ags@pucc-i (Seaman) (02/24/84)
I am still getting a few mail responses to the "balls in the bowl" problem. Many trick answers have been suggested. In case anyone is still wondering, the answer I had in mind involves the "numbered balls" argument, already posted by someone else, which says that the answer is indeterminate. The answer could be zero, or infinity, or anything between. -- Dave Seaman ..!pur-ee!pucc-i:ags "Against people who give vent to their loquacity by extraneous bombastic circumlocution."
stekas@hou2g.UUCP (J.STEKAS) (02/27/84)
I am still unconvinced that the problem of the balls in the bowl can have
any other answer than infinity. At every time t=12:00-1/n there are 99n
balls in the bowl. Therefore the number of balls is monotonically increasing
without limit as noon is approached. One can play games with the terms in
an infinite series only if it converges.
Jimags@pucc-i (Seaman) (02/27/84)
> I am still unconvinced that the problem of the balls in the bowl can have > any other answer than infinity. At every time t=12:00-1/n there are 99n > balls in the bowl. Therefore the number of balls is monotonically increasing > without limit as noon is approached. One can play games with the terms in > an infinite series only if it converges. 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? Especially since it has already been pointed out that if you number the balls and assume that ball number N is removed at 1/N minute before noon for all N, then every ball is removed and none are left. As an aside, I have observed that a good way to get lots of discussion on an article is to label it "Final word" or the equivalent on the subject line. -- Dave Seaman ..!pur-ee!pucc-i:ags "Against people who give vent to their loquacity by extraneous bombastic circumlocution."
stekas@hou2g.UUCP (J.STEKAS) (02/28/84)
> 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
ags@pucc-i (Seaman) (02/29/84)
>> 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. ---------------------------------------------------------------------------- You can't choose a method simply because it is the only way to get an answer you like. The method also has to make sense. Your approach sounds rather like a creationist trying to defend his religious beliefs. (Flames to net.religion, please -- where I won't see them.) Your suggested modification of the problem (one ball in and one ball out each time) DOES give a unique answer of zero balls in the bowl at noon, despite the fact that the LIMIT is undefined -- more evidence that the limit is irrelevant here. Please explain to me how the numbered-balls argument can be used to arrive at any answer other than zero for this version of the problem: If the first ball in is numbered "1", then the first ball out is also numbered "1", simply because no other balls are available. The numbered-balls argument gives an ambiguous answer to the (100 in, 1 out) problem because the PROBLEM is faulty, not because the ANALYSIS is faulty. Consider the following variation on the (100 in, 1 out) problem: The balls are not numbered, but ONE of the first group of balls is green. All other balls are red. Question: Is the green ball still in the bowl at noon? Question: Does it make a difference if the green ball is not known to be one of the first 100 to go in? Question: Suppose there are N green balls. How many of them are in the bowl at noon? Question: Suppose one of each group of 100 balls is green. How many green balls are in the bowl at noon? *** Dudley Moore: "But those questions don't make any sense!" Questioner: "Correct." - NOVA program "It's About Time" (PBS) -- Dave Seaman ..!pur-ee!pucc-i:ags "Against people who give vent to their loquacity by extraneous bombastic circumlocution."
mwm@ea.UUCP (03/03/84)
#R:pucc-i:-22000:ea:10200001:000:1760
ea!mwm Mar 3 06:25:00 1984
/***** ea:net.puzzle / hou2g!stekas / 1:27 am Feb 27, 1984 */
I am still unconvinced that the problem of the balls in the bowl can have
any other answer than infinity. At every time t=12:00-1/n there are 99n
balls in the bowl. Therefore the number of balls is monotonically increasing
without limit as noon is approached. One can play games with the terms in
an infinite series only if it converges.
Jim
/* ---------- */
You are right, the answer is infinity, as in what you get when you divide
any non-zero number by zero. In that case, as in this one, we write down
the tripped-eight symbol to indicate that the answer is indeterminate.
Key fact to note: infinity, per se, is not a number, and cannot be manipulated
like one. The tripped-eight symbol we (meaing mathematicians) read as infinity
actually has two seperate and distinct meanings, depending on context. When you
confuse these contexts, you start getting answers where there aren't any.
The first meaning is the one above (indeterminate). The second occurs only
with limits and such, as the bound on a variable. This means (and should be
read as) "increases without bound." In other words, goes to Aleph-null
(which IS a number, and CAN be manipulated like one) and maybe beyond.
BTW - having a series converge IS NOT sufficient information to allow you
to (safely) play games with a series. The series must converge absolutely,
otherwise you can make it sum to anything (as in this case).
Sorry to be so long-winded, but you hit a tender spot.
<mike
"Anyone who cannot cope with mathematics is not fully human. At best he is a
tolerable subhuman who has learned to wear shoes, bathe, and not make messes
in the house."
-Lazurus Long