# A Stack of Books

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Imagine that you have a number of books, all the same size and shape. Can you stack them on a table, with the stack leaning over the edge of the table, such that the top book clears the edge of the table completely?

Before we can answer the question, there is a physical principle that you need to know. The stack of books will not topple if the following holds for each book in the stack: The centre of mass of all the books above any particular book is directly above any portion of that book.

So, for one book, the book can extend up to ½ its length past the edge of the table without falling on the floor.

For two books, the top book can extend up to ½ of its length past the bottom book. The centre of mass of the two books is at a point representing ¾ of the length of the bottom book, so the remaining ¼ of the book can overhang the edge of the table, as shown in the following diagram: So, for two books, the top book of the stack can overhang the table a distance of ¾ of the length of the book.

You may wish to investigate what happens when there are more than two books in the stack before reading further.

All done? If you've investigated this further, you may have found that the nth book from the top of the stack can overhang an amount equal to ½(1n) of its length beyond the books below. So, for three books, the top book can overhang ½ of a book length beyond the other two books, the middle book can overhang ¼ of a book length beyond the bottom book, and the bottom book can overhang 16 of a book length beyond the table. So, the top book is ½ + ¼ + 16 ≈ 0.92 book lengths beyond the edge of the table.

So, for n books, the amount that the top book can overhang the table is given by the formula ½(1 + ½ + ⅓ + ... + 1n. Playing around with the formula, we find that, for a stack of four books, the top book can be made to clear the table completely, since ½(1 + ½ + ⅓ + ¼) > 1. For the top book to overhang the table by two book lengths, 31 books are required. For the top book to overhang the table by three book lengths, 227 books are required.

Since books aren't ideal objects of uniform density, you may find that it takes more books than this in practice; however, you shouldn't find it too difficult for the top book in a stack of five to clear the edge of a table.

You might ask: Is it possible, at least in theory, to make a stack where the top books juts out any length away from the table, or is there a maximum limit as to how far the top book can be beyond the table's edge? Well, looking at the formula ½(1 + ½ + ⅓ + ... + 1n), the sum in brackets is simply the harmonic series. It has been proven that the harmonic series diverges, so you can have a stack that juts out as far as you want. The series diverges very slowly, though, so, for example, to have the top book overhang 50 book lengths, you will require 1.5×1044 books.

Sources used (see bibliography page for titles corresponding to numbers): 57.