Maximum overhang

TL;DR

This paper proves that the maximum overhang of a stack of n identical blocks is proportional to n^{1/3}, resolving a long-standing problem by showing the previous conjecture of logarithmic overhang was incorrect.

Contribution

The paper establishes the optimal order of maximum overhang as n^{1/3}, confirming Paterson and Zwick's construction is asymptotically best possible.

Findings

Maximum overhang scales as n^{1/3}

Paterson and Zwick's construction is asymptotically optimal

Resolved the long-standing overhang problem

Abstract

How far can a stack of $n$ identical blocks be made to hang over the edge of a table? The question dates back to at least the middle of the 19th century and the answer to it was widely believed to be of order $\log n$. Recently, Paterson and Zwick constructed $n$-block stacks with overhangs of order $n^{1/3}$, exponentially better than previously thought possible. We show here that order $n^{1/3}$ is indeed best possible, resolving the long-standing overhang problem up to a constant factor.