The famous exchange of letters between Blaise Pascal and Pierre de Fermat in 1654 settled centuries of controversy surrounding the “Game of Points”, which asks how to divide the prize for a game of chance that has been untimely interrupted, and established the basic concepts of probability theory. Some modern controversies remain about this exchange, what was achieved, and who contributed which ideas. We review Pascal’s solution methods and relate them to previous attempts. A long-standing mistranslation of one of Pascal’s letters is corrected, suggesting he had a subtler and more modern grasp of the problem than previously believed. The different approaches to the problem taken by Pascal and Fermat ultimately can be seen, we argue, as epitomizing the differences between how the applied (Pascal) and pure (Fermat) mathematician frame mathematical questions and understand their solutions.