The Block Stacking Problem

Summary

The Block Stacking Problem John D. Norton Department of History and Philosophy of Science University of Pittsburgh https://sites.pitt.edu/~jdnorton/jdnorton.html February 21, 2025 1. Introduction In the block stacking problem, a collection of blocks are to be stacked at a table edge. If we stack the blocks so that they protrude past the edge of the table, how far can they go horizontally? The curious and then surprizing result is that we can extend the stack horizontally arbitrarily far. We just need to have a high enough stack of many blocks. Until we think more about it and perhaps do some sums, that just seems wrong. Surely, we expect that there is a limit to how far the stacks can go before the whole stack topples off the edge. A good general exploration is given by John F. Hall, "Fun with Stacking Blocks" American Journal of Physics, 73, No. 12, December 2005, pp. 1107-16. There is a long standing literature on the problem that extends back to the mid-19th century. This literature has given the physics in much detail for both the simple case to be dealt with here and for more complicated cases. We can do the calculations that establish the stability of the various stacks and thereby affirm that the stacks are stable and have the properties claimed. However it can still remain puzzling that the stack just does not fall since it extends so far past the edge of the table. My goal here is to develop an intuitive sense of comfort with the behaviors of these stacks. If I succeed, you will not just understand that the physics allows the stacks to be stable, but you will feel that it is proper and just. There will be a small bonus. Once we are comfortable with stacks of blocks that extend arbitrarily far beyond the table edge, we might think that we can just "take the limit to infinity" and have an infinite stack that extends infinitely far beyond the table edge. Limits to infinity, we shall see, are curious procedures. In this case they produce the oddest result: eith...