I highly recommend reading Jason Dyer’s description of the Erdős discrepancy problem, the subject of the most recent Polymath project (the Polymath projects are an experiment in massively collaborative mathematics, where anyone at all can contribute something towards a solution). The Erdős discrepancy problem in particular is fascinating because it is quite easy to understand the statement of the problem with only a knowledge of basic arithmetic—but it has been unsolved for almost 80 years!! Jason’s explanation of the problem is clear and engaging, and has cute pictures—go give it a read!