Balancing the Books

Thanks to David Tempel for suggesting this puzzle, which asks how big a stack can you make before it goes over the edge.

You can put a book on a table with part of it over the edge. As long as enough of the book is on the table, it will not fall. How much is "enough"? Just the tiniest bit more than half.

Now we can pile two books on the table so that the top one extends over the edge by more than half its length.

Is it possible to stack more books in this fashion so that the top one is fully beyond the edge of the table? If it is possible, how many books do you need?

First take a guess. Then try it. And finally, give an explanation of your result.

Course:

Math [1]

Result/Solution(s)

Here’s our stack of
three books,

four,

and five.

With four books, the top one is just about fully beyond the edge. We’ve added a line to the photo to help make this clear.

How does this work? In order for one book to remain on the table at least half of its mass must be on the table. In this diagram, the distances d1 and d2 must be equal.

When we added the second book, we arranged them this way:

1/4 of the bottom book and
1/2 + 1/4 = 3/4 of the top book
are beyond the edge of the table.

If we add up these pieces 1/4 + 3/4 = one book is beyond the edge of the table. Since there are two books all together, pieces adding up to one book are on the table. These are equal, so we are in balance.

3/4 of the top book is hanging beyond the edge of the table.

Now we add the third book.

The top book overhangs the middle book by 1/2. The middle book overhangs the bottom book by 1/4. The bottom book overhangs the table by 1/6.

Let’s add up the pieces that overhang the table:

Bottom book	1/6
Middle book	1/6	+	1/4
Top book:	1/6	+	1/4	+	1/2

The total amount of book beyond the edge of the table is

1/6

1/4

1/6

1/4

1/2

2/12

3/12

2/12

3/12

6/12

18/12

1 6/12

1 1/2

Since there are three books in total, half the mass, one and one-half books, is beyond the edge of the table and the other half of the mass, one and one-half books, is on the table. We are in balance.

How much of the top book is hanging over the table? Let’s add up the pieces:

1/6	+	1/4	+	1/2	=
2/12	+	3/12	+	6/12	=	11/12

Most of the mass of the book is over the edge, but we’re not quite there.

Let’s add a fourth book.

Now let’s add up the pieces of book that are beyond the table edge, as we did as before.

Bottom book:

1/8

3/24

Second from
bottom:

1/8

1/6

3/24

4/24

7/24

Third from
bottom:

1/8

1/6

1/4

3/24

4/24

6/24

13/24

Top:

1/8

1/6

1/4

1/2

3/24

4/24

6/24

12/24

25/24

48/24

So it appears that the pieces beyond the edge of the table add up to two books. Since there are four books in total, this stack remains stable: half on the table and half beyond the edge.

But there’s a problem here. The pieces of the top book add up to 25/24, which is more than a book. This can’t be. The pieces of the book can’t add up to more than one book.

Our miscalculation was to include the full 1/8 part of the top book. In fact, that book is beyond the edge of the table by 1/24 of a book. So if we correct this and again add up the pieces, we get

3/24 + 7/24 + 13/24 + 24/24 = 47/24

The pieces of book that are on the table add up to 49/24, just over two books. Together we have

47/24 + 49/24 = 96/24 = 4 books

Since slightly more than half the mass of the stack is on the table, we are in balance.

We have solved the problem. With four books in a stack, it is possible to have the top one beyond the edge of the table.

Math Puzzle [2]