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From: stekas@hou2g.UUCP (J.STEKAS)
Newsgroups: net.puzzle
Subject: Re: Balls in the bowl: Final word.
Message-ID: <181@hou2g.UUCP>
Date: Tue, 28-Feb-84 09:50:08 EST
Article-I.D.: hou2g.181
Posted: Tue Feb 28 09:50:08 1984
Date-Received: Wed, 29-Feb-84 09:42:32 EST
References: <220@pucc-i> <178@hou2g.UUCP>, <224@pucc-i>
Organization: AT&T Bell Labs, Holmdel NJ
Lines: 26

> Your arguments show quite convincingly that the limit of the number of balls
> in the bowl, as time-->noon, is infinity.
> 
> What makes you think the ACTUAL number of balls in the bowl at noon has
> anything to do with the limit?  ...

Well, because the limit is the only meaningfull way of getting a unique answer.
Dave's technique of numbering terms in the series can be used to generate any
final result you want because it is an invalid operation. Example...

Starting with an empty bowl, suppose at every t=12:00-1/n the number of balls
in the bowl did not change.  Taking the limiting case, at 12:00 the bowl would
be in the same condition as when we started - empty.

Taking Dave's approach, the number of balls can be made to "not change" by
adding and removing a ball at each t=12:00-1/n  giving a total number at
noon = Sum( 1 + -1).  Now cancell the -1 in every term against the +1 in the
following term to get Sum(1 + -1)=1.  Conclusion - if the number of balls is
not changed at every t=12:00-1/n there will be 1 ball in the bowl.

I'll leave it to the readers to decide which approach is correct.

                                                          Jim