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
More Thoughts by bob909)
Though the conclusion of this reasoning matches the actual solution, it is ultimately flawed. You assumed that a near infinite number of gaps implies that the rack would be significantly uncovered. However, the size of these gaps becomes infinitesmal.
You would need some kind of step in reasoning of the form "(f(x)g(x)) approaches a nonzero number whenever f(x)->infinity" (in this case, g(x) is like size of a gap, f(x) is like the number of gaps). But the counterexample of f(x)=x, g(x)=(1/x^2) shows that this is not necessarily the case.
Edited on September 17, 2004, 5:28 pm
re: More Thoughts
