The Paradox of Brick Masonry (Leaning Stack of Bricks)
How far can a stack of identical bricks, each balanced on the one below it, be made to overhang the edge of a table before it topples — is there any theoretical limit to how far the top brick can stick out?
History and references
Settled in 1955 by Paul B. Johnson (rigorous treatment using the divergent harmonic series).
The block-stacking (or 'leaning tower') problem has circulated in recreational-mathematics circles for over a century and was given a clean physics treatment by Paul B. Johnson in a 1955 American Journal of Physics note titled 'Leaning Tower of Lire,' establishing the (1/2)(harmonic sum) overhang formula. Lange's Chapter 3 ('Equilibrium of Forces') problem on 'brick masonry' almost certainly uses this or a closely related setup, since it is one of the few classic equilibrium puzzles whose 'obvious' bounded answer is wrong for a genuinely surprising, non-intuitive mathematical reason (divergence of the harmonic series).
The harmonic series is famous for diverging 'just barely' — this problem turns that abstract fact about an infinite sum into a physically constructible (if impractical) leaning tower.
Explanations that circulated
- Assuming that because each additional brick can only be shifted by an ever-smaller amount to keep the stack balanced, the total possible overhang must be bounded and finite, like a finite sum of shrinking terms.
- Failing to recognize that the maximum safe shift for the k-th brick from the top is proportional to 1/k, and the running total 1 + 1/2 + 1/3 + ... + 1/n (the harmonic series) grows without bound as n increases — so in principle, with enough bricks, the overhang can exceed ANY given distance, even though each individual increment shrinks toward zero.
References
- Leaning Tower of Lire, Paul B. Johnson, American Journal of Physics, 1955
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