Maximizing Brick Overhang: A Mathematical Exploration

Let's consider a scenario where we have two bricks and our goal is to stack them in such a way that the top brick extends as far out as possible without toppling. To achieve this, the top brick must be positioned perfectly at the center of the bottom brick, as illustrated in Figure A, under the assumption that both bricks are of equal length.

In doing so, the center of mass of the upper brick, being uniformly distributed, aligns exactly with the lower brick's edge. But what if we had three bricks? How should they be arranged to ensure the maximum overhang without causing a collapse? Through basic computations, we can determine that the top brick should be positioned halfway over the middle brick, and the middle brick should overlap the bottom brick by a quarter, as shown in Figure B.

As the number of bricks increases, the most effective strategy to achieve maximum overhang is the following: the top brick should overlap the second by half, the second should overlap the third by a quarter, the third should overlap the fourth by a sixth, the fourth should overlap the fifth by an eighth, and so forth. This generates a leaning tower of bricks, akin to the one depicted in Figure C.

Mathematically Explained

The total overhang for this configuration, which is the sum of all individual overhangs, given n+1 stacked bricks, is 1/2+1/4+1/6+1/8+...+1/(2n). This can be restructured as 1/2(1+1/2+1/3+1/4+...+1/n). The expression within the parentheses is the n-th partial sum of the harmonic series, a series that diverges positively. This creates a paradox: with an unlimited supply of bricks, the maximum overhang of a stack equals half the sum of the harmonic series, which is essentially half of infinity; hence, it is effectively infinite!

Hypothetically, a sufficiently tall stack of bricks could span the Golden Gate! However, it's crucial to remember that this series diverges very slowly: three bricks yield an overhang of 11/12, four bricks give 25/24, and ten bricks slightly over 1.46. With a hundred bricks, the overhang is approximately 2.29 and with a thousand, it's around 3.45. Considering factors like wind, vibrations, and brick imperfections, it's theoretically possible to build a tower with a large overhang, but practically it's a challenging endeavor. To achieve a 50 unit overhang, a tower of 15×10^(42) bricks would be needed, a height surpassing the distance from here to the edge of the observable universe.

The harmonic series, the sum of the reciprocals of positive integers (1+1/2+1/3+...+1/n+...), even though the terms progressively shrink towards 0 (1/n approaches 0 as n approaches infinity), diverges positively (meaning its sum is infinite). This result, initially deemed "pathological" by mathematicians, was first proven by Nicole Oresme (1323-1382), Bishop of Lisieux. The proof was later lost and independently rediscovered by Pietro Mengoli in 1647 and Johann Bernoulli in 1687. The divergence of the harmonic series underpins the so-called "paradox of infinite overhang".