The problem of allocating children to public daycares differs from the school choice problem in two fundamental ways: there is entry and exit of agents over time, and the priorities of schools over children are history dependent. We illustrate with the Danish case and show that there is no mechanism that is strategy-proof and yields a stable matching. We propose an algorithm in which parents sequentially choose menus of schools, ordered bythe child's birth date. This mechanism is strategy-proof, Pareto efficient, eliminates ex-post uncertainty, and may be considered fair: parents face similar choice sets, which increase over time.