I've come up with this question and I've been playing with it for a couple weeks by now without any definite breakthrough; it seems there should be a better approach than straight naive brute force, but I haven't come up with anything conclusive yet. (The work and partial answers I have come up with, I am posting in a self-answer so as not to clutter the question.)
If you're not familiar with block design as mentioned in the title, then this Stack Overflow discussion of the game "Spot It!" is a good place to look for immediate familiarization. It's not strictly necessary to know for purposes of this question, but may be interesting for its own sake.
Consider a directed graph with 13 vertices and 52 edges, where each vertex has in-degree 4 and out-degree 4, and the edges are defined in a particular way as described below. (I'll call the vertices by uppercase letters A through M.)
In the following list, each row represents four edges; for example, the row "ABCD" represents that there are directed edges (A,B), (B,C), (C,D), (D,A).
ABCD AEFG AHIJ AKLM BEHK BFIL BGJM CEIM CFJK CGHL DEJL DFHM DGIK
(Notice that each row in the above list shares one letter and only one letter with each other row in the list. This is the connection to block design.)
The target is to maximize the number of 3-cycles (cycles of length 3) without disturbing the graph's relationship to block design.
The sequence of the four letters on each line may be changed arbitrarily to specify another similar graph, without disturbing the property that each row shares one and only one letter with each other row.
(Also note that as the vertex ending each row in the list connects to the vertex at the beginning of the row, there are 6 possible permutations per row, not 24; that is,
BCDA would be equivalent as they represent the same four edges.)
Now, the particular graph specified above has only 4 cycles of length 3; namely:
DAK GAB JAE MAH
DAK is shorthand for
D->A, A->K, K->D.)
Here is the puzzle:
By permuting each row in the manner described, what is the maximum number of cycles of length 3 that can be achieved in the resulting graph?