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 A pool rack filled with balls (Posted on 2004-09-17)
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?

 See The Solution Submitted by Federico Kereki Rating: 3.3636 (11 votes)

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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.

 Posted by Steve Herman on 2004-09-18 20:53:33

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