In network theory, a triad census is a method designed to categorize and enumerate the various types of subgraphs with three nodes and their connecting edges within a network. Triads serve as fundamental building blocks for comprehending the structure and dynamics of networks, and the triad census offers a systematic approach to their classification. Typically, triad counts are obtained numerically, but lesser-known methods have been developed to precisely evaluate them without the need for sampling. In our study, we build upon Moody's matrix approach, presenting general diagrammatic rules that systematically and intuitively generate closed formulas for the occurrence numbers of triads in a network.