Several methodologies to design water systems in process plants are based on minimizing freshwater consumption. The objective is the appropriate one when water is scarce and costs are not a big issue. It has also been used as a substitute for cost in the belief that water and final treatment costs overwhelm other fixed capital and operating costs of regeneration processes. Among graphical and algorithmic methods, the popular "Pinch Technology"-based procedure has early proponents and contemporary advocates who consider and defend it as a good method to provide "insights" into the right answer. In this paper, we show that the minimum freshwater problem has sometimes a large number of alternative degenerate solutions, something that graphical and algorithmic procedures like the aforementioned pinch technology can hardly identify systematically. The fact that the degenerate solutions are many makes it worse. We provide a mathematical programming method to identify these solutions and to point out those that make more economical sense. We also provide means to identify suboptimal solutions that are very close to the optimum one. We illustrate these degeneracies in several cases of single and multiple contaminants.