We present new numerical tools for geophysical inversion and uncertainty quantification (UQ), with an emphasis on blocky (piecewise-constant) layered models that can reproduce sharp contrasts in geophysical or geological properties. The new tools are inspired by an "old" and very successful inversion tool: regularized, nonlinear inversion (Occam's inversion, Constable et al. (1987)). We combine Occam's inversion with total variation (TV) regularization and a split Bregman method to obtain an inversion algorithm that we call blocky Occam, because it determines the blockiest model that fits the data adequately. To generate a UQ, we use a modified randomize-then-optimize approach (RTO) and call the resulting algorithm RamBO (randomized blocky Occam), because it essentially amounts to running blocky Occam in a randomized parallel for-loop. Blocky Occam and RamBO inherit computational advantages and stability from the combination of Occam's inversion, split Bregman and RTO, and, therefore, can be expected to be robustly applicable across geophysics.