Research into computational jigsaw puzzle solving, an emerging theoretical problem with numerous applications, has focused in recent years on puzzles that constitute square pieces only. In this paper we wish to extend the scientific scope of appearance-based puzzle solving and consider ''brick wall' jigsaw puzzles - rectangular pieces who may have different sizes, and could be placed next to each other at arbitrary offset along their abutting edge - a more explicit configuration with properties of real world puzzles. We present the new challenges that arise in brick wall puzzles and address them in two stages. First we concentrate on the reconstruction of the puzzle (with or without missing pieces) assuming an oracle for offset assignments. We show that despite the increased complexity of the problem, under these conditions performance can be made comparable to the state-of-the-art in solving the simpler square piece puzzles, and thereby argue that solving brick wall puzzles may be reduced to finding the correct offset between two neighboring pieces. We then move on to focus on implementing the oracle computationally using a mixture of dissimilarity metrics and correlation matching. We show results on various brick wall puzzles and discuss how our work may start a new research path for the puzzle solving community.