Perhaps, if I find the answers in someone else's code, I won't have to play the darned game anymore.
It turns out that the game is NP-hard; the The Complexity of Flood-Filling Games (from Raphael Clifford, Markus Jalsenius, Ashley Montanaro, Benjamin Sach).
I don't feel so bad about it any more.
Some more hard games: Mastermind, Minesweeper.
Edward Vielmetti's Wordpress weblog 
NP-hard means that it is part of a class of problems for which nobody has ever found an algorithm whose run time doesn’t increase exponentially as the size of the grid increases. However the size of the grid doesn’t increase. All the puzzles at the site I found have a grid of 14×14. This result doesn’t tell us anything about the difficulty of solving 14×14 puzzles.
Thanks Jan.
Here’s one solver
http://www.madpickles.org/rokjoo/2011/03/09/c-o-l-o-r-f-l-o-w/
Note that there’s an opposite problem: instead of minimizing the number of turns you take to clear a board, maximize it.
I’ve just noticed that my bathroom floor is a giant four color Flood-It puzzle. I’ll never be able to use the John again without driving myself mad trying to solve it.