Many American classrooms have adopted a strategy which mostly focuses on rote memorization to teach math to its students. While this may sometimes help students pass multiple choice exams, we argue that this focus will not help students develop problem solving and critical thinking abilities. We created different activities targeted towards a wide range of age groups to see how students would respond to difficult problems that require perseverance and critical thinking. Each activity fit in with the Common Core Standards laid out by the Colorado Department of Education. Different activities catered to different age groups, but all of them were meant to be challenging for the audience. The culminating activity was shaped around a classic problem of Hercules fighting the Hydra. Hercules cuts off a head of the multi-headed dragon, and a certain amount of heads grow back based on rules we will define. We asked the students how the Hydra would behave as Hercules continued to attack. To help them with this problem, we compared it to a special group of fast growing sequences called Goodstein Sequences.