The quadrilaterals problem is a particularly rich task, applicable to in-service teachers as well as to the students they teach. The ways in which students engage in this task, the strategies they use, and the mathematics that results align well with this "MT" Focus Issue on flexible mathematical thinking. The authors offer their observations of various groups of in-service teachers as they investigate this problem and then extend these insights by showing how middle-grades students approached the same task. The purpose of sharing these experiences is to expose the rich potential of the task and to highlight the flexible nature of students' thinking about the task. (Contains 9 figures.)