Block-stacking problem

In statics, the block-stacking problem (sometimes known as The Leaning Tower of Lire (Johnson 1955), also the book-stacking problem, harmonic staircase, or a number of other similar terms) is a puzzle concerning the stacking of blocks at the edge of a table. The block-stacking problem is the following puzzle: Place N {\displaystyle N} identical rigid rectangular blocks in a stable stack on a table edge in such a way as to maximize the overhang. Paterson et al. (2007) provide a long list of references on this problem going back to mechanics texts from the middle of the 19th century. The single-wide problem involves having only one block at any given level. In the ideal case of perfectly rectangular blocks and uniform gravitational acceleration (value of g decreases with increase in height), the solution to the single-wide problem is that the maximum overhang is given by ∑ i = 1 N 1 2 i {\textstyle \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 N {\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 N {\displaystyle N} block-lengths past the edge of the table is 4, 31, 227, 1674, 12367, 91380, ... (sequence A014537 in the OEIS).