  Driven by physical questions pertaining to quantifying particle dynamics,
microscopy can now resolve complex systems at the single particle level, from
cellular organisms to individual ions. Yet, available analysis techniques face
challenges reconstructing trajectories in dense and heterogeneous systems where
accurately labelling particles is difficult. Furthermore, the inescapable
finite field of view of experiments hinders the measurement of collective
effects. Inspired by Smoluchowski, we introduce a broadly applicable analysis
technique that probes dynamics of interacting particle suspensions based on a
remarkably simple principle: counting particles in finite observation boxes.
Using colloidal experiments, advanced simulations and theory, we first
demonstrate that statistical properties of fluctuating counts can be used to
determine self-diffusion coefficients, so alleviating the hurdles associated
with trajectory reconstruction. We also provide a recipe for practically
extracting the diffusion coefficient from experimental data at variable
particle densities, which is sensitive to steric and hydrodynamic interactions.
Remarkably, by increasing the observation box size, counting naturally enables
the study of collective dynamics in dense suspensions. Using our novel analysis
of particle counts, we uncover a surprising enhancement of collective
behaviour, as well as a new length scale associated with hyperuniform-like
structure. Our counting framework, the Countoscope, thus enables efficient
measurements of self and collective dynamics in dense suspensions and opens the
way to quantifying dynamics and identifying novel physical mechanisms in
diverse complex systems where single particles can be resolved.
