Harmonic series (mathematics)
Stacking blocks

In the block-stacking problem, one must place a pile of n {\displaystyle n} identical rectangular blocks, one per layer, so that they hang as far as possible over the edge of a table without falling. The top block can be placed with 1 2 {\displaystyle {\tfrac {1}{2}}} of its length extending beyond the next lower block. If it is placed in this way, the next block down needs to be placed with at most 1 2 ⋅ 1 2 {\displaystyle {\tfrac {1}{2}}\cdot {\tfrac {1}{2}}} of its length extending beyond the next lower block, so that the center of mass of the top two block is supported and they do not topple. The third block needs to be placed with at most 1 2 ⋅ 1 3 {\displaystyle {\tfrac {1}{2}}\cdot {\tfrac {1}{3}}} of its length extending beyond the next lower block, and so on. In this way, it is possible to place the n {\displaystyle n} blocks in such a way that they extend 1 2 H n {\displaystyle {\tfrac {1}{2}}H_{n}} lengths beyond the table, where H n {\displaystyle H_{n}} is the n {\displaystyle n} th harmonic number. The divergence of the harmonic series implies that there is no limit on how far beyond the table the block stack can extend. For stacks with one block per layer, no better solution is possible, but significantly more overhang can be achieved using stacks with more than one block per layer.
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