This is purely a factual probability question, but since it’s about a game, I’m posting it here, since the last time I asked a question of a similar nature, it got moved here anyway.
Bejeweled Blitz is a game played on an 8x8 grid. There are gems in each of the squares of one of seven colors. You can switch two adjacent gems, provided that the swap forms at least one row or column of at least three of the same colored gem in a row, which then disappear and are replaced by falling gems.
In this version of the game, it is not possible to create a board without any possible switches. How much, if at all, does the game engine need to manipulate otherwise random probabilities to get that to happen? I’m specifically wondering about new gem creation.
When you make a match, the game randomly generates new gems to drop in. It then checks to see if that new board has any matches. If it doesn’t, it throws out the new gems it generated and randomly generates a new batch. It keeps doing this until it finds a new board with at least one match.
Even if just three new gems drop into the top row, there are always multiple combinations that can be randomly generated that will give you a match. Even in the worst case (an empty slot of 3 in an upper corner) there are about 30 combinations of new gems that will allow the player to make a match. (Out of 343 combinations total). Even in the worst case, on average the game will need to “reroll” less than six times to find a workable set of new gems. Most of the time it will only “reroll” once or twice.