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This book is an alternative and highly engaging introduction to the highlights of a typical undergraduate mathematics course. Starting from very simple levels or principles, it develops in an intuitive and entertaining way these mathematical highlights, which every rounded mathematician knows. The aim is to motivate and inspire the reader to discover and understand some of the truly amazing mathematical structures and ideas which are often swamped, or usually pass unnoticed or are not fully grasped, in a higher mathematics course. For the experienced reader the book offers refreshing, often instructive hindsight. For the novice it is an exciting intellectual journey.