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
Brian's solution by Larry Settle)
Larry,
It is possible that the three random points turn out to be something like... (0,1) (0,2) and (0,3) (or any three points with rational distance to the origin, I can think of at least four more :-) ).
Then, clearly this doesn't cover the origin.
So, "Any three randomly selected distinct points will ink the entire plane with probability one." seems to be faulty.
Your continuation of "Brian's A,B,C works just fine if he selects x to be transcendental." is not sufficient. (π+1,0) (π+2,0) and (π+3,0) will not cover point (π,0).
What's more, if he selects the x then this violates your earlier notion of randomly selecting distinct points.
re: Brian's solution
