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(5): Brian's solution by SilverKnight)
I agree with Larry.
Instead of picking 3 points in the plane, pick a single point in R^6 (the cartesian product of the plane 3 times). For every point (a,b,c,d,e,f) in R^6, color it either white or black: black if stamping the points (a,b), (c,d), (e,f) results in covering the whole plane, and white otherwise.
What Larry has done is argue that the "white" set in R^6 has measure 0. That doesn't mean there aren't a lot of points in it, only that they are relatively rare compared to black points. It is like saying "the odds of a random number being rational is 0" which is true.
re(6): Brian's solution
