The unsteady one-dimensional heat conduction equation is transformed into an ordinary differential equation called Kummer's equation unifiedly in the linear, cylindrical and spherical coordinate systems. Kummer's equation is solved in terms of the confluent hypergeometric functions and thus the similarity solutions are obtained. These solutions exist on the conditions that boundaries lie at the origin and infinity, or otherwise move with their positions proportional to the square root of time, and that the strength of heat source is a power function of time. For the already known similarity solutions expressed in terms of other functions, the corresponding confluent hypergeometric expressions are shown. If the conduction similarity solutions are applied to solve moving boundary problems with phase change, only one solution exists in each coordinate system.