A jigsaw is a recently proposed generative model that describes an image as a composition of non-overlapping patches of varying shape, extracted from a latent image. By learning the latent jigsaw image which best explains a set of images, it is possible to discover the shape, size and appearance of repeated structures in the images. A challenge when learning this model is the very large space of possible jigsaw pixels which can potentially be used to explain each image pixel. The previous method of inference for this model scales linearly with the number of jigsaw pixels, making it unusable for learning the large jigsaws needed for many practical applications. In this paper, we make three contributions that enable the learning of large jigsaws – a novel sparse belief propagation algorithm, a hybrid method which significantly improves the sparseness of this algorithm, and a method that uses these techniques to make learning of large jigsaws feasible. We provide detailed analysis of how our hybrid inference method leads to significant savings in memory and computation time. To demonstrate the success of our method, we present experimental results applying large jigsaws to an object recognition task.