We present results of a grounded analysis of individual interviews in which students play Vector Unknown -- a digital game designed to introduce visualizing vectors, scaling vectors, vector addition, and vector equations, attending to the geometric and algebraic representations of vectors. The game was designed to be used at the beginning of a linear algebra course, and participants in the study were students who had not previously taken such a course. We categorize the strategies the students employed while playing the game and analyze how these strategies evolved throughout the students' gameplaying. They range from less anticipatory button-pushing to more sophisticated strategies based on approximating solutions and choosing vectors based on their direction. We found that student focus alternated between numerical and geometric aspects of the game interface, which provided additional insight into their strategies. These results have provided revisions to the game and also informed our team's plans for incorporating it into classroom instruction.