Many problems can be formulated as questions about networks. We introduce a general strategy for embedding such problems in Euclidean space and finding discrete solutions in this space, with examples in the Boolean satisfiability (3-SAT) problem, the problem of finding a dominating set of a network, and the problem of encoding a binary data set as the output of a Boolean generative network. Then we introduce a novel directed percolation inverse problem, provide an algorithm for solving it, and investigate how the time it takes to find a solution is related to the percolation properties of the network. Next, we find a compact network representation of movie and viewer attributes from a set of Netflix users' ratings (useful for predicting whether a viewer will like a movie they have not seen). Our approach can match the performance of similar methods but requires far fewer attributes and only a fraction of the training data. Finally, we turn to natural language processing, where the most impressive recent results come from deep neural networks with hundreds of billions of continuous parameters. These networks are too complicated for humans to interpret and require mountains of training data, making it difficult to diagnose and correct unwanted behavior. We demonstrate two approaches that embrace the discreteness of syntax in order to learn grammar from a hundred or even fewer training sentences, produce interpretable output of their inferences, and generate novel sentences from the inferred grammar in a transparent way.