15.15 A paper punch can be placed at any point in the plane, and when it operates, it can punch out points at an irrational distance from it. What is the minimum number of paper punches needed to punch out all points in the plane?
(51st Putnam Mathematical Competition, 1990)
［Solution］Obviously, two hole punchers are not
enough.
Below we prove that three hole punchers are sufficient.
For example, in a Cartesian coordinate system, place the three hole punchers at points $A(0,0), B(d, 0)$, $C(2d, 0)$, where $d=\sqrt[4]{2}$.

Let $P$ be any point on the plane, and the distances from $P$ to $A, B, C$ be $a, b, c$ respectively. Thus, we have
$$
\begin{array}{l}
a^{2}+c^{2}=2 b^{2}+2 d^{2}, \\
a^{2}-2 b^{2}+c^{2}=2 \sqrt{2} .
\end{array}
$$
$a, b, c$ cannot all be rational numbers, so point $P$ must be punched by at least one hole puncher.