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 by Benjamin J. Ladd)

I hypothesize that we will achieve comprehensive inkage by choosing points A,B,C such that in cartesian coordinates both of A's x and y values are rational, both B's x and y values are irrational, and C's x value is rational while its y value is irrational.

If there exists a point which is not inked after we haved stamped at points A, B, and C, then it must be a rational distance from all three points. (Otherwise, of course, it would have been inked)

Let me define our points:

A = (ax,ay) B = (bx,by) C = (cx,cy) and introducing P = (px,py)

as per my assertion;
let ax, ay, and cx be rational and bx, by, and cy be irrational.

Then the three distances in question are: AP, BP, and CP
AP = √((ax-px)²+(ay-py)²)
BP = √((bx-px)²+(by-py)²)
CP = √((cx-px)²+(cy-py)²)

I think it will be possible to prove that it is impossible that all three of these distances are simultaneously rational. I am going to go keep working on it, but I hope that someone can pick up from here.

Cheers!

Posted by Eric on 2003-12-01 23:41:24