We present the Shape Part Slot Machine, a new method for assembling novel 3D shapes from existing parts by performing contact-based reasoning. Our method represents each shape as a graph of ``slots,'' where each slot is a region of contact between two shape parts. Based on this representation, we design a graph-neural-network-based model for generating new slot graphs and retrieving compatible parts, as well as a gradient-descent-based optimization scheme for assembling the retrieved parts into a complete shape that respects the generated slot graph. This approach does not require any semantic part labels; interestingly, it also does not require complete part geometries -- reasoning about the slots proves sufficient to generate novel, high-quality 3D shapes. We demonstrate that our method generates shapes that outperform existing modeling-by-assembly approaches regarding quality, diversity, and structural complexity.