The book is designed to introduce students of mathematics and related disciplines to fundamental techniques and principles of independent mathematical problem-solving. It begins by presenting specific proof principles such as the pigeonhole principle, invariants, induction, and working backwards through examples and detailed explanations, before moving on to more general strategies like observation and pattern recognition, generalization, specialization, and analogy. The solutions to the numerous example problems focus on explaining how one could have arrived at the respective proof steps independently. The strategies learned are then tested in various mathematical areas, namely elementary number theory, graph theory, finite combinatorics, linear algebra, and analysis. Each chapter concludes with a summary.