation of the edge splitter/joiner move given earlier.

I've a feeling that the four 'face cubies' can be put together in a number of different ways, which all look the same, but have a knock-on effect on the edges and corners. If there were some way to identify the individual face cubies, such as if there were a picture on each face, I think the cube would be easier solvable.

Labelling the face cubies

If I put a circular spot at the centre of a face, each face cubie will have one quarter of this spot at its central corner. Face turns don't affect the face cubies; central slices will split up the face cubies of a face, but note that the spot corners still stay in the centre. This indicates that although the face cubies can move around the cube, they always keep the same corner pointing towards the centre of the face.

If we label the four face cubies of a particular side as A, B, C, D, then there are six possible arrangements:

Obviously there are arrangements which don't have A at the top left corner, but these can easily be turned to produce one of the above six.

We don't need to worry about which way the face cubies are oriented on the face, as they will always have their "spot" corner towards the face centre.

Are all six of these actually achievable?

Consider the following sequence:

[ρ²φ] [UR'DL'] [φ'λ²]

(check this against notes

It splits up the face cubies of the top face, putting them on four separate faces, then moves them around those faces and recombines them onto the top face. If we track the individual face cubies we find it does in fact rearrange them. It goes from:

to:

So at least two arrangements are possible.

Now consider what happens to the face cubies and the corners when we work on the cube using only central slice moves (ρ λ υ δ φ β). Each face cubie will be attached to a corner cubie and will stay in the same position relative to that corner cubie.

We can carry out a sequence of moves which positions the corner cubies of the top face in any position, and can then do another sequence which rotates them to the correct orientation. This will also move the top face's face cubies into any position you like. So all six arrangements of face cubies are possible.

Speculation

If the face cubies are on the right faces but in the wrong positions within the face, it won't be possible to solve the edges. (The corners can be always be solved because they are not affected by the central slices.)

It's certainly true that if I take a fully solved cube and shuffle the face cubies on the top face using the sequence given above, then applying my full solving method, I eventually get to a position with two edges in the wrong place and everything else correct. This can't be solved using standard 3x3x3 methods.

But there's no way of knowing what way the faces cubies should be unless I mark them. What I need is a simple sequence which will jumble the face cubies while leaving the edges and corners untouched.

To make further progress, I think I'll need to look at various moves that manipulate the faces.