Figure a, b shows the ensemble energetic driving force for forward and reverse sintering steps, respectively. Forward sintering refers to the movement of monomers from small to large clusters, while reverse sintering refers to the movement of monomers from large to small clusters. For each cluster size in Figure , we consider the energetics of monomer transfer between the isomers of said cluster size, and all isomers of all other cluster sizes. We then calculate the energy difference, and weight it by the probability of the given combination of isomers occuring, based on their Boltzmann populations, then normalized by the number of Pt atoms in the cluster. This is represented by the magnitude of the bar in Figure , split into unfavorable (red, positive, preventing sintering), and favorable (green, negative,driving sintering) contributions. For a breakdown of each cluster size combination, and how we obtained the ensemble-averaged energy values, see Supplementary Note Analytic Methods. Figure shows that for all Pt n , except for Pt 4 and Pt 7 , forward sintering (forming larger clusters) is favorable, while Figure shows that reverse sintering (forming smaller clusters) is generally less favorable, but again is particularly unfavorable for Pt 4 and Pt 7 . The energetics predict that these two cluster sizes should be particularly stable with respect to both shrinking and growth. This explains the prevalence of Pt and most of the other cluster sizes and isomers is energetically unfavorable, both cluster sizes build up regardless of the cluster size composition of the starting point of the Monte Carlo simulations. Hence, in the absence of adsorbates, these two cluster sizes are "magic" in the sense of resistance to thermal sintering. Note that in the experiments, the lowest final ISS intensities were for Pt 1 and Pt 2 , suggesting that they sintered to particularly large multilayer structures, whereas larger clusters such as Pt 5 or Pt 6 also sintered, but with a larger fraction of the Pt atoms remaining in the top-most layer, i.e., forming fewer or smaller multilayer clusters. This propensity of the smallest Pt n to sinter to larger final cluster reflects the relative instability of atoms (and dimers) with respect to diffusion on the surface. Note that monomer motion on the support was explicitly accounted for in our MC simulations, however motion of larger cluster sizes was not.