Interesting and engaging mathematics problems can come from anywhere. Sometimes great problems arise from interesting contexts. At other times, interesting problems arise from asking "what if" questions while appreciating the structure and beauty of mathematics. The intriguing problem described in this article resulted from the second context, where trying to create an interesting example to support student thinking produced a rich investigation in its own right. The main purpose of the activity presented in this article is for students to find cases of triangles where the three heights and the lengths of the three sides are all integers. The task is intended to focus primarily on the Common Core State Standard for Mathematical Practice (SMP) 1--"Make sense of problems and persevere in solving them"--while also following SMP 5--"Use appropriate tools strategically." In addition to being an interesting mathematical problem in its own right, this problem about finding at least one triangle with integer values for all the lengths of its bases and heights can help students see the need to persevere in looking for an existence case and not simply apply empirical reasoning to support why no such cases exist. Providing opportunities for students to observe that the area of specific triangles remains consistent regardless of the base and height pair chosen is important and can then lead to further discussions about why these calculations are equivalent for any triangle. Using these triangles can be productive to illustrate whether it matters which side is chosen as the base so that students establish the equivalence of the three ways to calculate the area of a triangle.