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.

This is given explicitly by

(1)

where is a harmonic number. The first few values are

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

(OEIS A001008 and A002805).

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

(6) (7) (8)

(Derbyshire 2004, p. 6).

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, ... (OEIS 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).