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Multi-resolution techniques are essential for the performance of large-scale simulations of computational fluid dynamics with particle methods. In this study, a novel adaptive multi-resolution scheme has been developed for the least squares moving particle semi-implicit (LSMPS) method. With the proposed technique, particles are dynamically resized based on a local error estimate. The error estimate is defined by how well the velocity of a particle can be approximated by the Taylor-series expansions of its fluid neighbours, and vice versa for wall neighbours. Due to the adaptiveness of the multi-resolution technique, time-consuming optimization of predefined particle size targets is avoided. The adaptiveness also enables particle resizing which tracks transient resolution changes of the flow. Therefore, the adaptiveness should improve the computational efficiency of the multi-resolution method. In this study, the multi-resolution technique was tested for a two-dimensional eccentric rotating cylinder problem with a small clearance and a known steady-state solution. As expected, initially uniform particle sizes quickly decreased around the clearance. The particle size distribution evolution was smooth in both time and space throughout the simulations. Consequently, the multi-resolution method gave significantly more accurate results than a single resolution method with the same number of particles and time-step length. A drawback with the multi-resolution scheme is that the restrictions on time-step lengths become tighter. This issue is considered by an ongoing development of a multi-time stepping scheme.