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During the Second World War, estimates of the number of tanks deployed by Germany were critically needed. The Allies adopted a successful statistical approach to estimate this information: assuming that the tanks are sequentially numbered starting from 1, if we observe k tanks from an unknown total of N, then the best linear unbiased estimator for N is M(1+1/k)-1 where M is the maximum observed serial number. However, in many situations, the original German Tank Problem is insufficient, since typically there are l > 1 factories, and tanks produced by different factories may have serial numbers in disjoint ranges that are often far separated.Clark, Gonye and Miller presented an unbiased estimator for N when the minimum serial number is unknown. Provided one identifies which samples correspond to which factory, one can then estimate each factory's range and summing the sizes of these ranges yields an estimate for the rival's total productivity. We construct an efficient procedure to estimate the total productivity and prove that it is effective when log l/log k is sufficiently small. In the final section, we show that given information about the gaps, we can make an estimator that performs orders of magnitude better when we have a small number of samples. 

In 1946, Erd\H{o}s posed the distinct distance problem, which seeks to findthe minimum number of distinct distances between pairs of points selected fromany configuration of $$n$$ points in the plane. The problem has since beenexplored along with many variants, including ones that extend it into higherdimensions. Less studied but no less intriguing is Erd\H{o}s' distinct angleproblem, which seeks to find point configurations in the plane that minimizethe number of distinct angles. In their recent paper "Distinct Angles inGeneral Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolfuse a logarithmic spiral to establish an upper bound of $$O(n^2)$ on the minimumnumber of distinct angles in the plane in general position, which prohibitsthree points on any line or four on any circle. We consider the question of distinct angles in three dimensions and providebounds on the minimum number of distinct angles in general position in thissetting. We focus on pinned variants of the question, and we examine explicitconstructions of point configurations in $$\mathbb{R}^3$$ which useself-similarity to minimize the number of distinct angles. Furthermore, westudy a variant of the distinct angles question regarding distinct angle chainsand provide bounds on the minimum number of distinct chains in $$\mathbb{R}^2$$and $$\mathbb{R}^3$$.