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The present communication revisits the almost century-and-a-half-old problem of some identical small magnets floating freely on the water's surface under the action of a superimposing magnetic field created by a stronger magnet placed above them. Originally introduced and performed by Alfred Marshall Mayer and reported in a series of articles starting from 1878 onward, the proposed experiments were intended to provide a model (theoretical and educational) for the building block of matter that, at a microscopic level, is the atom. The self-organizing patterns formed by the repelling small magnets under the influence of a single attractive central force are presented in a slightly different reenactment of the original experiments. Although the set-up is characterized by an axially symmetric magnetostatic structure, and the floated magnets are all identical, the resulting equilibrium patterns are not necessarily symmetrical, as one would expect. To the authors' best knowledge, the present communication proposes for the first time a quantitative approach to that extremely complex conceptual problem by providing a methodology for computing the equilibrium point coordinates in the case of n = 1...20 floating magnets, as proposed by the original A.M. Mayer experiments. A good agreement between the experiments and computed data was demonstrated for n = 2...15 (1st variant), but it was less accurate while still preserving the experimental set-up configurations for n = 15 (2nd variant)...20. Finally, this study draws the conclusions from the performed experiments and their corresponding computer simulations, identifies some open issues, and outlines possible solutions to address them, as well as future developments concerning the subject in general. [ABSTRACT FROM AUTHOR]