- The Lonely Runner Problem
At time zero, k participants depart from the origin of a unit length
circular track to run repeated laps. Each runner maintains a constant
nonzero speed. Is it true that regardless of what the speeds are,
there exists a time at which the k runners are simultaneously at least
1/(k+1) units from the starting point? The term "lonely
runner" respects an equivalent formulation in which there are
k+1 runners with distinct speeds. Is there a time at which a
given runner is "lonely", that is, at distance at least
1/(k+1) from the others?
The Lonely Runner Conjecture, stated in the '60, has been proved for k<7 in 2001. It is now a theorem since June 2016 (see here for the complete solution).
- Limits of Mathematics
One normally thinks that everything that is true is true for a reason.
I've found mathematical truths that are true for no reason at all.
These mathematical truths are beyond the power of mathematical
reasoning because they are accidental and random.