When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially come across as more puzzling or even mysterious. In this article we explore three specific statements from abstract algebra that involve the number three, whose proofs are explanatory in the sense of Steiner. Explanatory proofs transform what might initially seem mysterious or even magical into lucid mathematics. We propose possible ways to use these statements and others similar to them in a course in abstract algebra emphasizing proofs, or in an introductory course on proofs based on group theory.