Television host Monty Hall used to ratchet up the suspense on his game show, "Let's Make a Deal," by offering contestants a last-minute chance to change their minds. His nail-biting, probability-based dilemma supplies the title for this intriguing book. Assembled by one of the world's top puzzle-makers, its wealth of complex but fun-filled challenges range from riddles of ancient geometry to modern issues of traffic patterns and interstellar communications. Readers of all ages can appreciate these imaginative puzzles and their colorful characters, which include traveling salesmen, Chinese postmen, and flea circus performers. "Police Chase" examines the definition of the center of a triangle. A series of "Magic Arrow" games offer variations on the classic magic squares. And traditional topics such as Euler's problem, Ramsey theory, polar and spherical coordinates, and Golomb rulers take on fascinating new twists. A treat for the eye as well as for the mind, the puzzles are accompanied by dynamic illustrations that help define underlying mathematical concepts. Complete solutions appear at the end.