Card Stacking ProblemDate: 06/10/2006 at 11:12:30 From: Anthony Subject: Stack of overhanging objects and max. overhang I've read somewhere that if you were to stack objects (cards for example) on top of each other (flat like a deck of cards), and have each card overhang the card below by a little bit, you could theoretically place a card who's trailing edge is actually past the front edge of the bottom card. The amount of each overlap would have to get smaller and smaller as you stack them up. My common sense would say that the stack would tip over and that there is a max, but according to what I read there actually was no maximum, and you could theoretically create a stack of cards that cantilevers as far as you want (it would be ridiculously high though!). My thought is that once the center of mass of cards 2 through x passes the front edge of card 1, the stack would tip over. Which, I think, means that'd you'd never get very far. Date: 06/12/2006 at 16:15:59 From: Doctor Douglas Subject: Re: Stack of overhanging objects and max. overhang Hi Anthony. Perhaps it goes against common sense, but your thinking is good common sense--it has everything to do with the center of mass of the top portion of the stack. Once the center of mass of any sub-stack passes the edge of the card immediately below that sub-stack (note that this card is not necessarily the bottommost card), then the sub-stack is out too far and it will tip over. The top card should be pushed halfway out (or for stability, just a teensy bit less). The second card should be pushed out so that the combined center of mass of the two top cards moves just short of the edge of card #3. How far is this? Since the top two cards have a combined length of 3/2 card lengths, the center of mass is at the 3/4 point. So you could push the second card out by an additional 1/4 of a card length. At this point the top card overhangs the edge by 3/4 of a card length (does this agree with your intuition?), but needs the second card as an intermediary to do so (if it were pushed out 3/4 of its length all by itself, it'd clearly tip over). Using this type of reasoning, you can work out how far to push the stack of the top 3 cards (you can push it an additional 1/6 of a card length, after which the top card has (1/6 + 1/4 + 1/2) = 11/12 of its length overhanging the edge. The fourth card adds 1/8 of a card more length, after which the top card is wholly off the edge: (1/8 + 1/6 + 1/4 + 1/2) = 25/24 and this is greater than 1. You can see that we are indeed adding up smaller and smaller quantities each time, but the sum *is* growing, and provided we take enough terms, it can be made as large as we want. The infinite sum 1 + 1/2 + 1/3 + 1/4 + ... is known as the Harmonic series, and it is associated with deep and very interesting mathematics. The sum diverges, meaning that it can be made larger than any finite number. Our card stacking problem deals with half of this sum, but that too diverges. For more on this problem, see the following web pages: Book Stacking Problem http://mathworld.wolfram.com/BookStackingProblem.html Harmonic Series http://mathworld.wolfram.com/HarmonicSeries.html At the first web page, there is a nice graphic of what could be achieved by a perfect arrangement of a 52 card deck. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ |
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