From: karr@cs.cornell.edu (David Karr)
Subject: Re: Balls in the urn puzzle (LONG explanat
Organization: Cornell Univ. CS Dept, Ithaca NY 14853
Date: Mon, 9 Nov 1992 19:37:50 GMT

In article <1992Nov6.223828.19474@lmpsbbs.comm.mot.com> johnson@sleipnir.mnesouth.corp.mot.com writes:
[... Lots of just-not-quite-getting-the-point verbiage deleted ...]
>I thought I had implied that you would also change the writing on the
>corresponding ball in the urn, so that ball 2 became 20, 1 became 10,
>and so on.  I see on rereading that I failed to make this clear, and
>apologize.

It was clear enough, but here we are back to the game where I say, "If
you mess up the procedure that decides which balls stay in and which
are removed, you are likely to change the problem in such a way that
you get a different result from the original," and you respond, "But
see, if I mess around with the numbers this way, I get these two
problems that YOU thought were DIFFERENT to come out the same."
To which I can only respond, "Oy vey."

>  But given that, you can see that the *numbers* are handled
>identically,

Not at all.  For example, right after you add the "0" to ball number
1, there are now TWO balls labeled "10" in the urn.  That would never
have happened if you hadn't started playing with numbers.

> and that is all that defines a particular ball.  

Not any more, see previous comment.  Which "10" ball is which?

>The balls
>themselves are interchangeable. 

Only if the person who is removing them can't read.

>Are you saying, then, that writing a number on a ball initiates some
>subtle change so that that particular ball can no longer legitimately
>bear any other number?  The balls are the medium, here...if the balls
>themselves are important, all this set theory is so much rhetoric,
>since it deals only with the numbers.  I agree that the labels are
>the mechanism, and the only importance of the balls is that it provides
>an intuitive means of considering a number of labels.  If this is the
>case, then moving labels from one ball to another is entirely legitimate
>as long as you keep to the rule of one ball per label.

If you want to SWAP the labels between two balls, I don't mind.  It will
make no difference in the results.  But all your proposals involve
deleting some labels altogether or introducing new labels or duplicating
existing labels.  And not surprisingly, it changes results.

Suppose I programmed a robot to perform a two-step task.  On step 1 it
will drop 10 balls labeled 1 through 10 into an urn.  On step 2 it
will search the urn and remove any balls labeled with even numbers.

Now if I allow the robot to operate on an empty urn without
interference, clearly there will be 5 balls in the urn at the end.
But if I sneak into the urn between steps 1 and 2 and append a "0"
to the label of each ball, the robot clearly will see balls labeled
10, 20, ... , 100 and will remove all of them, so 0 balls are left
at the end.

If you would think carefully about your various attempts at sleight
of hand with the ball numberings, you would see how they all do
something similar to the above, only because there are infinitely
many balls instead of just 10, it's easy to get confues.

>   =)etc.  What is a reasonable definition of lim S_n?
>=)
>=)Phrased in this way, there is unlikely to be any controversy.  The
>
>If you mean that it is obvious that the size of S_n is greater than the
>size of S_(n-m) (n and m positive, n>m) then I agree entirely.

And n and m finite, of course.

>Here it is.  This whole error is based on the misconception that an infinite
>set of integers contains all integers.  This is obviously wrong in the case
>of the set of even integers, or the set of integers not divisible by 10 as
>in the first problem.  It is also wrong, although not so obviously, in the
>case of sequential integers.

Sequential integers being 1, 2, 3, ..., I presume.  In other words, you
are saying that the set of all positive integers does not contain the
set of all positive integers.

>An earlier poster noted the tenacity with which people maintain their
>opinions on this problem.  I am also astounded by it--I am entirely
>sure I am right, and I gather my detractors feel the same.

I couldn't agree more with these two sentences.

-- David Karr (karr@cs.cornell.edu)


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