cantilever books

 
Stack 5 or 6 books at the edge of a tabletop. All the books are the same size. What is the maximum distance you can get any one of the books in the pile to extend out over the table's edge? If you slide the pile of books toward the edge they will not fall if less than half the pile is over the edge. Is about one half a book the maximum that the pile can be extended? No. To get the maximum extension begin by sliding only the top book out as far as possible. Then slide out the next one down so the top two move as a unit. When they are as far out as possible, slide out the third one from the top. Continue down the stack. If done carefully the top book in a stack of 6 will be entirely beyond the edge of the table.
 
Here's the explanation. Consider only the top book. Its center of gravity is at its center. It will not fall if its center of gravity is not beyond the edge of the next book down. Now consider the top two books.
 
Their combined center of gravity is half way between the centers of gravity of each of the two. Those two books will not fall as long as their combined center of gravity is not beyond the edge of the third book down. That lets the second book extend out about 1/4 book beyond the third book.
 
Now for the top three books. Where's their combined center of gravity? The top two books together weigh twice as much as the third book alone. So the combined center of gravity will be twice as close to the center of graviy of the top two as it is to the center of graviy of the third. That puts it 1/6 of a book in from the outside edge of the third book. You can continue the same logic down through the pile with the maximum extension for each book being 1/2, 1/4, 1/6, 1/8, 1/10 and 1/12. Add the fractions and you get 147/120 or more than one full book.