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?
Interestingly enough, the limit of the series of geometrical figures
formed by continuously decreasing the ball size IS A SOLID TRIANGLE,
without any spaces! The area of this limit is the same as (100%
of) the area of the pool rack. This is true in the mathematical
meaning of a limit, because every point inside the pool rack can be
determined to be arbitrarily close to to some pool ball, by decreasing
the pool balls sufficiently. Therefore, in one sense the triangle
IS completely covered.
"Paradoxically", I agree with Charlie and others that the limit of the
area asympotically approaches 90.69..% of the pool cue.
Therefore, in a different sense, the triangle IS NOT completely covered.
The problem is inherent in the problem statement. In general,
where f and g are functions, the limit(f(g)) is not the same as the
f(limit(g)). In this case, think of f as the area function and g
as the nth geometric shape formed by decreasing the pool ball size.
Then, the limit of the area of the series = 90.69..%, but the area of
the limit of the series = 100%. The fact that the limit of the
area of the series is less than 100% DOES NOT prove that the limiting
shape is not solid (which it in fact is). So which are we asking
about when we ask if the area is completely covered? I think it
is the limiting shape, and my answer is yes, it is covered.
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Consider a simpler case, which is the line segment from 0 to 1.
First cover the point at 1/2.
Then the in between points at 1/4 and 3/4.
Then the in between points at 1/8 and 3/8 and 5/8 and 7/8.
etc.
etc.
In the limit, is the line covered?
Well, in this case the limit is a solid line segment, but at every
iteration the area covered is 0 (just a bunch of isolated points).
The limit of the length of the series = 0 , but the length of the limit of the series = 1 (100%)!
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By the way, I just discovered this site about an hour ago, and
this is my first submission. I promise that future submissions
won't be as long.
Ambiguity leads to Paradox
