We explain extensive computer-aided searches that have been carried out for many years to find new ways of constructing abelian square-free words over four letters. These structures have turned out to be very rare and hard to find. We also encountered highly nonlinear phenomena that considerably affected our calculations and usually made them hard to accomplish. However, quite recently, we gained new insight into why these structures are so very rare. Consequently, the present work has the potential to make future explorations easier. The rarity of long words that avoid abelian squares can be explained, at least partly, by using the concept of an unfavorable factor. The purpose of this article is to describe the use of Mathematica in searching for and suppressing these factors. In principle, the same programs can be used with slight modifications for other kinds of word patterns as well.