the effective repulsion. This is qualitatively similar to previous results for vacuum-separated perfect-metal spheres/plates [@Lopez09] in the following sense. Previously, the attractive interaction between a sphere and a plate was in general found to enhance the attraction between two identical spheres as the plate became closer [@Lopez09].(There are certain regimes, not present here, where the attractive interaction decreases) Here, we observe that the sphere–plate interaction changes the sphere–sphere interaction with *the same sign* as $h$ becomes smaller: if the sphere–plate interaction is repulsive, the sphere–sphere interaction becomes more repulsive (larger $d$), and vice-versa for an attractive sphere–plate interaction. Since the spheres are not identical, the three-body effect is dominated by the sign of the stronger sphere–plate interaction out of the two spheres. Thus, examining the signs and magnitudes of the pairwise interactions in all cases of [Fig. \[fig:fig2\]]{} turns out to be sufficient to predict the sign of the three-body interaction, although we have no proof that this is a general rule. (In contrast, for non-spherical objects such as cylinders, there can be competing three-body effects that make the sign more difficult to predict, even in vacuum-separated geometries where all pairwise interactions are attractive, which can even lead to a 