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Book Stacking Problem

Posted by Diego Fonstad
Diego Fonstad
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on Saturday, 05 March 2011
in Great Explanations

Teachable Moment: Great visual way of thinking of and understanding harmonic series.

How far can a stack ofn books protrude over the edge of a table without the stack falling over? It turns out that the maximum overhang possible d_nfor n books (in terms of book lengths) is half the nth partial sum of the harmonic series.

 

 

BookStackingOverhangs This is given explicitly by

d_n=1/2sum_(k=1)^n1/k=1/2H_n,
(1)

where H_n is a harmonic number. The first few values are

(Sloane's A001008 and A002805).

BookStackingCards

When considering the stacking of a deck of 52 cards so that maximum overhang occurs, the total amount of overhang achieved after sliding over 51 cards leaving the bottom one fixed is

(Derbyshire 2004, p. 6).

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.

When more than one book or card can be used per level, the problem becomes much more complex. For example, using cards stacked in the shape of an oil lamp, an overhang of 10 is possible with 921 blocks (Paterson and Zwick 2006).

SEE ALSO: Harmonic NumberHarmonic Series

REFERENCES:

Boas, R. "Cantilevered Books." Amer. J. Phys. 41, 715, 1973.

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, pp. 3-8, 2004.

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.

Gamow, G. and Stern, M. Puzzle Math. New York: Viking, 1958.

Gardner, M. Martin Gardner's Sixth Book of Mathematical Games from Scientific American. New York: Scribner's, pp. 167-169, 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.

Hall, J. F. "Fun with Stacking Blocks." Amer. J. Phys. 73, 1107-1116, 2005.

Havil, J. "Maximum Possible Overhang." §13.11 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 132-133, 2003.

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

Paterson, M. and Zwick, U. "Overhang." In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, Held in Miami, FL, January 22-24, 2006Philadelphia, PA: SIAM, pp. 231-240, 2006.

Pickover, C. A. "Some Experiments with a Leaning Tower of Books." Computer Language 7, 159-160, 1990.

Pickover, C. A. Computers and the Imagination. New York: St. Martin's Press, 1991.

Pickover, C. A. The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, p. 238, 2002.

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 A001008/M2885, A002805/M1589, and A014537 in "The On-Line Encyclopedia of Integer Sequences."

Sutton, R. "A Problem of Balancing." Amer. J. Phys. 23, 547, 1955.

Walker, J. The Flying Circus of Physics with Answers. New York: Wiley, 1977.

d_(51) = 1/2H_(51)
(6)
= (14004003155738682347159)/(6198089008491993412800)
(7)
= 2.25940659073333...
(8)
d_1 = 1/2=0.5
(2)
d_2 = 3/4=0.75
(3)
d_3 = (11)/(12) approx 0.91667
(4)
d_4 = (25)/(24) approx 1.04167
(5)

Zombie Cat References: Weisstein, Eric W. "Book Stacking Problem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BookStackingProblem.html

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