In the nineteenth century, field theory brilliantly resolved a number of questions that had taxed mathematicians for centuries; for example, ‘The circle cannot be squared’ by straight edge and compass, and solving polynomial equations by radicals is not always possible. These successes have continued to be held up as superb examples of the power of mathematical thought, and are demonstrated at an undergraduate level. The purpose of this article is to provide another such natural example which leads to a concrete realisation of the free group on 2 generators.