# NxNxNxN¶

4x4x4x4

Shape: Tesseract

NxNxNxN, or N4 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 N4 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 N4 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.