Book Stacking Problem

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

where is the Digamma Function and is the Euler-Mascheroni Constant. The first few values are

(Sloane's A001008 and A002805).

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

References

Dickau, R. M. The Book-Stacking Problem.'' http://www.prairienet.org/~pops/BookStacking.html.

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.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.