Figure 1: One Block
Imagine that we have a supply of toy blocks of uniform density, all of the same size and weight, each one foot long. We also have a strong table, anchored securely to the floor. Our job is to arrange a stack of blocks on the table so that the stack extends out beyond the right edge of the table. When building the stack, the following rules apply:
Here is the question that we will study in this lesson: How far out beyond the right edge of the table can the right edge of the topmost block extend? Let's consider two simple examples to be sure that you understand the problem.
For the first example, let's place a single block so that it extends halfway off the table. It is thus perfectly balanced on top of the table. Unfortunately, no matter how little beyond the right edge of the first block the second block extends, the first block will ``lose its balance'' and the pile will collapse. Try this with a couple of identical books if you don't believe this.
For the second example, let's place one block so that one third of it extends off of the table, and a second block so that one third of it extends beyond the first block. Although this stack is stable, the bottom block is now perfectly balanced. The next block that is added at the top of the stack will cause the bottom block to fall. Again, it may be helpful to do an experiment with books.
In the first example, we succeeded in extending the top block half of a foot away from the table; in the second example, two thirds of a foot. Can we do better? If so, how much better?
As we study this question we will:
