I first encountered the block-stacking problem, also called The Leaning Tower of Lire, in a calculus course in college. The demonstration is useful for giving an intuitive feel for and natural example of the convergence or divergence of an infinite series.
This week in math class, I posed to the class a simple question about the nature of stacking identical blocks. "By stacking several blocks on top of one another on a table, can you set up the blocks such that the top block has its entire length beyond the edge of the table?"
The question has a fascinating mathematical answer that is most easily understood with a picture:
*CS20 students, Jacob B and Helena, work together to solve the block-stacking problem.
In groups of pairs, with ten blocks each, the students were asked to tackle this problem and consider some additional questions, like "How far out could you make the top block hang if you had as many blocks as you wanted?" and "What is the best strategy to use to maximize the total overhang distance?" Students were also provided with a ruler and encouraged to take measurements of their stacks. Beyond this, I intentionally made the work-time as open-ended as possible, to allow students to explore whatever questions and interests at which they naturally arrived.
For the next half hour of exploration, many groups discovered the critical realization that the intuitive method of stacking from the bottom up (adding one block on top extending further each time) didn't work very well. Instead, they found that going from the top down is a better strategy. Maximizing the overhang of the top block, and the overhang of the second-highest block, and so on.
In our discussion at the end of class, we talked about why the top-down approach works better. We can use the idea of "center of mass" to support this strategy. At most, the center of mass of our stack can be directly over the edge of the table — any further out, and the stack will fall. Since all of the blocks are uniform in density and length, then the furthest that one block could extend is simply half of its length. From there, we can calculate the center of mass of one block stacked hanging halfway over another block, and use that point to place the stack at the edge of the table. If we keep going with this method, we get a tower with the most significant overhang distance.
*Here, you can see how a pattern emerges in the amount of overhang of each block past the one below it. The real measured values are slightly shorter than the idealized ones, caused by factors like the wood blocks not being perfectly uniform in density or sharp at their edges.
In this set-up, the sum of the ratios of each overhang distance to the total block length follows a clear mathematical pattern -- ½ + ¼ + ⅙ + ⅛ + 1/10 + ... -- which happens to be half of a well studied mathematical series -- 1, ½, ⅓, ¼, ⅕ + … -- called the "harmonic series." In class, we went through a simple proof that this series does not converge; that is, it does not have a finite limit. The implication of this is that in an idealized world with idealized blocks, there is no limit to how far out our tower could overhang if we had unlimited blocks.
Photos contributed by Max Trostel, Teaching Fellow and Caitlin Lemley, Math Teacher.