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We study diffusion of (fluorescently) tagged hard-core interacting particles of finite size in a finite one-dimensional system. We find an exact analytical expression for the tagged particle probability density function using a Bethe ansatz, from which the mean square displacement is calculated. The analysis shows the existence of three regimes of drastically different behavior for short, intermediate, and large times. The results are in excellent agreement with stochastic simulations (Gillespie algorithm).