Following the death of Baron Loránd von Eötvös in 1919, his collaborators Desiderius Pekár and Eugen Fekete co-authored a paper in 1922 containing the results of a series of earlier experiments testing the identity of inertial and gravitational mass, the Weak Equivalence Principle (WEP). Although the so-called “EPF” paper made no claim for any WEP violations, a subsequent 1986 reanalysis of the EPF paper revealed a pattern in their data suggesting the presence of a new (“fifth”) force in nature. Although the EPF data, and the 1986 reanalysis of these data, present fairly compelling evidence for such a fifth force, many contemporary experiments have failed to detect its presence. Here we summarize the key elements of this “Eötvös paradox,” and suggest some possible paths to a resolution. Along the way we also discuss the close relationship between Eötvös and Einstein, and consider how their respective contributions may have been influenced by the other's.