Part a) of the following problem appeared in one of the Putnam Exams (sorry, don't know which year exactly).
If you want to solve Part a) don't read Part b).
You have a painting device, which given the co-ordinates of a points in the 2D plane, will colour all points on that plane black, which are at an irrational distance from the given point.
Initially you start out with the 2D plane being white.
a) You want to colour the whole plane black. What is the minimum number of points you need to feed to the painting device?
b) Show that it is sufficient to feed $(0,0), (1,0), (\sqrt{2},0)$.