X-Cube

The X-Cube is not a cube - it isn't even a regular polyhedron. You can see this "cube" here.

At first glance, the X-Cube appears to be a subset of a 5x5x5 cube, and one might assume that techniques for solving a 5x5x5 would apply equally well to the X-Cube. However, the 4 outer faces of the cube may only be rotated when the outer face is complete. Given the cube's asymmetry, many 5x5x5 manipulations will rapidly create a state from which outer moves are impossible (but would have been fine on a normal 5x5x5 cube). Clearly another approach is necessary.

One important observation is that the X-Cube can be treated as a 3x3x3 cube with extra outer faces attached to 4 of the 6 natural faces of a cube. There are 6 "inner" moves (Front, Back, Left, Right, Up, Down) which operate on this 3x3x3 core just like a normal cube. If you restrict yourself to these inner moves and mentally ignore all the extra stuff protruding from the core, then you can work the X-Cube just like a 3x3x3 cube. It may look a bit strange in the middle of a solution, but as long as no outer moves were performed you'll be able to restore the X-Cube with your favorite 3x3x3 algorithm.

Now let's talk about the "extra stuff". In a solved X-Cube, 2 of the faces are shaped like a cross (see the illustration above), while the other four faces are 3x5 rectangles. Note that since the centers (and their colors) are fixed, the orientation of the colors is also fixed and thus the same two colors will always be cross faces, while the other four are outer faces. On my cube, Yellow and White are cross faces, while Red, Blue, Green, and Orange are outer faces.

Each corner of the core has two outer corners attached to it, and this pair of outer corners shares the same three colors. On of these colors belongs to a cross face, the other two are outer face colors. Core edges are a little more complicated. 8 of these reside on a cross face and have a single outer edge attached. We'll call these "single edges". Four of the edges lie between outer faces and are attached to 2 identically colored outer edges. We'll call these "double edges".

Notation

Orient the cube with the cross faces to the front and back. Normal cube moves will refer to rotating the cross faces (F, B) or the inner moves of the other faces (U, D, L, R). Outer moves will use lower case letters (u, d, l, r). One center slice operations is also somewhat useful: H will stand for rotating the horizontal middle layer clockwise when viewed from above. This carries the right center to front, front to left, left to back, and back to right. Note that this operation is the only one that changes the orientation of the fixed centers.

It is often useful to refer to a core piece and its associated outer pieces as a single group. When the group is referred to as X, then the core piece itself is X0, and the outer piece(s) are X1 and X2.

The notation X/Y refers to swapping pieces X and Y. Similarly, X/Y/Z is a cycle that carries X to Y, Y to Z, and Z back to X.

Solution Overview

1. Solve the "shape" of the puzzle while ignoring color. The goal is to get it back into the basic cross shape so that the four outer moves are possible. During this phase use only inner moves, and it is helpful to think in terms of the 3x3x3 core. Use edge swaps to get the four double edges into position. Then use edge flips and corner rotations to smooth out the cross faces. It is possible to reach a state where a single edge flip is required on the cross faces. Since edge flips must occur in pairs, flip the appropriate single edge and also flip one of the double edges (this won't change the shape at all).
2. Use outer edge 3-cycles to match up the four sets of double edges. For example: (R'HR u R'H'R u') will cycle rD/uR/uB. Similar cycles can be performed by starting with L instead of R', using H' instead of H, or u2 or u' instead of u. All of these combinations cycle something in either the right or left outer face with two edges in the top face. Once all four double edges are matched up, they can be used as a quick visual check that outer moves have been restored properly during the rest of the solution.
   1. Match up the remaining 8 outer edges with an appropriately colored edge. The same 3-cycles used in step 2 should work here as well.
   2. Use outer corner 3-cycles to match up pairs of outer corners along with an appropriately colored corner. One such cycle is (R'D'R u R'DR u'). At the completion of this step all of the outer pieces should be matched up with a suitable core piece.
3. Using only inner moves, solve the core 3x3x3 using your favorite algorithm. The only caveat is that due to parity problems it is possible to reach states that are not solvable on a 3x3x3 cube, such as requiring a single edge flip or edge swap. These can occur because some permutations of outer corners or edges are equivalent to edge flips or swaps.
   • If you are left with a single edge flip F and a swap of single edges G and H: Pick a double edge D and perform the a double swap of outer edges D1/D2 and G1/H1. This will solve G and H and leave you with both D and F requiring flips, which is a legal 3x3x3 state.
   • If you are left with a single corner swap of A and B: perform a double outer corner swap of A1/B1 and A2/B2, which will solve A and B.
   • If you are left with a single edge swap then you can reduce it to a single corner swap by making any inner move, which will then result in an odd permutation for the corners and an even permutation of edges. Restore the edges, then all but one of the corners, and then fix the single corner swap as detailed above.
   • A lone edge flip is the most tedious problem to solve. Fix the flip with a double outer edge swap as detailed above, which will in turn create an extra edge swap. Then fix the odd edge permutation with normal moves at the expense of creating an odd corner permutation. Finally use a double outer corner swap to deal with the final corner swap.