The traditional Stefan problems concerning solidification or liquidisation phenomena of a phase changing bar are revisited in this paper. Both single and double phase models are taken into consideration while analyzing the problems. The physical phase transition process is permitted to take place under the imposed movement of the material along or reverse directions with a constant speed. Some interesting mathematics are presented, particularly revealing the appearance of multiple solutions for the coefficient which determines the movement of the phase change interface. It is shown that the solid-liquid interface location is enhanced growing relatively faster assisting the phase change process when the semi-infinite slab moves in the direction of melt or freeze. However, the phase change process is expectedly deterred when the bar moves in the opposite direction. When the boundaries are controlled with flux conditions such that the initial position is permitted to release a heat flux inversely proportional to the square root of time, unique interface locations for both the single and two phase models are found up to the Peclet number Pe <= root 2 corresponding to a solid-liquid interface location factor lambda = 1/root 2. Otherwise solutions are tripled in one phase model and multiple solutions are encountered for the two phase model. For the Stefan problems involving the initial boundary as fixed temperature or heat flux with latent heat a power function of position, explicit phase change solutions appear to be unique for the entire action of the phase change material, no matter whether the single or double phases are considered.