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The Impossible Problem in Mathematics
When I was in high school, we were going through things that were considered to be impossible in mathematics. One of them was a trisection of the angle only using a compass and a straight edge. So I worked on it for a while and it came up with a method which is not an exact solution. It involved an asymptotic approach where you start with a rather crude trisection. And actually you can get down to a point where there's no difference or at least an infinitesimally small difference between the actual trisection and what you provide it.