While visiting the Calculus and Physical Sciences Tutorial Lab at Grand Rapids Community College, a question was posed: for what values of $n$ will the sum of the first $n$ positive integers be a perfect square? A thorough investigation of the problem and the introduction of the concept of an isosceles "almost" right triangle yielded a number of interesting results. One of the results involves a sequence of rational numbers that converges to $\sqrt{2}$, yielding some excellent approximations.