A smoke ring is a torus of spinning air that pushes itself forward — and a smaller ring moves FASTER than a big one. Fire two rings down the same line and they leapfrog: the trailing ring narrows, speeds up, and threads clean through the middle of the one ahead, which has widened and slowed — then they swap and do it again. Slide time and watch the doughnuts take turns.
A thin vortex ring of circulation Γ, ring radius R and core radius a self-propels at U ≈ (Γ/4πR)·(ln(8R/a) − ¼) — crucially, U grows as R SHRINKS, so a smaller ring is faster. Two coaxial rings therefore leapfrog: the rear ring, in the forward flow of the front ring, contracts and speeds up and passes through it; the front ring, in the rear ring’s inflow, expands and slows — then the roles swap, periodically (Helmholtz, 1858). A fail-loud self-check throws unless the smaller ring’s U exceeds the larger ring’s. ◆ real fluid dynamics, node-verified.
The thin-core self-induction law and a periodic radius exchange are modelled (idealised, inviscid); real rings lose energy to viscosity and eventually stop leapfrogging — the mechanism (smaller = faster → threads through) is the exact, classical result.