The Block-Stacking Problem

The scenerio: you have some rectangular blocks all stacked at the edge of a table. Just by nudging blocks over, how far can you get the stack to reach beyond the edge of the table?

Well, we know that if the center of mass of a block is over space, it will topple and fall. So one block can be pushed halfway out over space before falling. It is also true that if the center of mass of any group of blocks taken together as a unit is over space, they will topple as that unit. Now it's just a matter of calculating those centers of mass. The result is that the top block, Block 1, can reach over 1/2 the length of a block (let's call the length of a block 1 lob). The block below Block 1, Block 2, can overhang 1/4 lob. Block 3 can overhang 1/6 lob. There is a pattern here. If n is the block number, it's overhang is 1/(2*n) lob.

n	Overhang (lob)	Total overhang (lob)
1	1/2	0.5
2	1/4	0.75
3	1/6	0.917
4	1/8	1.04

Note that at four books, you can overhang by a whole block, so the top block is not over the table at all anymore. Can we reach two blocks over? The overhang is getting smaller each time, but each added block still gets us a little farther. Incidentally, those incremental overhangs are related to the harmonic series (1, 1/2, 1/3, 1/4, 1/5...). It turns out that to reach 2 lob would take 21 blocks. Theoretically, any overhang can be achieved with enough blocks, but practically there will be limits. We're assuming the blocks are perfectly rigid, actually they will flex some small amount and topple sooner. Also, the numbers quickly blow up. To make it 3 lob would require 227 blocks, and 4 lob would require 1,674 blocks. For five lob: 12,367.