Tips for Writing the UKMT Challenges

Here are some tips for writing the UK Junior, Intermediate, and Senior Mathematical Challenges organised by the United Kingdom Mathematics Trust (UKMT):

• These challenges are multiple-choice tests, so most of the tips on the multiple choice test tips page apply.
• Set realistic goals for yourself, and don't be too worried about doing "badly." These tests are designed such that the average test-taker gets somewhere around half of the marks available. Given that most people writing these contests are above-average in mathematics, even a rather modest score is still creditable. If you are interested in getting a certificate, you can estimate what score you'll need by looking at the UKMT website, but don't be so focused on that that writing the contest ceases to be any fun.
• It may not be to your advantage to attempt every question; it may be better to concentrate on relatively easy questions and ignore the last several questions, giving you more time to work on the remainder. Questions 16–25, and especially questions 21–25, are designed to be rather difficult, so it may be best to focus on the first 15 questions.
• On the Junior and Intermediate competitions, the hard questions are only worth slightly more (6 points as compared with 5) and each question is weighted equally on the Senior competition, so it isn't to your benefit to rush through the easy questions so quickly that you make careless mistakes.
• If you can't answer a question, should you guess randomly? That depends:
  • There's no penalty for a wrong answer on the first 15 questions of the Junior and Intermediate challenges, so you should definitely guess if you can't come up with an answer.
  • For questions 16–20 of the Junior and Intermediate challenges, you get six marks for a correct answer and lose one for an incorrect answer, so it's to your benefit to guess.
  • For questions 21–25 of the Junior and Intermediate challenges, you get six marks for a correct answer and lose two for an incorrect answer, so it's not to your benefit to guess (unless you can eliminate at least one answer with certainty).
  • For each question on the Senior challenge, you gain four marks for a correct answer and lose one for an incorrect answer, so, on average, it will neither help nor hurt you to guess randomly.
  (the recommendations as to whether to guess or not are based on fairly elementary calculation; the proof is left as an exercise to the reader). Also keep in mind that, as discussed on the multiple-choice tests tips page, you can improve your odds of getting the correct answer by eliminating obviously incorrect answers before guessing.
• You don't have to study for these competitions, but if your results are important to you, some preparation is probably a good idea. If you decide to do so:
  • Don't prepare in the same manner that you would prepare for a mathematics test. These challenges test originality and insight, not how well you can remember and apply formulae.
  • Working through problems on old copies of tests is the best way of preparing. At the United Kingdom Mathematics Trust's website you can find a copy of the past year's papers, as well as options for purchasing older issues. Your school may also have some copies of old contests.
  • While most of the content in the questions is of the sort that you have learned in your classes, the content is often employed in unusual or unfamiliar ways that require insight and originality. It can help to study problem solving skills as well as strategies for discovering how to solve problems.
  • This isn't strictly necessary, but since you aren't allowed to use a calculator, you may want to brush up on your mental arithmetic skills and learn some calculation shortcuts.
• Be sure to read the instructions before starting to write the test; they are fairly helpful (of course, if you did prepare by using old contests, you'll probably already be familiar with the instructions in advance).
• Since you aren't allowed to use a calculator, it's highly unlikely that the answer to any question will involve complex calculation. If it appears that it does, look for a shortcut.
• Diagrams are not necessarily drawn to scale, but they appear to be drawn reasonably accurately most of the time. So, it's often helpful to eyeball the diagram to estimate the quantity you're asked to find. This should usually give you a rough idea of what the answer might be.