It is well documented in both the British press and mathematics education literature that many consider A-levels (qualifications for secondary school leavers in England, Wales and Northern Ireland) to be insufficient preparation for undergraduate mathematics study in the UK. A-level Mathematics and Further Mathematics are not solely intended for those wishing to go on to study undergraduate mathematics, though this is often the only mathematics study and preparation available to, and done by, most students. A number of British universities now require applicants to perform well on entrance examinations or "extension" papers in addition to achieving certain grades in A-level Mathematics and, sometimes, Further Mathematics. This article outlines research which used the Mathematical Assessment Task Hierarchy (Smith et al., 1996) and builds upon the work of Darlington (2014) to describe the mathematical skills required to answer the questions in extension papers. These are then contrasted with skills required at A- and undergraduate level. Each of the three extension papers analysed (Advanced Extension Awards, Sixth Term Examination Papers and university admissions tests) were found to differ from each other, as well as from A-level and undergraduate examinations. This suggests that there are benefits for students doing such papers before university mathematics study. The advantages of doing these papers are not only for admissions tutors, who can use them as additional indicators of candidates' aptitudes, but also for the students themselves as they gain insights into the nature of undergraduate mathematics and related further mathematical challenges.