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
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.
© 1996-9 Eric W. Weisstein