In this paper, the concept of maximization of productive entropy flux is introduced and used to arrive at an exact analytical solution for determining the network with the minimum total area. The term productive entropy is used to denote entropy generated as a result of heat transfer which is the desired outcome. Other irreversibilities, like friction, and mixing result in undesirable entropy production and therefore they need to be minimized, making the network with the minimum total area, the one that has the maximum net productive entropy. Maximum productive entropy and minimum area also results in maximum productive entropy flux. Optimizing a Heat Exchanger Network (HEN) is generally accomplished either heuristically, using second law based rules, or numerically, using different optimization techniques. No exact mathematical solution is available even for optimizing the seemingly simple problem of a network of four heat exchangers used to heat two cold streams using one hot stream. For this network, it is mathematically proven that maximizing productive entropy flux results in the minimum area network. This result is deductively extended to other networks, and is valid for other heat exchanger networks we have tested, regardless of the number of hot and cold streams, and perhaps is generally valid.