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(6): Brian's solution by Brian Wainscott)
I agree with Larry too... it just doesn't address the problem.
There ARE many more irrational than rational points.
But this isn't about the odds of finding a particular point (or a set of points). It is about the existence of a point (or a set of points) that satisfy or refute the problem.
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What if I paint the WHOLE plane black, except for a small circle of radius .00000001 centered on the origin, which I paint white?
What are the odds that if I take a random point, the point will be white...?
The problem with this argument (which is analogous to Brian's, Brian's, and Larry's) is that the probability isn't the issue.
Existence of a white point is the issue. And, in this case, the whole plane is not black.
re(7): Brian's solution
