Overhang

TL;DR

This paper challenges the classical belief that the maximum overhang with n frictionless blocks is achieved by harmonic stacks, showing instead that significantly larger overhangs proportional to n^{1/3} are possible.

Contribution

The paper demonstrates that the classical harmonic stack solution is far from optimal by constructing stacks with overhangs growing as n^{1/3}, revealing new potential for maximizing overhang.

Findings

Classical harmonic stacks are not optimal for maximum overhang.

Constructed stacks achieve overhangs proportional to n^{1/3}.

Maximum overhang can be exponentially larger than previously believed.

Abstract

How far off the edge of the table can we reach by stacking $n$ identical, homogeneous, frictionless blocks of length 1? A classical solution achieves an overhang of $1/2 H_n$, where $H_n ~ \ln n$ is the $n$th harmonic number. This solution is widely believed to be optimal. We show, however, that it is, in fact, exponentially far from optimality by constructing simple $n$-block stacks that achieve an overhang of $c n^{1/3}$, for some constant $c>0$.