Improving a Search Heuristic for symmetric number-board game

[Shaitan00]

Board (4x3)
A B C D
E F G H
I J K L

You have 11 chips [1,1,1,1], [2,2,2,2], [3,3], [4] (for example) placed randomly on the 4x3 board in slots A,...,L and one empty slot (that moves) that we'll call X, the movements are done similarly to the 8-puzzle game where X (empty) can move either up/down/left/right.

The Goal is to ensure that the top row and bottom row are symmetric (middle row is irrelevant) - so the following would be valid solutions:
1 1 2 2
3 3 4 X
1 1 2 2
or
1 1 2 3
X 2 4 2
1 1 2 3
As you can see in both cases the TOP and BOTTOM rows are symmetric (identical)...

So - just to help understand the game a little - assume you have the following flow from initial state to goal state:
STATE(0)
3 3 1 1
4 2 1 2
3 3 1 X
--> Move LEFT

STATE(1)
3 3 1 1
4 2 1 2
3 3 X 1
--> Move UP

STATE(2)
3 3 1 1
4 2 X 2
3 3 1 1
! GOAL !

Therefore - my problem is to find out what is the BEST MOVE (up, down, left, right) to make.
Assume you are at STATE(0) in the example above, where do you move X to get the best result thus leading you closer to a GOAL? X could be moved UP or LEFT (two options) so why pick one over the other? The solution is to evaluate the two possible outcomes and pick the better one ... this is the part where I need help, the EVALUATION algorithm needed to choose the next best move.

My current method works rather well but not well enough to win first place (I think - needs to be under 3-seconds to solve puzzle) - I call it "Number of non-Symmetric Pairs" and it works as follows:
- For each state calculate the number of non-symmetric pairs and make the move that gives you the lowest number
- If you have equal lowest-numbers then choose one at random

Let me illustrate - for starters this would be the value for the states listed in the example above:
STATE(0) = 1 (as the 4th pair isn't symmetrical)
STATE(1) = 1 (as the 3rd pair isn't symmetrical)
STATE(2) = 0 (as all pairs are symmetrical)

Where this works - assume we are at STATE(1), we have 3 choices: LEFT, UP, RIGHT - which would each yield the following values:
STATE(1)->LEFT = 2
STATE(1)->UP = 0
STATE(1)->RIGHT = 1
Therefore the lowest number is 0 and we should move UP because that will generates less non-symmetrical pairs (and lucky for us in this case it is also the goal).

Where this doesn't work so well - assume we are at STATE(0), we have 2 choices: LEFT, UP - which would each yield the following values:
STATE(0)->UP = 1
STATE(0)->LEFT = 1
As you can see the values are identical, so in this case I cannot determine which move is best so I must try both routes, which in turns costs me a lot of time I was hoping I wouldn't have to waste.


With that said - I am looking to find a more "informed" (efficient, intelligent) function/algorithm to help me solve my problem quicker and in less moves which therefore yields a more favorable results.
Along the same lines I am trying to see if there is another or better way to evaluate how close one move is to a goal when compared to another...

Any ideas, hints, help you might come up with would be much appreciated.
Note that the cells could have words, colors, numbers - this was just an example to help illustrate how it works.

Thanks,

 

[KataKoniK]

I would approach this problem by breaking it down into smaller, more manageable parts and using a combination of mathematical modeling and algorithmic techniques to find the most efficient solution.

First, I would create a mathematical model for the board and the chips, assigning values to each chip and its position on the board. This would allow me to quantify the symmetry of each potential move and determine which move would bring the board closer to the goal state.

Next, I would use an algorithm such as A* search to systematically explore the possible moves and evaluate them based on the mathematical model. A* search uses a heuristic function to guide the search towards the most promising moves, significantly reducing the number of moves that need to be evaluated.

In addition, I would also consider incorporating techniques from artificial intelligence, such as machine learning, to improve the efficiency and accuracy of the algorithm. This could involve training a model on a large dataset of board configurations and their corresponding optimal moves, and using this model to make informed decisions about which moves to make in each step.

Furthermore, I would also explore the possibility of using parallel computing techniques to speed up the search process and find the best solution in the shortest amount of time.

Overall, by combining mathematical modeling, algorithmic techniques, and artificial intelligence, I believe it is possible to find a more efficient and intelligent solution to this problem.

 

[piercas]

my response would be that the problem described above is a classic example of a search problem in artificial intelligence. The goal is to find the most efficient path from an initial state to a goal state, taking into account the constraints of the problem. In this case, the goal is to achieve a symmetric board by moving the empty slot in a certain way.

One potential approach to improving the search heuristic would be to incorporate domain knowledge into the evaluation function. This means using specific knowledge about the problem domain to guide the search towards more promising solutions.

For example, in this particular problem, we know that the top and bottom rows must be symmetric in the goal state. Therefore, we could assign a higher weight to moves that result in a more symmetric top and bottom row, while also considering the number of non-symmetric pairs. This would make the evaluation function more informed and potentially lead to better solutions in a shorter amount of time.

Another approach could be to use a more sophisticated search algorithm, such as A* search, which combines both depth-first and breadth-first search techniques. This algorithm uses an evaluation function, similar to the one described above, to guide the search towards the most promising solutions. This could potentially lead to a more efficient and intelligent search.

In conclusion, there are various ways to improve the search heuristic for this symmetric number-board game problem. By incorporating domain knowledge and using more advanced search algorithms, we can find more efficient and intelligent solutions, and potentially achieve the desired goal of solving the puzzle in under 3 seconds.