The Fence Challenge

Problems in mathematics can often seem unapproachable, requiring years of education and specialization beyond to even understand the statement of a question. For the purpose of democratizing mathematics, we seek problems which can be gameified, so that anyone can interact with and understand them, regardless of age or educational history.

One such problem is that of the fence challenge. Consider the pentomino, in essence a 5 square domino. There are 12 different ways to assemble a pentomino which are distinct up to rotation and reflection. The question of the fence challenge is thus: How much area can be enclosed using the 12 distinct pentominos? We think this question is accessible and interesting; it bears both mathematical depth and entertaining accessibility. This problem, in the form of a game, has been presented to various audiences of both school-age children, working mathematicians, and computer scientists. Even though not everyone approaches the problem with the understanding that they attempt the solution to an isoperimetric problem in the setting of pentominos, we can still observe human approaches to spacial reasoning and critical thinking, providing an accessible, fun mathematical experience.

Beyond a tool for mathematical outreach, we see the fence challenge as a worthy method for measuring human performance in problems with exponentially ballooning solutions. While the maximal area is known, computing maximal areas for all given clockwise orders of the pentominos is both not known and computationally infeasible. Thus, we offer the fence challenge as a method to evaluate human solutions against methods from ML which may be applied to solve the fence challenge. We are especially interested in how solutions from individual players evolve as they play, and which methods they employ to approach optimality. We hope to understand what allows human solutions to problems such as the fence challenge to go beyond approaches from machine learning, and in what circumstances they do so.

Play the fence challenge here!