Find a point that minimizes the sum of the distances to N given points on a number line.
(In reply to solution (spoiler)
Clearly the point itself constitutes the answer.
Adding two points, one to the left to the existing point,
another to the right does not affect the solution.
Add another two points...etc
So much for the odd case.
SImilar reasoning for the even case, except starting
anyplace between the two initial points.