Concrete Math
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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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. However, at SODA'06, Paterson and Zwick constructed n-block stacks with overhangs of order n1/3. Here we complete the solution to the overhang problem, and answer Paterson and Zwick's primary open question, by showing that order n1/3 is best possible. At the heart of the argument is a lemma (possibily of independent interest) showing that order d3 non-adaptive coinflips are needed to propel a discrete random walk on the number line to distance d. We note that our result is not a mainstream algorithmic result, yet it is about the solution to a discrete optimization problem. Moreover, it illusrates how methods founded in theoretical computer science can be aplied to a problem that has puzzled some mathematicians and physicists for more than 150 years.