From: mk452a@oktext.sbc.com (Mark Kamphaus)
Subject: Re: Balls in the urn puzzle
Reply-To: mk452a@oktext.UUCP (Mark Kamphaus)
Organization: Southwestern Bell
Distribution: na
Date: Thu, 5 Nov 92 20:43:28 GMT

I received a couple of replies in e-mail and have
kept up with the posts on this "puzzle" but I
still do not like the answers.
One reason is that they do not seem to be answering
the question as asked, but manipulating it into
a different problem and then attempting to answer
that one. 
The problem establishes discrete intervals during
which an action which produces a finite result is
performed.
During the first interval, 10 balls numbered 1-10
are placed in the urn.  In the second interval,
10 balls are added numbered 11-20 and the ball 
numbered two is removed.  This process begins at
one minute till midnight and every 1/2 of the
remaining period giving an infinite number of
periods. The question asks How many balls are in
the urn after midnight.

First, the number of balls in the urn is
not given by 10i-i because there is no
removal in the first period.  The number
of balls in the urn is given by
f(i)=9i+1 where i is the interval number.
Examination of the problem will show that
the following statements are true.
1) All balls put into the urn will be removed
   from the urn . [ This is trivial, tell me
   the number of the ball and I will tell you
   the interval that it was removed, which will be
   #+1=i ]
2) No removal of a ball will ever empty the
   urn.  [At the time of the removal I can
   also tell you the number of balls remaining
   in the urn (9i+1 or from (1) 9#+10)
For both of these statements to be true, we must
examine the third given, and that is that we have
a finite action occuring within a discrete interval.
[ consider the set of positive numbers, every element
is discrete and finite, but the set is infinite ]
Since no removal of a ball ever empties the urn, consider
the time that each ball is in the urn.  It can
trivally be shown that each ball spends a finite time
in the urn, however, as i->oo t->0.  I'm sure there is
some function name for it but we can tell
which interval the ball was placed into the urn by
noting that balls A1,A2,..B0 are placed in the
urn during interval B and removed during the
interval defined above.  What this leads to is the
observation that as midnight approached we have an infinite
number of balls spending zero time in the urn.
So does that make it virtually empty or
virtually full?  
The question has not been answered yet.  We know what
the function does as it approaches midnight, but
we don't know what happens at midnight.  
My own feelings is that this can't be dismissed as
impossible in reality.  If we start considering
real world constraints, then at some point in time
before midnight, the function can no longer be
perfomed and midnight happens with a finite
number of balls in the urn (if the assumption
that both the addition and subtraction stop
when either can not be perfomed).  Different
answers for other assumptions. But theoretically,
let us examine real world requirements. Infinite
resources are not available, but infinite resources
are required for the task.  The speed of the balls moving
into and out of the urn becomes important.  It doesn't
take to long for the balls to reach a faster than light
speed.  Also, we should consider what happens when the
intervals decrease below 1/c in the observation frame.
What can be observed then?  I think that if we make the
assumption of infinite resources being available, that
we can not make the assumption that midnight ever occurs.

mark