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]]]]]]]

Consider one card: its center of gravity, at 1/2 the card, should be above the table, or it will fall.
Consider two cards: The top card's center of gravity should be above the end of the card under it, or it will fall.  Also, the center of gravity of the two cards should be above the table. For two cards, the lowest card sticks out 1/4 and the top card sticks out 1/2 over the underlying card (Or 3/4 over the table).
Consider three cards: the two top ones behave as the above two cards.  The center of gravity of the three is above the table if the lowest card sticks out 1/6 (Then 36/72 of the mass is above the table).  The top card now sticks out 1/2+1/4+1/6

In general: (1+1/2+1/3+...+1/52)/2 for 52 cards, which is abouth 2.269, multiplied with the 3.5 inches, this gives 7.942 inches

Posted by Hugo on 2005-02-04 15:19:38