# NxNxNxN¶

NxNxNxN, or N^{4} is a generic term for a 4-dimensional twisty puzzle in the shape of a hypercube with N layers per axis. It is a direct higher dimensional analogy of the NxNxN Rubik’s Cube.

This page is concerned with the case where N is greater than 3. The 2x2x2x2 and 3x3x3x3 have their own pages.

## Pieces¶

The N^{4} has \(N^4 - (N-2)^4\) hypercubies. If N is even, all hypercubies are movable, and if N is odd, all but 8 are movable. It has \(8(N-2)^3\) 1c, \(24(N-2)^2\) 2c, \(32(N-2)\) 3c, and \(16\) 4c pieces. These pieces come in many subtypes.

- 1c
- Centers: These pieces are at the centers of the facets. When \(N \geq 3\) is odd, 8 of these pieces exist and they are immovable. They are not present when \(N\) is even.
- T-centers: These pieces exist in orbits of 48 between the facet centers and the ridge centers. When \(N \geq 5\) is odd, there are \(\frac{N-3}{2}\) orbits. They are not present when \(N\) is even.
- Y-centers: These pieces exist in orbits of 96 between the facet centers and the edge centers. When \(N \geq 5\) is odd, there are \(\frac{N-3}{2}\) orbits. They are not present when \(N\) is even.
- X-centers: These pieces exist in orbits of 64 between the facet centers and the corners. When \(N \geq 5\) is odd, there are \(\frac{N-3}{2}\) orbits. When \(N \geq 4\) is even, there are \(\frac{N-2}{2}\) orbits.
- Semi-oblique centers: These pieces exist in orbits of 192. There are several subtypes, each of which have \(\frac{(N-3)(N-5)}{4}\) orbits when \(N \geq 7\) is odd, and \(\frac{(N-2)(N-4)}{4}\) when \(N \geq 6\) is even.
- TY-centers: These pieces are between the facet centers, ridge centers, and edge centers.
- TX-centers: These pieces are between the facet centers, ridge centers, and corners.
- YX-centers: These pieces are between the facet centers, edge centers, and corners.

- Oblique centers: These pieces exist in orbits of 192 off all hyperplanes of symmetry. They come in two chiralities. When \(N \geq 9\) is odd, there are \(\frac{(N-3)(N-5)(N-7)}{8}\) orbits of each chirality. When \(N \geq 8\) is even, there are \(\frac{(N-2)(N-4)(N-6)}{8}\) orbits of each chirality.

- 2c
- Middle ridges: These pieces are at the centers of the ridges. When \(N \geq 3\) is odd, they come in one orbit of 24. They are not present when \(N\) is even.
- T-ridges: These pieces exist in orbits of 96 between the ridge centers and the edge centers. When \(N \geq 5\) is odd, there are \(\frac{N-3}{2}\) orbits. They are not present when \(N\) is even.
- X-ridges: These pieces exist in orbits of 96 between the ridge centers and the corners. When \(N \geq 5\) is odd, there are \(\frac{N-3}{2}\) orbits. When \(N \geq 4\) is even, there are \(\frac{N-2}{2}\) orbits.
- Oblique ridges: These pieces exist in orbits of 192, but they are not chiral. When \(N \geq 7\) is odd, there are \(\frac{(N-3)(N-5)}{4}\) orbits of each chirality. When \(N \geq 6\) is even, there are \(\frac{(N-2)(N-4)}{4}\) orbits of each chirality.

- 3c
- Middle edges: These pieces are at the centers of the edges. When \(N \geq 3\) is odd, they come in one orbit of 32. They are not present when \(N\) is even.
- Wings: These pieces exist in orbits of 64 between the edge centers and the corners. When \(N \geq 5\) is odd, there are \(\frac{N-3}{2}\) orbits. When \(N \geq 4\) is even, there are \(\frac{N-3}{2}\) orbits of each chirality.

- 4c
- Corners: These pieces are at the corners. When \(N \geq 2\), they come in one orbit of 16.

## Turning¶

Each turn of the N^{4} is a rotation of one of its cubic cells, or a slice layer between two parallel cuts. Each layer turned can be oriented in any of 24 orientations of a cube.