Forced mechanical lifting through cold pool gust fronts can trigger new convection and, as previous work highlights, is enhanced when cold pools collide. However, as shown by conceptual models, the organization of the convective cloud field emerging from two versus three colliding cold pools differs strongly. In idealized dry large-eddy simulations we therefore compare collisions between two and three cold pools. The triggering likelihood is quantified in terms of the cumulative vertical mass flux of boundary layer air and the instantaneous updraft strength, generated at the cold pool gust fronts. We find that cold pool expansion can be well described by initial potential energy alone. Cold pool expansion monotonically slows but shows an abrupt transition between an axisymmetric and a broken-symmetric state mirrored by a sudden drop in expansion speed. We characterize these two dynamic regimes by two distinct power law exponents and explain the transition by the onset of "lobe-and-cleft" instabilities at the cold pool head. Two-cold pool collisions produce the strongest instantaneous updrafts in the lower boundary layer, which we expect to be important in environments with strong convective inhibition. Three-cold pool collisions generate weaker but deeper updrafts and the strongest cumulative mass flux and are thus predicted to induce the largest midlevel moistening, which has been identified as a precursor for the transition from shallow to deep convection. Combined, our findings may help decipher the role of cold pools in spatially organizing convection and precipitation.