We explore the varied uses of the uniform distribution on [theta - 1/2, theta + 1/2] as an example in the undergraduate probability and statistics sequence or the mathematical statistics course. Like its cousin, the uniform distribution on [0, theta], this density provides tractable examples from the topic of order statistics to hypothesis tests. Unlike its cousin, which appears in many probability and statistics books, this uniform is less well known or used. We discuss maximum likelihood estimators, likelihood ratio tests, confidence intervals, joint distributions of order statistics, use of Mathematica®, sufficiency, and other advanced topics. Finally, we suggest a few exercises deriving likelihood ratio tests when the range is unknown as well, or for the uniform on [theta - rho, theta + rho].