When faced with a problem in which there is no perfect algorithm and the number of permutations wouldn't permit doing a full tree search for the best answer, you may want to consider using a genetic algorithm. You may be shocked at how much more intelligent the solution is than what a human could come up with.
Melanie Mitchell wrote a book called Complexity: A Guided Tour which was released in April 2009. It contains a chapter on genetic algorithms in which she describes a simple problem and how a GA is used to solve it. I implemented a t-sql solution to the problem in about three hours. It took 20 minutes to run about one million simulations. It came up with solutions that are counter-intuitive and exceed human ability by leaps and bounds.
Here is a definition of the problem: There is a 10 x 10 grid surrounded by a wall. Half of the grid squares contain an empty soda can. A robot starts in the upper left square and executes 200 actions. An action can be to move North, South, East, West, or to try to pick up a can, or to pick a random direction in which to move. The robot can only see five squares, the current square it is on plus the four adjacent squares.
The robot scores 10 points for each can it picks up. It scores -5 points for trying to move into walls. It scores -1 point for trying to pick up a can from an empty square. If the robot executes a perfect run, 50 cans will be picked up without penalties for a total of 500 points. It is important to remember that the robot has no memory of previous moves nor access to problem solving routines. The intelligence will be due to an evolutionary process only.
Here is how to implement a genetic algorithm solution: Generate an initial population of children. Test them in a simulator. Kill off all but the best two children. Cross breed those as parents for a new generation of children by taking a top portion of DNA from one parent and combining it with a bottom portion of DNA from another parent. Generally a few random mutations are then introduced into each child. Test these children and repeat the process of selection and replication until a good enough answer occurs.
In our simulation, a square can be empty or a wall or a can. Since the robot can see five squares, the number of permutations is 3x3x3x3x3 which equals 243. Each permutation is a situation in which a gene will determine the action to take. Our robot just happens to have 243 genes to handle each of the possible permutations.
Each of the 243 genes is one of six action values to move in one of four directions, pick up, or move random. If we take 6 to the 243rd power, we see that we can have 1.23x10^189 unique children, way too many to ever test in a brute force manner. The included script checks only one million evolved children out of nearly an infinite possible number of children and comes up with an almost ideal answer.
Run the ScoreRobotFitness.sql script (in the Resources section below) to create the ScoreRobotFitness stored procedure. It will be called for each child that is created to score its performance.
Run the EvolveRobots.sql script (in the Resources section below) to create and evolve robots. Start with 100 NumGenerations and 100 NumOffspring to see results in seconds. Start with 1000 of each if you want the ideal robot after 20 minutes of calculations. Copy the results to Excel for a quick graph on its learning curve.
Run the DecipherDNA.sql script (in the Resources section below) after replacing @s with your ideal robot's DNA to decipher the rules it came up with. Most likely it came up with heuristics that a human wouldn't. For example, it may have avoided picking up a can in the current location, preferring instead to move to the left of a string of cans and start picking them up from the beginning. Picking up cans from the middle of a string would make it more difficult to backtrack to the stranded cans at the left.
Most humans would score about 350 after hand-coding the genes and optimizing for a few days. I'm not sure a human would ever be able to score as well as this GA did in only 20 minutes of evolution. Try implementing a randomly populated grid, tweaking the generations and offspring counts, allowing diagonal movements, or varying the mutations and let us know what interesting heuristics emerge, especially if you can speed up the evolution process.