# book-stacking problem

How much of an overhang can be achieved by stacking books
on a table? Assume each book is one unit long. To balance one book on a
table, the center of gravity of the book must be somewhere over the table;
to achieve the maximum overhang, the center of gravity should be just over
the table's edge. The maximum overhang with one book is obviously 1/2 unit.
For two books, the center of gravity of the first should be directly over
the edge of the second, and the center of gravity of the stack of two books
should be directly above the edge of the table. The center of gravity of
the stack of two books is at the midpoint of the books' overlap, or (1 +
1/2)/2, which is 3/4 unit from the far end of the top book. It turns out
that the overhangs are related to the harmonic numbers *H*_{n},
(see harmonic sequence) which
are defined as 1 + 1/2 + 1/3 + ... + 1/*n*: the maximum overhang
possible for *n* books is *H*_{n}/2. With
four books, the overhang (1 + 1/2 + 1/3 + 1/4)/2 exceeds 1, so that no part
of the top book is directly over the table. With 31 books, the overhang
is 2.0136 book lengths.