Hidden structures There is a lot of talk about "hidden structures" in the realm of mathematics: hidden structures in the ZFC system, hidden structures in the natural number system, and so on. Saunders Mac Lane poignantly formulated:

Mathematics is a network of hidden structures.  [Mac Lane 1980,
  p. 362].

I'd like to know how this manner of speaking is to be understood concretely:

How and why can a structure be hidden? 

Can this be made comprehensible to a layman?
 A: There are lots of times when a set admits more structure than you might have originally noticed.  A pretty elementary example is when a combinatorial or set-theoretical device turns out to admit an action by a group $G$, making it into a $G$-set.  This is what underlines the unreasonable effectiveness of group theory in combinatorics.  So a laymen's example might be something as simple as the 15-puzzle or the Rubik's cube.  Or maybe you run into a group that itself admits a "hidden" action from some other group, further illuminating its structure.  Or maybe your abelian group turns out to be an $R$-module for some ring $R$ of interest.  All of these phenomena have occurred many times in the development of mathematics.
On a (only slightly) different front, maybe you find that the set of points on a geometric curve admits a shockingly important group structure.  Ditto for other geometric objects like the space of line bundles on a projective variety.  Or maybe a topological space (or a group) admits the structure of a Lie group.  Or maybe a space parameterizing some interesting collection of objects turns out to actually be a fine moduli space for such objects.  
The list goes on and on.  In fact, I think it would be a reasonable stance (among many other valid such stances) that the search for hidden structures is the process of doing mathematics.
