Block-stacking problem

The single-wide problem involves having only one block at any given level. In the ideal case of perfectly rectangular blocks, the solution to the single-wide problem is that the maximum overhang is given by ${\displaystyle \sum _{i=1}^{N}{\frac {1}{2i}}}$ times the width of a block. This sum is one half of the corresponding partial sum of the harmonic series. Because the harmonic series diverges, the maximal overhang tends to infinity as ${\displaystyle N}$ increases, meaning that it is possible to achieve any arbitrarily large overhang, with sufficient blocks.

The number of blocks required to reach at least ${\displaystyle N}$ block-lengths past the edge of the table is 4, 31, 227, 1674, 12367, 91380, ... (sequence A014537 in the OEIS).[1]

Comparison of the solutions to the single-wide (top) and multi-wide (bottom) block-stacking problem with three blocks

Multi-wide stacks using counterbalancing can give larger overhangs than a single width stack. Even for three blocks, stacking two counterbalanced blocks on top of another block can give an overhang of 1, while the overhang in the simple ideal case is at most 11/12. As Paterson et al. (2007) showed, asymptotically, the maximum overhang that can be achieved by multi-wide stacks is proportional to the cube root of the number of blocks, in contrast to the single-wide case in which the overhang is proportional to the logarithm of the number of blocks.