You have an ink stamp which is so amazingly precise that, when inked and pressed down on the plane, it...

analysis/irrational.stamp

Problem

You have an ink stamp which is so amazingly precise that, when inked
and pressed down on the plane, it makes every circle of irrational
radius (centered at the center of the stamp) black.

Question:  Can one use the stamp three times and make every point
in the plane black?  [assume plane was white to begin with, and
ignore the fact that no such stamp is physically possible]

Show solution

Yes.  Center the stamp at (0,0), (1,0), and (0,pi).

Suppose there is a point (x,y) which is not covered.
Then there are rational numbers a,b,c satisfying the following equations:

  (1)   x^2   +   y^2     =  a
  (2) (x-1)^2 +   y^2     =  b
  (3)   x^2   + (y-pi)^2  =  c

Subtract (2) from (1) and solve for x.  Thus x is rational.
From equation (2), y is algebraic.  But equation (3) implies
that y is transcendental, contradiction.