Recently, many AI researchers and practitioners have embarked on research visions that involve doing AI for "Good". This is part of a general drive towards infusing AI research and practice with ethical thinking. One frequent theme in current ethical guidelines is the requirement that AI be good for all, or: contribute to the Common Good. But what is the Common Good, and is it enough to want to be good? Via four lead questions, I will illustrate challenges and pitfalls when determining, from an AI point of view, what the Common Good is and how it can be enhanced by AI. The questions are: What is the problem / What is a problem?, Who defines the problem?, What is the role of knowledge?, and What are important side effects and dynamics? The illustration will use an example from the domain of "AI for Social Good", more specifically "Data Science for Social Good". Even if the importance of these questions may be known at an abstract level, they do not get asked sufficiently in practice, as shown by an exploratory study of 99 contributions to recent conferences in the field. Turning these challenges and pitfalls into a positive recommendation, as a conclusion I will draw on another characteristic of computer-science thinking and practice to make these impediments visible and attenuate them: "attacks" as a method for improving design. This results in the proposal of ethics pen-testing as a method for helping AI designs to better contribute to the Common Good.