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: by SilverKnight)

True, points CAN be chosen, but there are relatively few such. Given a random real number in [0,1] it MIGHT be rational, but the odds are it won't be. There are relatively few rationals (compared to the irrationals) in [0,1].

As for my proof, which part is a problem? I proved that S(A,B) has measure 0 in the plane. So not only does it have a non-empty complement from which D can be chosen, amost every point in the plane is in the complement. The assertion is an immediate result of the proof of the claim. I'm confused about what else you are expecting.

Posted by Brian Wainscott on 2003-12-09 15:33:18