These problems typically involve "constraint satisfaction" and require the creation of organizational constructs. This is a fundamental type of problem in biological research.
A classic example is sudoku (click here to download a nice pdf guide). Students are often surprised to learn that there are strategies for attacking sudokus. Some of the more devilish sudokus can require more than simple logic. At that point, one can resort to Ariadne's thread, a methodically organized trial-and-error, similar to how a computer plays chess.

I find the most satisfying "organizational" problems to be those that benefit from the invention of organizational tools or constructs in their development. Two classics are "Who owns the zebra?" and "Pair-wise cells".
Organizational Problems