When it comes to non-linearity, even the simplest mechanical systems become intractable or even inaccessible by analytical means. As a consequence, it may be hard to decide which one of two given numerical methods would be more appropriate to solve a certain non-linear problem, since both methods may have to be considered as good or as bad as the other. Not always will numerical considerations and tests for plausibility be available to make the best choice. The more welcome are instances where a mechanical system allows insight by means that are understandable not only by the mathematician or theoretical physicist, but anybody with a programmable calculator at hand. In this article a visual proof is given that for a certain simple, yet analytically challenging mechanical system a unique solution exists, which can be found by simple fixed-point iteration as easily to be performed by the reader. As turns out, this system is mostly intractable by means of the finite element method, yet being easily manageable by means of the distinct/discrete element method. Thus, this article gives evidence to the assumption that systems exist which may yield almost arbitrarily wrong results when treated with the finite element method, while giving arbitrarily accurate results when handled by means of the discrete element or similar method. Our visual proof is based on a well-known fixed-point theorem, which can be applied to the problem after the system's geometry has been translated from the real plane to the complex plane. Once the simple contraction principle has been understood, it will only take a look on our main figure to see why – in this special case – fixed-point iteration will be successful and why – in this special case – it provides us with a reference to decide between discrete and finite elements. On our way, with Archimedes of Syracuse an old acquaintance will be met, whose so-called arbelos happens to provide the visual background to our analytical considerations.