The Overhang Problem (2007) [pdf]

  • "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."

    (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.

    https://www.maa.org/sites/default/files/pdf/upload_library/2...

  • These diagrams remind me of Lemmings and the builders that would build staircases over gaps you needed to cross.

  • 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.

    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: https://arxiv.org/pdf/0707.0093.pdf

  • I often use this example when teaching calculus: https://youtu.be/BjmypdFCl-Q

  • So some scientists get to play with jenga on taxpayers' dime and students' tuition fee.

    (I'm not angry, I'm envious.)

  • I wonder how close the historical examples come to the 20 or 30 block 'ideal'?