homework and exercises - Stacking Shelves with Overhang

I want to stack some boxes which are 14" x 10" with some 12" records inside. This means that there will be a 2" overhang outside of each box.

I know that to have an item to balance on a edge 50% of an item needs to be supported. So I think I'm safe to load the boxes in this way.

If I were to stack boxes loaded in this way on top of each other is there a limit to how high they can go?

Answer

While the standard answer to this problem is usually given in terms of harmonic series, (see for instance this page at MathWorld) which results in stack looking like this:

I would like to share the nonstandard answer I found following one of the MathWorld links. A whole new class of solutions was found in the paper by Paterson and Zwick:

Paterson, Mike, and Uri Zwick. Overhang. Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm. ACM, 2006. arXiv:0710.2357.

The paper is quite accessible, so I suggest to have a look, but the main idea could be easily understood by looking at some of the solutions for various number of shelves:

So, by discarding assumption that only one shelf could be placed per level we are able to produce much more solutions: overhang now scales like $c\,n^{1/3}$ with number of shelves, instead of $\frac 12 \ln n$ for harmonic series solution.