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?

(In reply to re(4): Solution by David Shin)

Yes, one cannot write an encyclopedia as a solution. However, it doesn't seem to have strained you too much to give the method you gave. Since there are n unit balls in the bottom row, the distance between centers of the outer balls is then 2(n-1) and the two leftover pieces of the bounding triangle are each sqrt(3), making the total length of the bottom side of the bouding triangle 2[n-1+sqrt(3)]. The area of the bounding triangle is thus

[sqrt(3)/4]{2[n-1+sqrt(3)]}^2=sqrt(3)[n-1+sqrt(3)]^2

just as Bractals said.

However, I think there is an assumption being made here about how the balls fit inside the triangle for general values of n. That is, there is still an unsolved math (or mathematical physics) problem here.

Posted by Richard on 2004-09-24 13:17:16