In May, Haag was finishing her first year of graduate school at the University of Colorado, Boulder, where Kertzer was an undergraduate. Both looked forward to a break from classes. Haag planned to explore new hikes and climbing routes. Kertzer, a Boulder native, wanted to play soccer and prepare his grad school application. But as aspiring research mathematicians, they had also applied for a half-time summer research program in the group of the mathematician Katherine Stange. Stange is a number theorist who describes herself as a mathematical "frog" ?" someone who delves deep into one problem's intricacies before hopping to another. She is interested in "simple-seeming questions that lead to a richness of structure," she said. Her projects often poke at number theory's elusive open problems by using computers to generate large data sets.