In statics, the block-stacking problem (sometimes known as The Leaning Tower of Lire (Johnson 1955), also the book-stacking problem, 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 ${\displaystyle N}$ 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 above formula for the maximum overhang of ${\displaystyle n}$ blocks, each with length ${\displaystyle l}$ and mass ${\displaystyle m}$, stacked one at a level, can be proven by induction by considering the torques on the blocks about the edge of the table they overhang. The blocks can be modelled as point-masses located at the center of each block, assuming uniform mass-density. In the base case (${\displaystyle n=1}$), the center of mass of the block lies above the table's edge, meaning an overhang of ${\displaystyle l/2}$. For ${\displaystyle k}$ blocks, the center of mass of the ${\displaystyle k}$-block system must lie above the table's edge, and the center of mass of the ${\displaystyle k-1}$ top blocks must lie above the edge of the first for static equilibrium.^[3] If the ${\displaystyle k}$th block overhangs the ${\displaystyle (k-1)}$th by ${\displaystyle l/2}$ and the overhang of the first is ${\displaystyle x}$,^[4]

${\displaystyle (k-1)mgx=(l/2-x)mg\implies x=l/2k,}$

where ${\displaystyle g}$ denotes the gravitational field. If the ${\displaystyle k-1}$ top blocks overhang their center of mass by ${\displaystyle y}$, then, by assuming the inductive hypothesis, the maximum overhang off the table is

${\displaystyle y+{\frac {l}{2k}}=\sum _{i=1}^{k}{l/2i}\implies y=\sum _{i=1}^{k-1}{l/2i}.}$

For ${\displaystyle k+1}$ blocks, ${\displaystyle y}$ denotes how much the ${\displaystyle k+1-1}$ top blocks overhang their center of mass ${\displaystyle (y=\sum _{i=1}^{k}l/2i)}$, and ${\displaystyle x={\frac {l}{2(k+1)}}}$. Then, the maximum overhang would be:

${\displaystyle {\frac {l}{2(k+1)}}+\sum _{i=1}^{k}l/2i=\sum _{i=1}^{k+1}l/2i.}$

Hall (2005) discusses this problem, shows that it is robust to nonidealizations such as rounded block corners and finite precision of block placing, and introduces several variants including nonzero friction forces between adjacent blocks.