A deck of 52 playing cards is placed over-hanging the straight edge of a table. The cards are identical and uniformly dense. Suppose these cards are 3.5 inches in length, and this side is perpendicular to the table's edge. If the individual cards can be pushed forward or pulled back from the edge as you please, what is the farthest the cards can reach beyond the edge without any card tipping or leaning off of the card or table immediately below it?

Example of what a three-card deck may look like:
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[[[[Table]]]]]]]

(In reply to re(2): Ballast (pointer) by ajosin)

It is a very clever design, but as far as I can see, it won't work.
I wrote down the force-moment sums.  Use 1 unit to be the length of the domino with 1 unit weight).
The left part (left of Xu) then works out:
9*3/4*3/8 (9 domino's with weight 3/4 where the center of gravity is at 3/8 of the balancing Xu line)
1*1/4*1/8
Gives a total of 82/32
The right part work out at
9*1/4*1/8
1*3/4*3/8
1*3/4
1*5/4
giving a total of 82/32
So far so good.  Now, in order to avoid dominoes 6 and 10 to fall off, the 9 domino is cleverly moved to the right, and 8 is moved the same amount to the left (In order to keep the center of gravity above the table)..
Here arises the problem: domino 10 (+ domino 6) will create an upward (counter-clockwise) moment on 9.  Further more: 8's counter-clockwise moment will also be larger and the pile (1,2,3,4,5,7,8,9,11) will fall off  12.  And then the rest falls off the table.

Posted by Hugo on 2005-03-18 09:29:46