Fibonacci is one of the best-known names in mathematics, and yet Leonardo of Pisa (the name by which he actually referred to himself) is in a way underappreciated as a mathematician. When hearing the name we are most likely to think of the Fibonacci sequence, and perhaps Leonardo's problem about rabbits that began the sequence's rich history. Leonardo's role in bringing the ten-digit Hindu-Arabic number system to the Christian nations of Europe might also come to mind. While these two contributions are undoubtedly enough to guarantee him a lasting place in the story of mathematics, they do not show the extent of Leonardo's enthusiasm and genius for solving the challenging problems of his time, and his impressive ability to work with patterns of numbers without modern algebraic notation. In this article, we will try to shed light on this side of Leonardo's work by discussing some problems from Liber quadratorum, written in 1225, using the English translation, The Book of Squares, made by L. E. Sigler in 1987. All page references in what follows are to that book. Questions for student investigation are at the end of this article, on page 7.