This demonstration project is like the game you mention:
http://demonstrations.wolfram.com/PicturePuzzle

A really typical way of solving a problem like this is to write a function that recursively searches for a solution. While you probably can't find an example for your particular problem, you can find it for other problems. I would encourage you to look online for sudoku solvers written in Mathematica and other similar languages. Most sudoku solvers work by recursively searching for a solution. Once you understand how the sudoku solver works, you should be able to apply the same technique to write a solver for your problem.

thanks Sean Clarke,it is also very true what you mention, I will begin to review what recursion.