Statistics and optimization of random pan stacking

We consider a minimal model of random pan stacking. A single pan consists of a V-shaped object characterized by its internal angle α . The stack is constructed by piling up N pans with different angles in a given, random order. The set of pans is generated by sampling from various kinds of distributions of the pan angles: discrete or continuous, uniform or optimized. For large N the mean height depends principally on the average of the distance between the bases of two consecutive pans, and the effective compaction of the stack, compared with the unstacked pans, is 2 log 2/ π . We also obtain the discrete and continuous distributions that maximize the mean stack height. With only two types of pans, the maximum occurs for equal probabilities, while when many types of pans are available, the optimum distribution strongly favours those with the most acute and the most obtuse angles. With a continuous distribution of angles, while one never finds two identical pans, the behaviour is similar to a system with a large number of discrete angles.