These are some university-interview-type problems discussed in the “pre-STEP” sessions at the City of London Academy Southwark on 7, 14, and 21 November 2019.
With university-interview-type problems:
• the professors expect you not to get a complete solution
• they don’t expect you to write out your working in detail or neatly
• they’re looking for you to be able to think of approaches which might solve the problem (wrong approaches can be nearly as good as correct ones), to be able to learn from it and correct yourself if your first approach turns out to be wrong, and at least to be able to work through your approaches sufficiently to be able to tell whether they are right or wrong.