The question is simple: what is the longest line of gems that we can get in a row with the match-3 games such as Bejeweled? Aside from allowing for an infinite grid of jewels the rest of the normal rules of Bejeweled apply. Most Importantly, if at any step in the jewel removal process there are three or more contiguous matching jewels then they are removed.

With that in mind I came up with a schema where a line of 13 jewels is formed. You start with the following configuration:

Exchange the two jewels indicated by the arrows and you’re off! That gems cascade into this pattern:

Which then fall into this formation:

And at this point it’s obvious the green and yellow gems will disappear leaving a row of 13 red gems. The key to this step was creating the staggered columns of yellow and green gems to keep the red gems from forming a contiguous line during this middle step.

Now, I would not be surprised if this basic method could be extended to create longer sequences. However, is there an upper bound or is there no limit to the length of a gem row? Since there is a finite number of gem colors I forsee some problems with later iterations with unintended contiguous sequences.

Any mathgeeks out there wanna tackle this?