# Can one make a stack of books that spreads infinitely beyond the edge of a stable?

Suppose you have infinite books of the same dimensions and weight. Can you make a stack of books that spreads beyond the edge of a table infinitely?

I reasoned that the concept of a center of gravity will not allow this since spreading beyond the edge of the table means that the center of gravity is off the table.

I am trying to formulate a proof to justify or debunk my hunch. I think this may involve geometric series since just a fraction of a new book on top of the stack remains on the table.

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"Can one make a stack of books that spreads infinitely beyond the edge of a stable?" - No, because the horse will always knock them over. – Joshua Shane Liberman Apr 17 '12 at 18:27
Didn't I see this a long time ago in Martin Gardner's column? – Andrea Mori Apr 17 '12 at 18:38
See this. – J. M. Apr 17 '12 at 18:43
You can do much better than the harmonic series referenced in the answers in terms of overhang for a given number of bricks. The overhang grows as $n^{\frac 13}$, rather than $\log n$. See math.dartmouth.edu/~pw/papers/maxover.pdf – Ross Millikan Sep 12 '12 at 22:24

## 2 Answers

Take four congruent books.

Place the first book so that (just less than) 1/8 of its length juts out over the table.
Place the second book so that (just less than) 1/6 of its length juts out over the first book.
Place the third book so that (just less than) 1/4 of its length juts out over the second book.
Place the fourth book so that (just less than) 1/2 of its length juts out over the third book.

Now the fourth book extends out over the table by (just less than) 25/24 book lengths! And if you use enough books, you can make this extension arbitrarily large.

In practice, books aren't very good for this, because they are not stiff enough for their weight. I find unembossed credit cards to be an excellent substitute (I work for a credit-card-embossing company).

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I remember somebody on MO saying they tried this out with Springer books... they seem to work well. – J. M. Apr 17 '12 at 18:57

Yes, you can do this infinitely! The sum of the overhung lengths is a harmonic series.

http://www.robertdickau.com/BookStacking.html

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Actually, this way you can't make an infinite overhang, but an arbitrarily large one. Even if you think the center of gravity of infinitely many books is defined, you have to build the stack from the bottom, so you start with some $\frac1{2n}$, not with $\frac1\infty$. – Hagen von Eitzen Sep 12 '12 at 22:07
Thanks for pointing out that subtlety. – PatEugene Sep 19 '12 at 10:55