Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site pucc-i Path: utzoo!watmath!clyde!burl!ulysses!gamma!exodus!mhtsa!mh3bs!eagle!harpo!ihnp4!inuxc!pur-ee!CS-Mordred!Pucc-H:Pucc-I:ags From: Pucc-H:Pucc-I:ags@CS-Mordred.UUCP Newsgroups: net.puzzle Subject: Re: Balls in the bowl: One Last time. Message-ID: <228@pucc-i> Date: Fri, 2-Mar-84 09:38:02 EST Article-I.D.: pucc-i.228 Posted: Fri Mar 2 09:38:02 1984 Date-Received: Sun, 4-Mar-84 00:20:15 EST References: <220@pucc-i> <178@hou2g.UUCP>, <224@pucc-i> <181@hou2g.UUCP>, <227@pucc-i> <186@hou2g.UUCP> Organization: Purdue University Computing Center Lines: 32 > Since the # of balls is well defined only on [11:59, 12:00), taking > the limit can't tell me what the state is at 12:00. But the numbering > technique cannot be used unless the precise order of removing the balls > is known, and that information wasn't available (faulty probelm?). You almost have it, but there is one small point. It is true that the order of removal was not specified, which is precisely what makes the problem ill-defined. The very purpose of the numbered-balls argument was to point out that the order of removal makes a difference. Many people have been claiming that the answer has something to do with an infinite series (100-1) + (100-1) + ..., in which "terms can be rearranged to get different answers". This explanation is invalid for two reasons: 1. The value of an infinite series can be defined only in terms of the limit of partial sums, and I have already pointed out that limits have nothing to do with the Balls in the Bowl Problem. 2. The infinite series argument fails to explain why the (1 in, 1 out) version of the problem has a unique answer, which the numbered-balls technique handles successfully. One last variation on the problem to think about: what happens in the (1 in, 1 out) version if the bowl is not initially empty? -- Dave Seaman ..!pur-ee!pucc-i:ags "Against people who give vent to their loquacity by extraneous bombastic circumlocution."