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If it turns out that a problem is equivalent to a known open problem, then the open-problem tag is added. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it before, and with what results?"
84
votes
Not especially famous, long-open problems which anyone can understand
It is currently unknown if all triangles have a periodic billiard path. (See, for example, https://en.wikipedia.org/wiki/Outer_billiards#Existence_of_periodic_orbits)
165
votes
Not especially famous, long-open problems which anyone can understand
The moving sofa problem: What rigid two-dimensional shape has the largest area $A$ that can be maneuvered through an L-shaped planar region with legs of unit width?
So far the best results are $2.2195 …