I recently stumbled upon the following problem:
Consider the plane: You may color each point either red or blue. Is there a way to color it such that each unit circumference (centred anywhere) contains exactly one blue point? And two?
I solved it relatively easy: the "one" case has no solution, and the "two" case is solved placing the blue points in a set of parallel lines, at a distance of two from their neighbours.
Of course I couldn't resist the temptation to consider the $n$ case: For $n$ even, the solution is easily extended considering lines at a distance of $4/n$.
My question is: any help for the odd $n$ case?
If there is any justice in the world, there should be no solution, but I honestly don't know where to start.
EDIT: I have had this idea: if I have a solution for an $m$ case, and another for the $n$ case, such that they do not intersect (that is, whenever a point is blue in a solution it is not in the other) then simply superimposing the solutions yields a valid $m+n$ solution. It also stands to reason that if a $k$ solution exists, and contains an $m$ solution, then removing it yields a $k-m$ solution.
So all that I need to prove is that an odd $k$ solution must contain a $k-1$ solution, and therefore also a $1$ solution, which I have shown cannot exist, and therefore also the generic odd solution cannot exist.
For reference: I know geometry up to the basics of manifolds, analysis up to (but not including) Lebesgue measure and integration, and some group theory.
Here is the proof for the $n = 1$ case: Obviously, there must be at least one blue point: otherwise any circumference would contain no blue points and break the condition. Consider the circle centred in that blue point: it must have one blue point. We therefore have two blue points at a distance of 1, therefore there is a circle that contains both, and violates the condition. Thus, there cannot be a successful configuration of blue points, QED.
I also have this other proof: Take one blue point. Consider all circles containing that point. Those circles cannot have any other blue point, because they must have exactly one. All those circles cover a disc of radius 2 centered in the blue point, and said disc cannot contain any other blue points. Therefore the circle centered in the blue point contains no blue points and breaks the condition, QED.