For a mental exercise I decided to try and solve the bubble breaker game found on many cell phones as well as an example here:Bubble Break Game

• The random (N,M,C) board consists N rows x M columns with C colors
• The goal is to get the highest score by picking the sequence of bubble groups that ultimately leads to the highest score
• A bubble group is 2 or more bubbles of the same color that are adjacent to each other in either x or y direction. Diagonals do not count
• When a group is picked, the bubbles disappear, any holes are filled with bubbles from above first, ie shift down, then any holes are filled by shifting right
• A bubble group score = n * (n - 1) where n is the number of bubbles in the bubble group

The first algorithm is a simple exhaustive recursive algorithm which explores going through the board row by row and column by column picking bubble groups. Once the bubble group is picked, we create a new board and try to solve that board, recursively descending down

Some of the ideas I am using include normalized memoization. Once a board is solved we store the board and the best score in a memoization table.

I create a prototype in python which shows a (2,15,5) board takes 8859 boards to solve in about 3 seconds. A (3,15,5) board takes 12,384,726 boards in 50 minutes on a server. The solver rate is ~3k-4k boards/sec and gradually decreases as the memoization search takes longer. Memoization table grows to 5,692,482 boards, and hits 6,713,566 times.

What other approaches could yield high scores besides the exhaustive search?

I don't seen any obvious way to divide and conquer. But trending towards larger and larger bubbles groups seems to be one approach

Thanks to David Locke for posting the paper link which talks above a window solver which uses a constant-depth lookahead heuristic.