Represent the three-dimensional space as a grid of unit cubes. Is there a way to colour each cube in black or white so that each cube has half of its $26$ neighbours (sharing a common side, face, or just a vertex) black and half of them white? What about higher dimensions?
For two dimensions it is certainly possible - perform the standard chessboard colouring.
In three dimensions, stack two layers of standard chessboard together with matching squares in the third dimension. Now stack these bilayers with the squares mismatched. Each cube has $8$ neighbors in its plane, of which $4$ are each color. It has $9$ neighbors in the layer above and $9$ in the layer below. $5$ of the ones in the layer above match $4$ in the layer below.
I believe you can continue the pattern to as many dimensions as you want. For $4D$, take two of the above $3-$spaces and stack them with matching faces. Then take bilayers and stack them offset. The $26$ neighbors in a layer are equally split and the layer above and below are staggered so the remaining $54$ neighbors are evenly split.