This article contributes to an emerging body of research on the role of assessment design in the prevention and detection of contract cheating. Drawing on the largest contract cheating dataset gathered to date (see cheatingandassessment.edu.au), this article examines the types of assignments and exams in which students self-reported having engaged in some form of third-party cheating, and compares this with the types of assignments and exams in which staff self-reported "detection" of third-party cheating. The article outlines three key findings. Firstly, students most commonly reported cheating in the context of exams (particularly multiple-choice exams), yet staff reported the detection of cheating in exams relatively rarely. Secondly, students reported cheating in traditional written assignments, such as reports and essays, at slightly lower rates than exams, however staff detection rates for these assignments were far higher than for exams. So third-party cheating was reported by students as occurring most commonly in exams, yet it was detected most commonly by staff in assignments. And thirdly, staff detection rates relative to student cheating rates were typically highest for text-rich assessments -- regardless of whether they were invigilated (e.g., essay under exam conditions) or not (e.g., essay). These findings challenge simplistic advice in the literature and public debate that universities should move away from text-based assignments and towards invigilated exams as a means to prevent contract cheating. While text-rich forms of assessment are not immune to contract cheating, exams are not inherently secure. While staff appear to be practised at detecting cheating in the context of student writing, greater awareness is needed to improve the detection of cheating in exams.