One geometry rule makes every column and classic arch countable — and the community rule agrees with it inside 5%.
Columns and classic arches are layers of four balloons. Each layer's four balloons sit in a square; the next layer turns 45° and rests in the hollows. That geometry fixes the spacing: the horizontal offset is 0.541·d, so the height between layers is √(1 − 0.541²)·d ≈ 0.841·d — derived, and pinned by the engine's tests.
The classic-arch rule says 6 eleven-inch and 12 five-inch balloons a foot. The geometry at those sizes: 5.7 eleven-inch a foot, 11.4 five-inch — within 5% of the community's counts. Two roads, one answer.
A 6 ft column of 11 in balloons: the diameter is 0.792 ft, the spacing 0.841 × 0.792 ≈ 0.67 ft. The layers: 6 ÷ 0.67 ≈ 9 — ceil to 9, the convention that never shortens a build — so 9 × 4 = 36 balloons. The community check on the same column: 6 × 5.7 ≈ 34 — the 5% gap is the rounding, and the builder's spares absorb it.
| the mistake | what it costs |
|---|---|
| layers rounded down | the build lands short on install morning; the app's rule rounds up |
| the 0.841 replaced with a guess | gap-y layers or crushed ones; the geometry is the rule |
| mixing balloon sizes inside one layer | the square collapses; a size per layer, the mixes live in garlands |
The Counts sheet runs the layers, adds the double-stuffed splits and the spares, and carries the order to bags. The same arithmetic runs in the browser calculator.
Sourced: the layers-of-four geometry is derived and tested (the engine's BuildsTest), the community check is the decorator consensus, the rounds-up is the app's convention, and every label on screen matches this file.