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Some variants have no other goal than to make a fun game with some additional and interesting chess pieces, or to explore the possibilities of chess in three dimensions or on a hexagonal grid.
But many chess variants are instead aimed at correcting problems with the game of Chess as it now exists.
Thus, Chess 960 is aimed at addressing the fact that a lot of knowledge exists about chess openings.
One of the common complaints about chess is that there are too many draws. A number of schemes have been proposed to deal with this; for example, scoring a draw as 1/3 point in a tournament. I've suggested giving 100 points per game, a 60-40 split in the event of stalemate, and 51-49 for perpetual check by White... but 55-45 for perpetual check by Black; I drew some inspiration for this from the success of komidashi in Go.
In my opinion, though, the biggest complaint about chess is that games aren't as exciting any more ever since Steinitz came along and showed us how to play chess properly, in a sound and scientific manner.
I'm not at all sure how one could make a chess variant that would change that. Make it so complicated that positional judgment is impossible? Make at least some captures compulsory, so that dashing piece sacrifices would work?