The attachment and detachment of flames to cold surfaces is examined using mathematical and numerical solution methods for a simplified flame attachment/detachment problem. The flame structure considered here resembles flames near propellant surfaces, where the surface consists of alternating condensed-phase sectors of oxidizer and fuel species, which gasify and burn in the gas. The mathematical arguments originate from a constant-density flow analysis, focusing on the flow dynamics near the location on the diffusion flame (DF) arc near the cold surface where the DF ends. This location is defined as the flame nidus. Relationships are derived for the propagation speed and quenching distance of the flame (1) when it is strongly influenced by the wall and (2) when it is lifted from the wall. A criterion is produced to determine when a DF nidus is or is not attached to the surface. Numerical solutions demonstrate the validity and limitations of the simplified theoretical models by delineating the regions of applicability of theories based on a postulated premixed form of flame nidus propagation.