"How far can a stack of n identical blocks be made to hang over the edge of a table? The question has a long history and the answer was widely believed to be of order logn. Recently, Paterson and Zwick constructed n-block stacks with overhangs of order n^1/3 , exponentially better than previously thought possible. We show here that order n^1/3 is indeed best possible, resolving the long-standing overhang problem up to a constant factor."<p>(from intro)
Imagine being this author's parents at parties. "Yes, our son is a Computer Scientist. No, not the well-paid kind, the kind that solves problems. No, not that sort of problem; the ones like how many bricks it takes to make a wall sturdy. No, not a regular wall; a curved one. I don't know who wants a curved brick wall. Yes, I suppose he could just use rebar and mortar."
Just discovered this -- it contains the optimal constructions for up to N=20. They don't prove optimality but they do claim it.<p><a href="https://www.maa.org/sites/default/files/pdf/upload_library/22/Robbins/Patterson1.pdf" rel="nofollow noreferrer">https://www.maa.org/sites/default/files/pdf/upload_library/2...</a>
It mentioned that the harmonic stacks arise from the restriction that the blocks should be stacked in one-on-one fashion. A related restriction might be that the blocks must be stacked one-by-one, i.e. each additional block must maintain balanced state.<p>I think trying to reproduce some of the diagrams with physical blocks would be difficult unless multiple blocks are placed at the same time.
arxiv link, since the Dartmouth math website is having a bad evening: <a href="https://arxiv.org/pdf/0707.0093.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/0707.0093.pdf</a>
I often use this example when teaching calculus: <a href="https://youtu.be/BjmypdFCl-Q" rel="nofollow noreferrer">https://youtu.be/BjmypdFCl-Q</a>