## Determining when a stack of boxes will topple

Consider a tower of boxes which is stacked so that each box is offset slightly to the right of the one below it. How far can such a stack extend before it topples under its own weight? Of course, the answer depends on how many boxes there are in the tower; but, is there an absolute bound as to how far this satck can reach, regardless of how many boxes are used?

In answer this question, you should keep a few things in mind. To begin, a single box will topple if its centre of mass is not supported from below. Assuming that all the boxes are the smae size and are made of the same, homogeneoues material, this amounts to saying that a box will fall over if its centre extends beyond the box below it. On the other hand, two boxes will not topple if

• the top box's centre of mass lies above the box below it
• the common centre of mass of the two boxes lies above the box immediately below both them.
Similarly, a stack of of n boxes will be stable if
• the top box's centre of mass lies above the box below it
• the common centre of mass of the top two boxes lies above the box immediately below both them.
• ...
• the common centre of mass of the top n boxes lies above the box immediately below them.
Assuming that the bottom left corner of the lowest box is the origin, the centre of mass of the top box is a + b + c + d + e + f + 1/2 while the centre of mass of the box below it is a + b + c + d + e + 1/2. Since the boxes are all identical, the centre of mass of an n of them is just the average of all their centres of mass. Hence, for example, the centre of mass of the top two boxes is a + b + c + d + e + f/2 + 1/2. Similarly, the centre of mass of the top three boxes is a + b + c + d + 2e/3 + f/3 + 1/2. In order for these top three boxes to be stable, this centre of mass must lie above the therightmost edge of the box below it. This edge is positioned at a + b + c + d + 1.

These considerations lead to a family of n-1 equations for n boxes. Solving these simultaneously leads to a solution for the how far to the right the the boxes can be stacked. Use this information to answer the original question. How many boxes are required to extend out 1 unit beyond the right edge of the bottom box? What about 2 units? 3 units?

Juris Steprans