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determined that the maximum density of all arrangements could be found by minimizing a function with 150 variables. In 1992, assisted by his graduate student Samuel Ferguson, he embarked on a research program to systematically apply linear programming methods to find a lower bound on the value of this function for each one of a set of over 5,000 different configurations of spheres. If a lower bound (for the function value) could be found for every one of these configurations that was greater than the

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