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?
Let x be any positive number such that x^2 is irrational.
Let AC be any line segment on the plane with length 2x.
Let B be the midpoint of AC.
Let P be any point in the plane.
Let u be the length of PA.
Let v be the length of PB
Let w be the length of PC.
Let T be the measure of angle ABP.
/ \ \_
u/ \ \__
/ |v w\_
/ \ \_
/ T \ \
By the law of cosines:
u^2 = v^2 + x^2 - 2vx*cos(T)
w^2 = v^2 + x^2 - 2vx*cos(pi-T)
Since cos(pi-x) = -cos(x), the second equation can be written as
w^2 = v^2 + x^2 + 2vx*cos(T)
Adding the two equations together yeilds
u^2 + w^2 = 2v^2 + 2x^2
If we assume u, v, and w are all rational, then a contradiction is formed because if both u and w are rational then v^2 = (u^2 + w^2 - 2x^2)/2 is irrational which means v must be irrational.
So points A, B, and C are three points which allow the stamp to cover the entire plane.
Edited on December 9, 2003, 10:08 am