A deck of Zip disks

Pick a disk. Any disk.
Except that they’re going to be broken open and destroyed. I haven’t forgotten, Mags! :)

This reminds me of a puzzle or problem I’ve seen in two places. Today I found it in an old issue of the Pi Mu Epsilon Journal. Edward J. Arismendi, Jr. (Cal State University at Long Beach) proposed the following problem: “How far beyond the edge of a table can a deck of cards be stacked without the pile falling off the table?” Answer below.‡

This is also how Chapter One (“Card Trick”) of John Derbyshire’s 2003 book, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics begins:

“Like many other performances, this one begins with a deck of cards. Take an ordinary deck of 52 cards, lying on a table, all four sides of the deck squared away. Now, with a finger slide the topmost card forward without moving any of the others. How far can you slide it before it tips and falls? Or, to put it another way, how far can you make it overhang the rest of the deck?”

The “overhang problem” is also described in the Wikipedia entry for Derbyshire’s book.

Arismendi and Derbyshire do not formulate the overhang problem in terms of Zip disks.

~~~~~~~~~~~~~~
‡ Arismendi proposed this as Problem 649 in the Spring 1987 issue, p. 403;
Arismendi’s solution was printed in the Spring 1988 issue, pp. 546–547.

Answer “Since the mass m of a card is uniformly distributed, the center of mass of the card is at its geometrical centroid. Hence one card will (just) balance if it extends ½ its length over the edge of the table or over the edge of the next card below. Thus place the first card extending ½ its length over the edge of a second card. The center of mass of the two cards is midway between their respective centers, ¼ of the way from the edge of the second card (and ¾ of the way along the first card). So place these two cards so their center of mass is at the edge of a third card. The center of mass of the three cards is
    (2·0 + ½·1)/3 = 1/6
of the way from the edge of the third card, and so on. Hence the top card of n cards can protrude
    1/2 + 1/4 + 1/6 + 1/8 + … + 1/2n
card lengths over the edge of the table.” For a deck of 52 cards, the sum is approximately 2.269, so just over 2¼ card-lengths!

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February 29 posts
 4 years ago: No post
 8 years ago: “Wood grain”
12 years ago: “Abstract in gray”

[Mags]

Where do you come up with these things? Interesting capture and narrative. I'm waiting patiently. =)