Saramsha

No. 020  ·  6 October 2026

As Far As You Like

Flat blocks, stacked one on another over the edge of a table. Each one can shift sideways a little before it's placed — push far enough and the tower overhangs the edge; push too far and it falls. Drag the pale block out past the edge, then let go.

balance sits over its support — holds balance is past the edge — falls

About this piece

A sum that never stops

A block stays up exactly as long as its weight, and the weight of everything stacked above it, presses down somewhere over whatever is holding it — the block below, or the table. The moment that combined balance point drifts past the supporting edge, there is nothing left to press down on and the whole thing rotates off. That single rule, applied over and over, is the entire physics of this page: no friction model, no springs, just centres of mass and edges.

It also means a block never has to sit squarely on the one beneath it. It can hang out over the edge, right up to the point where its own balance point — not its edge, its centre — reaches the edge of its support. Push it a hair further and it topples; stop a hair short and it is holding on with room to spare.

Half, then a quarter, then a sixth

Set every block in the tower at exactly its limit, from the top down, and a clean pattern falls out. The top block can overhang the one below it by up to half a block's length. The second block from the top can overhang the third by up to a quarter. The third can overhang the fourth by a sixth. In general, the k-th block from the top can overhang whatever is beneath it — another block, or, at the very bottom, the table itself — by exactly 1 / (2k) of a block-length, and not a sliver more.

Add those overhangs up for a tower of n blocks and the total distance the top block can reach past the table's edge is ½ × (1 + ½ + ⅓ + … + 1/n) — one half of the first n terms of the harmonic series. Four blocks, arranged this way, overhang the table by slightly more than one whole block-length, which looks impossible until you have watched it happen.

The sum that outgrows every bound, given enough patience

The harmonic series is the textbook example of a sum whose individual terms shrink to nothing — 1/1,000,000 is a vanishingly small number — while the running total still grows without limit. Nicolas Oresme gave the standard proof around 1350: group the terms as 1, then (½), then (⅓+¼), then (⅕+⅙+⅞+⅛), doubling the group size each time. Every group adds up to more than ½, and there are infinitely many groups, so the total passes every number you could name — it is just extraordinarily slow about it. Reaching a running total of 20 (an overhang of 10 block-lengths on this page) takes on the order of e²⁰, roughly half a billion, blocks. Reaching 40 takes something like 10¹⁷ of them. The series does not stop growing. It just stops being something you could ever finish stacking.

Getting further, with help

Single straight towers like the one on this page are provably optimal for a single straight tower — but they are not the best overhang physics allows. In 2009 Mike Paterson and Uri Zwick showed that if you are permitted to use blocks as counterweights off to the side, rather than stacked in one column, the achievable overhang grows roughly as the cube root of the block count instead of its logarithm — a dramatically better rate, at the cost of a far stranger-looking structure. This page keeps to the single tower, because that is the version you can build with your eyes and believe without a proof.

Blocks here are idealised: perfectly rigid, perfectly flat, placed at their exact theoretical limit with no margin for error. Real bricks have friction and give, which is a small mercy — it is the only reason anyone can stack more than two or three of these in practice and have them survive a passing breath of air.