We introduce mathematical tools to describe the geometric problem of sieves, two-dimensional holes that admit certain three-dimensional objects to pass through them, but block others. This is achieved by formulating the sieve design problem as a two-player game where both players (the one that wants to pass, and the one that wants to block) try to find a set of rigid transformations to achieve their objective. We also introduce an algorithm for solving this game by solving a global optimization problem employing both differentiable rendering with gradient-based optimization as well as particle swarm optimization. Our procedure accounts for real-world manufacturing concerns, and we fabricate a variety of examples demonstrating the practical viability of our sieves. Our implementation takes advantage of GPUs and does not rely on any clean or manifold input geometry as long as it is a triangle mesh. We can produce intricate sieves that block an arbitrary set of shapes \(\mathcal {B}\) but admit another arbitrary set of shapes \(\mathcal {A}\) (if finding a solution is possible for our method).