This question was inspired by Math puzzles for dinner.

The arrow compatibility conditions in that problem can be considered an attempt to discretize the notion of a continuous vector field.

The Hairy-Ball Theorem states that there is no continuous nowhere-vanishing vector field on an even-dimensional sphere.

We are led to the following formulation of a discrete Hairy-Ball Theorem for two dimensions:

Instead of a $ 2 $-sphere, we consider unit squares on the surface of a $ (3 \times 3 \times 3) $-Rubik’s cube. Instead of searching for a continuous nowhere-vanishing vector field, we ask if there exists a ‘legal configuration’ of arrows on the squares of the cube.

A ‘legal configuration’ of arrows consists of the following:

- For any non-corner square, an arrow pointing in one of the eight cardinal directions is placed.
- For any corner square, there is one cardinal direction that does not point to an adjacent square. For these squares, the placed arrow should point in one of the other seven directions.

The following conditions are modified from the original formulation:

- Orthogonally-adjacent non-corner squares should be compatible in the sense that if they are flattened to lie in a plane, then the arrows should be at most $ 45^{\circ} $ apart.
- Orthogonally-adjacent corner squares should be compatible in the sense that if they are flattened to lie in a plane, the arrows are one rotation away from each other within the seven allowed directions. For example, if northeast is a prohibited direction on a corner square, then a northward pointing arrow on the square and an eastward pointing arrow on an adjacent square are compatible.

So, can you comb a hairy Rubik's cube discretely? Does a legal configuration of arrows exist? What about an $ (n \times n \times n) $-cube?