We start with a rhombille tiling of the plane:
These are the symmetries (glide reflections, reflections, rotations and translations) for wallpaper group p6m:
To wrap the tiling onto a cube, we identify the tiles with the faces of the cube:
This is the unit cell, i.e., the smallest unit which makes up a cube, and the tile which appears in the colored tiling.
This is the remaining wallpaper group of the colored pattern is p2. Reflections and glide reflections are gone, translations take a larger step, only centers of order 2 remain. The cube is obtained by dividing out this group.
Finally, the board is obtained by dividing every face of the cube into smaller cells, using an rectangular grid.
Another board type is the projected cube. It is constructed from the cube by identifying opposite points.
In this case we only have three faces, but their orientation changes.