Eigenrays between a source and receiver are classically determined by an initial value or “shooting” approach. In range-dependent environments the ray paths can be chaotic, putting a fundamental limit on the accuracy of classical ray tracing methods. An alternative approach can be based on Fermat’s principle of minimum propagation time. Using the so-called “direct methods” of the calculus of variations, the travel time integral can be minimized directly, rather than by means of the Euler–Lagrange differential equations used in shooting. In the current paper, examples are studied where the changes in a ray’s trajectory occur at discrete points. Discrete mappings, analogous to the Euler–Lagrange equations used in shooting techniques in continuous problems, are introduced. It is demonstrated that direct methods can be applied to the travel time summations in discrete problems, just as they can be applied to the travel time integral in continuous problems. It is shown that direct methods can be used to calculate certain arrivals (e.g., “head waves”) that cannot be produced by shooting. Using discrete examples, direct methods are compared to the more conventional discrete mapping (shooting) approach, without the complications of numerical analysis and infinite dimensionality found in continuous problems. Two examples are studied that are associated with a standard discrete mapping known to be chaotic. Direct methods are used to find eigenrays for these chaotic examples. The advantages and limitations of direct methods in discrete problems with chaos are discussed.