If you want a small book of well-chosen problems, complete with hints and solutions, I think The Stanford Mathematics Problem Book: With Hints and Solutions by George Pólya and Jeremy Kilpatrick is one of the best books you can buy for under £10. (The current price is around £5 on Amazon!)
There are twenty sets of three or four problems, all taken from the Stanford maths competitions, covering number theory, algebra, geometry and “counting” arguments. These problems have a similar feel to the UKMT maths challenge problems, with one key difference: these questions are not multiple choice and require students to present and justify their solutions. Some of the problems require A level Mathematics or Further Mathematics knowledge, for example, there’s one question about the half-angle identities and others that require induction, but many would be accessible to a strong GCSE Mathematics class. Students may need to look up one or two words, as some of the geometric terms are (sadly) not used widely any more.
Like many good problems, each question is stated concisely but allows a range of possible solutions and approaches, which helps make them accessible to more students. Students are encouraged to try their ideas and look for patterns and are rewarded for finding insights that simplify the problem. Students are often asked to hypothesise a result or generalise from some given pattern, both useful skills for students considering further study in maths.
The problems are made even more useful through the addition of the hints, which often take the form of questions such as, “can you separate the parts of the condition?”. If you’re familiar with Pólya’s problem-solving approach in How to Solve It, you’ll recognise the style. The solutions are short, but sufficient to reveal the key points and check the completeness of an answer, however it’s up to the students to provide a convincing argument for their result!
Some of the easier problems take only a few minutes if you spot the main idea and would make great starters. Others could be used for a whole lesson, especially as they try to move from a “convincing argument” to a genuine proof of the result. Individual problems make nice extension tasks for strong students and a full set of problems would be a valuable homework for an A level class.
Overall, this is an excellent selection of problems that can be used to get students thinking about and discussing maths. If you already use problems from NRICH, RISPS or similar, then this is another brilliant resource.