I think the answer is no. We consider a cube, and we define a board to be a half plane of cubes. We define a 3D coordinate system, with the x-axis stretching left to right, y-axis front to back, z-axis up to down, and the cube at the origin. We now connect a board perpendicular to the x-axis, joined to the cube at the top face with 2 half faces touching the cube. Then we connect another board at the bottom of the cube perpendicular to the x-axis, again not connected by a whole face to the origin cube. We repeat this by joining 2 more boards perpendicular to y-axis connected to the cube at left and right faces, and finally connect 2 more boards perpendicular to z-axis connected by front and back faces. The resulting figure would be the same as the set of cubes with with coordinates $(x,y,z),xyz=0$. The remaining empty spaces can easily be filled up. Note that the origin cube does not share a whole face with another cube in common. PS a board, if not clear, is defined to be the set of cubes $(x,y,z)$ such that $z=0,x\geq 0$.