A pool rack is an equilateral triangle, filled with 15 equal-sized balls. Seen from above, we'd see a triangle, with 15 circles within.

Imagine we used smaller and smaller balls. The more the balls, more area of the triangle would be covered.

In the limit, with infinite balls, would all of the triangle be covered?

Imagine each circle as insribed within a regular hexagon.  The tesselation of these hexagons corresponds to the way the balls fit within the rack (ignoring the edges and corners.)

The proportion of each hexagon filled by its circle is independent of its size.

This proportion of the triangle covered would not change, thus the triangle will never all be covered.

This problem actually has a false assertion: "The more the balls, more area of the triangle would be covered."

-Jer

Posted by Jer on 2004-09-17 13:13:32