Card Stacking Problem

Date: 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/