The object came out dirty. Why? How might I fix it?

I am a beginner with Blender using version 2.71. I noticed several other similar question posted here, but please move it if the place I posted is inappropriate. I will do better next time.

For a Mankalah game, I created a game board.

The spatial properties of the board seem perfect. But there are strange artifacts on the surface which appear also upon rendering. I have no idea what I did or did not do to cause them. The board was created from a cube which was elongated to the correct shape. Then an array of spheres and dirigible-shaped objects was intersected with the board and the difference was taken. When I tried to do it again for a sanity check, I repeatedly received the message “Modifier is disabled, cannot apply.” With the new, unsullied board positioned so the spheres and dirigibles intersect the ‘cube’ (Cube.004, in layer 2.)

The attachments are:

  1. The original work file with spheres and dirigibles which caused the discoloration
    Upload of .blend file reported “Upload of file failed.” How can I fix it? [file size: 4,385 kbytes]
  2. The current work file with the new try in layer 2 which raises the “Modifier disabled …” message. Also several copies of the dirty board, together with the array of spheres and the 2 dirigibles in various layers.
    Upload of .blend file reported “Upload of file failed.” How can I fix it? [file size: 4,391 kbytes]

Remove any faces inside the mesh
Remove any double vertices (select all vertices and W / remove doubles)
Recalculate face normals (select all faces and Ctrl+N)

An Edge Split modifier might also help.
Right now you seem to have smooth shading applied onto the whole mesh, which looks weird on the flat parts and can produce shading artefacts on the - let’s say: suboptimal - geometry the Boolean modifier created.

Drop idea of Boolean all - model it. All there is needed is half UVSphere and Array modifier temporarily: