In [1], Fernandez outlined an algebraic approach to the construction of Azulejo patterns. An open problem listed in this paper is “to describe the geometric features of these designs that account for their aesthetic properties.” This paper points to a direction to answer this problem. We identify aesthetic pleasantness with aesthetic challenge and present two definitions of aesthetic challenge: (1) Aesthetic challenge can be partially defined and quantified by the number of English phrases needed to describe an Azulejo pattern; (2) We can also define aesthetic challenge as equal to the number of primitive figures—isolated points and lines—in the fundamental octant of the Azulejo pattern. We show that these two definitions correlate and that they provide an alternative to Fernandez's purely algebraic approach for construction of Azulejo patterns. Several examples are presented showing that both approaches—the algebraic approach of Fernandez and the aesthetic challenge approach of the author—yield aesthetically pleasing designs. Future directions of research on the definition of aesthetic challenge are presented.