Humans come with a syntactic retina which lets us cognize our perceptions of strings of words, just as our visual retina allows us to cognize our perceptions of visual space.
The "solution" to the "puzzle" provides the algebraic machinery required to formalize the notion of a "syntactic retina". Thanks to John Burkardt for taking the time to examine a naive algebraic observation of mine, and seeing that there was a worthwhile puzzle and solution buried in this observation.
To see why John's "solution" to the "puzzle" provides the algebraic machinery required for a formal characterization of the syntactic retina, consider first these three flavors of syntactic "structures" or "objects".
Although it is not immediately obvious, there is a permutation lurking in each of these objects, as may be seen by considering this finite-state object.
More generally, there is a permutation lurking in ANY derivation tree of ANY derivation in any finite-state OR context-free grammar. And this fact suggests that John's mechanics may in fact provide the key to understanding the nature of the human syntactic retina.
But to see exactly how and why John's mechanics provide the proper foundation for formalizing the notion "syntactic retina", it is necessary first to drop back and reconsider some theoretical fundamentals involving classical finite-state and context free grammars, and their equivalent Merge-based grammars in UG.
We begin this reconsideration of fundamentals HERE