There is an intriguing maths conjecture called the lonely runner conjecture. It is intriguing because it is easily stated and on the face of it looks possible to prove. However, it has been unsolved for 45 years. Let me share it. There is a track of unit length upon which n runners run at n arbitrary different integer (constant) speeds. The runners never tire and run forever. The lonely runner conjecture says that at some point during this never-ending race each runner becomes lonely ( loneliness being defined as no closer than 1/n to the nearest runners).
A link to a possible proof is here. If its right its right but if its wrong it might be close enough for some genius to fill in the missing piece of the puzzle.
Any views on this or suggestions for alternative solutions.