This paper addresses the triangle width problem, which generalizes the classic 2-machine flexible job-shop problem (FJSP) with tooling constraints. This new problem can be studied from 3 different angles: scheduling, matrix visualization, and the order of vertices in hypergraphs. We prove the equivalence of the different formulations of the problem and use them to establish the NP-Hardness and polynomiality of several of its sub-cases. This problem allows us to find more elegant (and probably shorter) proofs for several combinatorial problems in our analysis setting. Our study provides an elegant generalization of Johnson’s argument for the two-machine flow-shop. It also shows if a matrix can be triangular by permuting rows and columns and a low k -visit.