Book Stacking Problem

The book stacking problem asks how far a balanced stack of identical books of uniform density can protrude over the edge of a table. In the idealized planar model, the books are horizontal rigid blocks and contact forces are vertical. With one book per level and a positive integer, the maximum possible overhang (in terms of book lengths) is half the th partial sum of the harmonic series (Paterson and Zwick 2009, p. 19).

The balancing condition is that the vertical projection of the center of mass of the books above each interface lie over the supporting region at that interface. In the harmonic arrangement attaining this bound, the projection lies exactly above the right edge of each supporting book. Since the books are identical, placing the top books in equilibrium adds of a book length to the overhang, which gives the harmonic sum below.

This is given explicitly by

(1)

where is a harmonic number. The first few values are

			(2)
			(3)
			(4)
			(5)

The numerators and denominators of are given by OEIS A001008 and OEIS A002805, respectively.

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, measured from the edge of the bottom card, is

			(6)
			(7)
			(8)

(Derbyshire 2004, p. 6).

For , the number of books required with one book per level is the smallest positive integer satisfying . For , 2, ... book-lengths of overhang, 4, 31, 227, 1674, 12367, 91380, 675214, 4989191, 36865412, 272400600, ... (OEIS A014537) books are needed.

When more than one book or card can be used per level, greater overhangs are possible and the problem becomes much more complex. For example, Paterson and Zwick (2009, p. 43) report an oil-lamp construction with an overhang of 10 using 921 blocks with additional point weights.