  Runge-Kutta methods are a popular class of numerical methods for solving
ordinary differential equations. Every Runge-Kutta method is characterized by
two basic parameters: its order, which measures the accuracy of the solution it
produces, and its number of stages, which measures the amount of computational
work it requires. The primary goal in constructing Runge-Kutta methods is to
maximize order using a minimum number of stages. However, high-order
Runge-Kutta methods are difficult to construct because their parameters must
satisfy an exponentially large system of polynomial equations. This paper
presents the first known 10th-order Runge-Kutta method with only 16 stages,
breaking a 40-year standing record for the number of stages required to achieve
10th-order accuracy. It also discusses the tools and techniques that enabled
the discovery of this method using a straightforward numerical search.
