The word 'mathematics' usually conjures up a world of more-or-less familiar problems to be solved by more-or-less familiar techniques. This book examines a very different aspect of mathematics, namely how one can begin to explore unfamiliar, fresh ideas and chance observations, how one can pursue them through various stages until the light eventually begins to dawn, and how this whole process invariably throws up other interesting questions one would otherwise never have thought of. The items have been designed to provide certain basic experiences from struggling to explain simple, but puzzling mathematical phenomena, to discovering for oneself some new and totally unexpected bit of mathematics. Readers should have no difficulty with the mathematical techniques which are required. They should therefore be in a position to reflect on some of the more striking features of the process whereby unsuspected mathematical relationships are uncovered, and on the way they emerge. The problems studied avoid the kind of sophistication which would put them out of reach of ordinary students, yet are sufficiently complex to capture the essential features of the process of mathematical discovery. No attempt has been made to reduce this process to the level of checklists of catchwords. Readers will want to identify and reflect on the significance of these essential features for themselves.