Anyone who has ever broken a dish or a glass knows that the resulting fragments range from roughly the size of the object all the way down to indiscernibly small pieces: typical fragment size distributions of broken brittle materials follow a power law, and therefore lack a characteristic length scale. The origin of this power-law behavior is still unclear, especially why it is such an universal feature. Here we study the explosive fragmentation of glass Prince Rupert's drops, and uncover a fundamentally different breakup mechanism. The Prince Rupert's drops explode due to their large internal stresses resulting in an exponential fragment size distribution with a well-defined fragment size. We demonstrate that generically two distinct breakup processes exist, random and hierarchical, that allows us to fully explain why fragment size distributions are power-law in most cases but exponential in others. We show experimentally that one can even break the same material in different ways to obtain either random or hierarchical breakup, giving exponential and power-law distributed fragment sizes respectively. That a random breakup process leads to well-defined fragment sizes is surprising and is potentially useful to control fragmentation of brittle solids. Fragmentation of breaking glass as a brittle solid is a problem of equal practical and theoretical importance. Kooij et al. demonstrate that the fragment size distribution can surprisingly be both, either power-law or exponential, depending on how a particular specimen is broken.