Ice multiplication (IM) processes can have a profound impact on cloud and precipitation development but are poorly understood. Here we study whether a lower limit of ice nuclei exists to initiate IM. The lower limit is found to be extremely low (0.01 per liter or less). A counterintuitive but profound conclusion thus emerges: IM requires cloud formation around a thermodynamic sweet spot and is sensitive to fluctuations in cloud condensation nuclei concentration alone.