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Book Stacking Problem

\begin{figure}\begin{center}\BoxedEPSF{BookStacking.epsf scaled 650}\end{center}\end{figure}

How far can a stack of $n$ books protrude over the edge of a table without the stack falling over? It turns out that the maximum overhang possible $d_n$ for $n$ books (in terms of book lengths) is half the $n$th partial sum of the Harmonic Series, given explicitly by

d_n={1\over 2}\sum_{k=1}^n {1\over k}={\textstyle{1\over 2}}[\gamma+\Psi(1+n)]

where $\Psi(z)$ is the Digamma Function and $\gamma$ is the Euler-Mascheroni Constant. The first few values are
$\displaystyle d_1$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}=0.5$  
$\displaystyle d_2$ $\textstyle =$ $\displaystyle {\textstyle{3\over 4}}=0.75$  
$\displaystyle d_3$ $\textstyle =$ $\displaystyle {\textstyle{11\over 12}}\approx 0.91667$  
$\displaystyle d_4$ $\textstyle =$ $\displaystyle {\textstyle{25\over 24}}\approx 1.04167,$  

(Sloane's A001008 and A002805).

In order to find the number of stacked books required to obtain $d$ book-lengths of overhang, solve the $d_n$ equation for $d$, and take the Ceiling Function. For $n=1$, 2, ... book-lengths of overhang, 4, 31, 227, 1674, 12367, 91380, 675214, 4989191, 36865412, 272400600, ... (Sloane's A014537) books are needed.


Dickau, R. M. ``The Book-Stacking Problem.''

Eisner, L. ``Leaning Tower of the Physical Review.'' Amer. J. Phys. 27, 121, 1959.

Gardner, M. Martin Gardner's Sixth Book of Mathematical Games from Scientific American. New York: Scribner's, p. 167, 1971.

Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science. Reading, MA: Addison-Wesley, pp. 272-274, 1990.

Johnson, P. B. ``Leaning Tower of Lire.'' Amer. J. Phys. 23, 240, 1955.

Sharp, R. T. ``Problem 52.'' Pi Mu Epsilon J. 1, 322, 1953.

Sharp, R. T. ``Problem 52.'' Pi Mu Epsilon J. 2, 411, 1954.

Sloane, N. J. A. Sequences A014537, A001008/M2885, and A002805/M1589, in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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© 1996-9 Eric W. Weisstein