A Puzzle to Flip Your Lid Over

Clicking on the tiles in the grid below will make them flip over. The idea is to make all the tiles the same colour. The catch is that not only the tile you click on flips, but those adjacent to it as well.

Some problems:

  1. Find a solution for the initial configuration given here. You can return to this configuration by pressing the Reset button.
  2. Devise an algorithm (preferably one which can be executed without the aid of a computer) which will work for an arbitrary initial configuration. You can generate a random configuration by pressing the Randomise button.
  3. Devise an algorithm which will find a minimal solution given an arbitrary initial configuration (a computer may be used for this one).
  4. I have reason to believe that all initial configurations of a 6x6 board are soluble, as are 5x5, 7x7 and 8x8, but not 4x4 or 9x9. Attempt to prove, disprove and/or generalise this conjecture.