Fisher’s Cube, solved.
Fisher’s Cube is a particularly interesting variant of the Rubik’s Cube
In your standard Rubiks Cube, you have a cube-shaped shell and a core that turns around three orthogonal axes. Importantly, the shell is aligned on the core so that the puzzle turns around the centre of each face. Thus the turning axes of the core are aligned with the 4-fold (90°) axes of symmetry of the cube, which means that rotations of the core will move the components of your shell from isometry to isometry, and your shell will always maintain its cube shape.
Fisher’s Cube, one side turned 90° — a move symmetric on the core but not on the shell.
The Fisher Cube has the same core as the Rubiks Cube, and like the Rubiks Cube the shell is a cube. What makes the Fisher Cube particularly interesting is the shell is rotated on the core, so that the axes of symmetry of the core do not match the axes of symmetry of the shell. In the Fisher Cube, only one of the core’s axes pass through opposite face centres of the shell. On that axis alone, the fourfold symmetry of the core matches the fourfold symmetry of the shell, so those two faces can be rotated through 90° without jumbling the cube. The other two axes of the core pass through the edge centres of the shell cube. Edge centres are axes of symmetry for the cube, but only two-fold. If we rotate a face around one of these axes by 180°, we move the pieces of the shell from isometry to isometry, and the cube retains its shape. But if we rotate a face around one of these axes by just 90°, we don’t reach an isometry, and the result is something that isn’t a cube any more. Thus the Fisher Cube has a nice combination of jumbling and non-jumbling operations.
Complexity
Fisher Cube, jumbled
Despite the apparent complexity of a jumbled Fisher Cube, the problem of solving it is almost identical to that of the Rubiks Cube. There are just two differences:
- On the Rubiks Cube, the face centres are a single colour, and do not need to be oriented — that is, we neither know nor care if a face centre is upside-down, as it looks the same regardless. On the Fisher Cube, some of these face centres now correspond to edge pieces, and it becomes necessary to ensure that they are placed the right way up.
- Conversely, on the Rubiks Cube all edges must be placed the right way up, but on the Fisher Cube some of these edges correspond to face centres, which need not be correctly oriented. This situation creates an interesting variation in the “endgame”. In the best known algorithm for solving the Rubiks Cube, the final task is correctly rotate two misrotated edge pieces. Or perhaps four. There will never be just one misrotated edge piece; this is impossible. On the Fisher Cube, this does occur… or it appears to occur. In fact, the underlying impossibility remains. When it appears that a Fisher Cube is solved except for one misrotated edge piece, it is because the second misrotated edge piece is a face centre, for which the misrotation cannot be observed!
Variations and related puzzles
The Master Pyramorphix — a Rubiks Cube variant that, like the Fisher Cube, has both jumbling and non-jumbling moves
As a shell variant of the Rubiks Cube, the Fisher Cube has much in common with the Rubiks Cube itself and all the many other cube variants. But in particular, the Master Pyramorphix shares with the Fisher Cube that fascinating property of having both non-jumbling and jumbling play. The Master Pyramorphix puts a Rubiks Cube axis inside a tetrahedron shell, aligning the six turning axes with the six edge centres. The edge centres are axes of symmetry for the tetrahedron, but only two-fold. Thus the Master Pyramorphix can be played with 180° turns only, in which case the shell retains its shape, or with 90° turns, in which case it jumbles.