According to the Maximin principle of distributive justice, one outcome is more just than another if the worst off individual in the first outcome is better off than the worst off individual in the second. This is often interpreted as a highly egalitarian principle, and, more specifically, as a highly egalitarian way of balancing a concern with equality against a concern with efficiency. But this interpretation faces a challenge: why should a concern with efficiency and equality lead us to a (...) concern only for the very worst off? Egalitarian proponents of Maximin tend to meet this challenge by showing that it possible to formally derive Maximin from surprisingly weak conditions relating efficiency and equality to justice. But in this paper, I argue that this reply fails. The issue is not that the proofs are invalid, but instead that every existing proof of Maximin transforms into an impossibility result as soon as we realize that any reasonable conception of equality must satisfy a certain formal property. The upshot is that even if it is possible to defend Maximin on other grounds, there is no known way to defend it on the grounds that it balances a concern with equality against a concern for efficiency. Further, more general implications of this result for egalitarian theories of distributive justice are also explored. (shrink)