Henry Ernest Dudeney/Modern Puzzles/129 - The Circle and Discs/Solution

by : $129$

 * The Circle and Discs

Solution

 * Dudeney-Modern-Puzzles-129-solution.png

The dotted lines represent the red circle and a regular pentagon inscribed within it.

The center of this circle is the point $C$.

Find the point $D$ which is equidistant from $A$, $B$ and $C$.

With radius $AD$, draw the circle $ABC$.

Similarly construct the points $E$, $F$, $G$ and $H$.

These points are the centers of the circles representing the discs with which the red circle is covered.

We are given that the red circle is $6$ units in diameter.

The diameter of the discs is then just under $4$ units in diameter.

Covering is possible if the ratio of the two diameters is greater than approximately $0.6094185$.

In the case above, where all five discs reach the center of the red circle, the ratio is the golden mean $\approx 0.6180340$.