A well-known story from the folklore of mathematical history tells how Archimedes bounded from his bathtub and ran naked through the streets of ancient Greece yelling, "Eureka!" [I have found it]. Apparently a period of intense meditation in the bathtub had provided the inspiration for Archimedes' discovery of the law of hydrostatics which now bears his name. Whether apocryphal or authentic, this legend captures the ecstasy which accompanies a flash of insight into a deep problem. It is to those who relish the joy of mathematical discovery that this book is dedicated. It is for those who chase relentlessly the thrill of eureka that this book is written.
Through the past three millennia, the mental giants of each generation have pitted their problem solving skills against the most intractable problems to challenge the human mind. When a problem yielded like a dragon to a deft sword, the successful strategy was refined, polished, and added to the growing body of intellectual technique known as mathematics. One of the greatest problem solvers of all time, Isaac Newton, characterized his favorite discoveries as "smoother pebbles or prettier shells than ordinary". From an assortment of the most beautiful pebbles and the prettiest shells of mathematical technique, Ravi Vakil, author of this book, has created an intriguing mosaic. It is a mosaic which links various branches of mathematics through powerful overarching ideas. Ravi himself is a preeminent problem solver, having won top honors on virtually every mathematics competition and olympiad. His remarkable achievements are outlined on page 9 and on the back cover.
Throughout this book, Ravi Vakil profiles several other recent winners of mathematical competitions and olympiads: young people who have honed their intellectual gifts to world class levels. As you read these human interest features, you will observe that these fine young minds all share a passion for the eureka sensation. It is something they all understand and something that bonds them together in friendly collegial competitions.
We are pleased to publish this celebration of the human intellect. It is offered as a salute to the problem solvers of the present and future and as a tribute to the intellectual giants of the past. To the reader, we issue a challenge. We dare you to flip through the pages of this book without finding any problems to pique your curiosity!