I am interested in common problems/puzzles that have a clear set of rules/constraints and objective that lends itself to modeled and solved with a linear program. Especially for people new to LPs, I think that translating a tangible problem into a mathematical formulation is one of the best ways to explain the applicability of LPS.
Some easy examples I can think of would be sudoku and the 8-queens puzzle. What are other examples that come to mind?