Consider the motion of a Brownian particle that takes place either in a two- dimensional plane or in the three-dimensional space. Given that only the distance of the particle to the origin is being observed, the problem is to detect the true dimension as soon as possible and with minimal probabilities of the wrong terminal decisions. This talk will discuss the solution to this problem in the Bayesian formulation under any prior probability of the true dimension when the passage of time is penalised linearly. This is a nice example of tackling an optimal stopping problem for a 2-dimensional coupled Markov process. The solution uses a measure change, a stochastic time-change, Mayer and Lagrange reformulations, and allowing for negative initial times, which could help provide ideas for solving other 2-dimensional optimal stopping problems.