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: Solution, I got confused. by Juggler)

If the larger balls were left in place and at each stage balls were added that were exactly tangent to the other balls, as much as possible, then at each stage, the unfilled area would decrease by some fraction.  It would probably plug more that 1/2 the remaining free area (even taking this as a 2-D problem so that the smaller balls don't partially go underneath the larger balls), but to be generous, let's say that only 1/3 of the unfilled area becomes filled.  That is, at each stage the unfilled area gets multiplied by 2/3.

Any tiny amount of uncovered area could then be achieved by raising 2/3 to a high enough power, so there is no limit as to how much of the area would be covered, and you might as well say that the whole base was covered.

In reality, as mentioned, probably more than 1/2 of exposed area would be covered at each stage and so converge even faster, especially if one then allows smaller balls to go under larger ones in 3 dimensions, avoiding boundary problems.

 

Posted by Charlie on 2004-09-17 14:27:45