This text aims to unify and inter-relate mathematical topics and explain how to design, run and analyse better algorithms. Many of the less common algorithms are included i.e. planarity, graph colouring, minimization of machine states. These are actually run so that students can see the importance of working through each step of an algorithm by hand. Topics are chosen for their contribution to the students ability to reason abstractly. For example, "Matrices" introduces the topic from the study of arrays, "Disjunctive Forms" does not rely on the typical Karnaugh Maps and Quine-McClusky Algorithm to find the minimal forms of any given proposition and "Planarity" presents a complete planarity algorithm allowing the student to master a tough, interesting procedure. Logic and proof are explained through example and technicalities and limitations of "real"computer languages are avoided. Topics are set in some historical framework whenever possible, within the overall studies from which they are derived. This book should be of interest to degree and diploma students on introductory mathematics courses and service courses for computer science.

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