You have an ink stamp that is so amazingly precise that, when inked and pressed down on the plane, it makes every circle whose radius is an irrational number (centered at the center of the stamp) black.
Is it possible to use the stamp three times and make every point in the plane black?
If it is possible, where would you center the three stamps?
(In reply to
re(3): Brian's solution by SilverKnight)
The first use of the stamp leaves uninked, concentric circles at rational radii. The second use of the stamp leaves uninked, intersection points of the two sets of concentric circles. Each concentric circle has a countable number of intersections (since the rationals are countable) and there are a countable number of circles(same reason). Therefore, the total number of uninked points is of the order of countable squared and is also countable. The probability(slightly tongue in cheek) that a third use of the stamp from a random point will not cover all the remaining points is the proportion of rationals in the continuum(i.e. 0).
Brian gets the credit because the problem only called for a specific solution and x=transcendental in his diagram is a solution.
His solution is easily generalizable by writing similar equations for non co-linear points. The central argument will be that the distance between two of the random points will be expressed as the solution of an algebraic equation with rational co-efficients which is a contradiction.
re(4): Brian's solution
