international competition founded in France in 1991 in which over 50 countries now take part. The idea came from the Australian Mathematics Competition, and was named the Kangaroo in recognition of this. The UKMT uses Kangaroo questions for four competitions that are open to students who do sufficiently well in the Challenges, as follows:
Junior Kangaroo for students aged 12 or 13;
Grey Kangaroo for students aged 14;
Pink Kangaroo for students aged 15 or 16;
Senior Kangaroo for students aged 17 or 18.
Each paper consists of 25 questions, for which 60 minutes are allowed. The Junior, Grey and Pink Kangaroo papers are made up of multiple-choice questions. In the Senior Kangaroo, students are asked to give numerical answers but explanations are not required.
Olympiad Papers
These papers are designed for students who do extremely well in the Mathematical Challenges. The heart of all these papers is a small number of tough questions for which fully explained answers are required. They are given the name Olympiad because they are part of a pathway that leads to the UKMT team for the annual International Mathematical Olympiad, a competition in which over one hundred countries take part.
The Olympiad paper that follows on from the Junior Mathematical Challenge is the Junior Mathematical Olympiad for students aged 12 or 13. This is a two-hour paper made up of 10 A-section questions (answers only) and 5 B-section questions (full explanations required).
There are three Olympiad papers that follow on from the Intermediate Mathematical Challenge. These are the Cayley Mathematical Olympiad for students aged 14, the Hamilton Mathematical Olympiad for students aged 15, and the Maclaurin Mathematical Olympiad for students aged 16. Each of these is a two-hour paper with six questions for which full explanations are required. (Before 2004 these papers had A and B sections.)
The Mathematical Olympiad for Girls (MOG) is for girls aged 16 or 17. It aims to encourage and inspire as many girls as possible to get involved in advanced mathematical problem solving. The paper lasts two and a half hours, and consists of five challenging mathematical problems for which full written solutions are required.
There are two British Mathematical Olympiad (BMO) papers – Round 1 and Round 2 – that follow on from the Senior Mathematical Challenge aimed at students aged 17 and 18. To qualify for Round 2, you have to do well in Round 1. The Round 2 questions are therefore very challenging. Students are given three and a half hours for these papers.
The Team Maths Challenge is an event for teams of four students aged 14 or younger. There are over 60 Regional Finals each year, leading to a National Final in June. There are four rounds: the Group Round, the Crossnumber, the Shuttle and the Relay. We have used many of the Crossnumbers for our interludes, where we explain how they work. Questions from other rounds have been used as some of our daily questions. The Senior Team Maths Challenge, which is organised in partnership with the Advanced Mathematics Support Programme, is very similar but aimed at students aged 17 and 18.
We have also included a few problems taken from the UKMT’s Mentoring Scheme. In the scheme students are given monthly problem sheets. They tackle these in their own time, with a mentor who can give them help and who comments on the solutions that are submitted. There is no element of competition.
UKMT publications
We list here the UKMT publications that cover the competitions described above. They contain full solutions to the problems and may be ordered from the UKMT website: www.ukmt.org.uk
Yearbooks
The UKMT has published a Year Book for every year from 1998–99 to 2016–17. Each Year Book includes the problems and solutions for that year’s competitions.
Mathematical Challenges
The following books contain all the papers for the Junior, Intermediate and Senior Mathematical Challenges for the years in question, together with short solutions:
Ten Years of Mathematical Challenges: 1997 to 2006, UKMT, 2006,
Ten Further Years of Mathematical Challenges: 2006 to 2016, UKMT, 2016.
Each of the next three books contains all the problems from the relevant Mathematical Challenge up to the date of publication, arranged by topic and difficulty. The problems are not in multiple-choice format, and the books include hints but not full solutions.
Junior Problems, Andrew Jobbings, UKMT, 2017
Intermediate Problems, Andrew Jobbings, UKMT, 2016
Senior Problems, Andrew Jobbings, UKMT, 2018
In addition, both short and extended solutions for all the Challenge papers for recent years, which include questions for further investigations, may be downloaded for free from the UKMT website.
Mathematical Olympiad
The following books give advice about tackling harder problems at different levels, and include the problems and solutions from different Olympiad competitions, as specified.
First Steps for Problem Solvers, Mary Teresa Fyfe and Andrew Jobbings, UKMT, 2015 – includes all the problems, with solutions, from the Junior Mathematical Olympiad papers from 1999 to 2015.
A Problem Solver’s Handbook, Andrew Jobbings, UKMT, 2013 – includes all the problems, with solutions, from the Intermediate Mathematical Olympiad papers from 2003 to 2012.
A Mathematical Olympiad Primer, 2nd edition, Geoff Smith, UKMT, 2011 – includes all the problems, with solutions, from the British Mathematical Olympiad Round 1 papers from 1996 to 2010.
A Mathematical Olympiad Companion, Geoff Smith, UKMT, 2016 – includes all the problems, with solutions, from the British Mathematical Olympiad Round 2 papers from 2002 to 2016.
1. How many van loads?
A transport company’s vans each carry a maximum load of 12 tonnes. A firm needs to deliver 24 crates each weighing 5 tonnes.
How many van loads will be needed to do this?
2. An L-ish puzzle
Beatrix places copies of the L-shape shown on a 4 × 4 board so that each L-shape covers exactly three cells of the board.
She is allowed to turn around or turn over an L-shape.
What is the largest number of L-shapes she can place on the board without overlaps?
