What is the simplest thing you can do to solve a problem? In the context of semi-supervised feature selection, we tackle exactly this—how much we can gain from two simple classifier-independent strategies. If we have some binary labelled data and some unlabelled, we could assume the unlabelled data are all positives, or assume them all negatives. These minimalist, seemingly naive, approaches have not previously been studied in depth. However, with theoretical and empirical studies, we show they provide powerful results for feature selection, via hypothesis testing and feature ranking. Combining them with some “soft” prior knowledge of the domain, we derive two novel algorithms (Semi-JMI, Semi-IAMB) that outperform significantly more complex competing methods, showing particularly good performance when the labels are missing-not-at-random. We conclude that simple approaches to this problem can work surprisingly well, and in many situations we can provably recover the exact feature selection dynamics, as if we had labelled the entire dataset.