I forget what it was called, but I remember a past post about trying to disprove very, very settled rules of math or science. A lot of the people who commented on it said that they had tried to do this as teenagers. (I never tried to construct unconstructable shapes, but I tried for a couple weeks to design a perpetual motion machine, once. I stopped after my middle school science teacher explained why a certain design wouldn't work - the explanation was what I needed to finally grok the laws of thermodynamics.)