Many of you I assume are familiar with Bejeweled, a puzzle game in which the player swaps neighboring, colored gems in an 8 x 8 grid to create rows and columns of ≥ 3 matching colors. My question pertains to what happens after a match is made.

Let's assume a horizontal match of 3 gems is made on the 5th row down. This causes these 3 gems to disappear, and for the gems in the 4 rows above them to "fall" into the empty space they leave behind. What I am trying to figure out is the probability that these falling gems (and the randomly generated new gems that appear on the top row) create a new match of 3 anywhere on the board.

Here is what we also need to know:

There are 7 gem colors.

There are 6 possible ways for a single, non-border gem to match with two of its neighbors.

Therefore the probability for three gems to match at random should be (1/7)^2, because the color of the first gem doesn't matter. Only the next 2 need to match each other and the first.

The formula I am trying to design would give the probability for a new 3-gem match resulting from a horizontal 3-gem match we make on the 5th row down.

So what I came up with was this: p = 3(((1/7)^2)*(6(r-1))) where r = the row on which the original horizontal match is made, and 6 is the number of ways a new, non-border gem can match with its neighbors.

Obviously, this is very wrong, and results in a value > 1 for our designated 5th row match. But I don't know how to fix it, or if there is a better direction entirely from which to attack this problem.

Thank you very much for your help on this!

Andrew