Colored Squares and Cubes

1. Color each square in a grid, such that no two neighboring squares have the same color, and each square neighbors any other color only once. (Neighboring squares share sides - don't consider corner-touching squares to be neighbors.) What is the minimum number of colors needed for:
   1. a 3x3 grid?
   2. a 4x4 grid?
   3. the plane?
2. Color each cube in a 3-D lattice such that no two neighboring cubes have the same color, and each cube neighbors any other color only once. (Neighboring cubes share faces - don't consider corner- or edge-touching cubes to be neighbors.) What is the minimum number of colors needed for:
   1. a 2x2x2 cube?
   2. a 3x3x3 cube?
   3. a 4x4x4 cube?
   4. 3-space?

Source: Original.