Abstract: A wheel rolls down steps under the action of gravity—will it roll faster and faster or slower and slower? This is an interesting question. In this paper, based on two mechanics contest problems, a unified mechanical model is established and an in-depth analysis is conducted. It is presented that complete conditions for the existence of a limit in the angular velocity sequence of a wheel rolling down steps. The paper also points out the oversights and deficiencies of the two mechanics contest problems.