The Ultimate Mathematical Challenge: Over 365 puzzles to test your wits and excite your mind
The Ultimate Mathematical Challenge: Over 365 puzzles to test your wits and excite your mind
Литагент HarperCollins
’Be warned: cracking puzzles releases a very addictive drug.’ – Marcus du SautoyHave you ever wanted to be a puzzle pro or logical luminary? Well, look no further!The perfect way to liven up your day, The Ultimate Mathematical Challenge has over 365 puzzles to test your wits and excite your mind. From starter puzzles to perplexing Olympiad problems designed to stretch even the strongest mathematicians, this book is the ideal forum to get your brain into gear and feed it with the challenges it craves.Specially curated from the UK Mathematics Trust’s catalogue of puzzles, most of these problems can be tackled using no more than a little numerical knowledge, logical thinking and native wit. Including interludes of crossnumber conundrums and shuttle challenges, space for your working out and a handy glossary for those obscure mathematical terms, this book has everything you need to solve captivating problems all year round.Do you have what it takes to conquer The Ultimate Mathematical Challenge?
Copyright (#ulink_76ca4698-1304-5af5-9590-269a5d3f3ad1)
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First published by HarperCollinsPublishers 2018
FIRST EDITION
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Source ISBN: 9780008316402
Ebook Edition © November 2018 ISBN: 9780008316419
Version 2018-11-22
Contents
Cover (#ua10b24fa-5137-5ed0-8232-74a38c96554e)
Title Page (#u45d0f140-b767-5816-955e-d4a3b407da0e)
Copyright (#ulink_c7989f6c-17e5-58e5-a3f7-45f7b2a2faa4)
Foreword by Marcus du Sautoy (#ulink_37cfa5b0-e885-5236-bf6f-0d291edc0e42)
Introduction (#ulink_ba90edcf-bd78-5597-9d9d-50d7e39e48ec)
The Problems (#ulink_1f63c861-a8c5-5e82-b5b3-59819c39051f)
Week 1 (#ulink_8b129db6-15e4-5249-94b8-96ea37e6c4c7)
Week 2 (#ulink_d75a4250-adf1-5973-bb56-19aaa81827bc)
Logic Challenge 1 (#ulink_5adef3ac-3021-5a73-a1ac-d2f7467f1b14)
Week 3 (#ulink_6d8c0dcf-79de-51ab-9552-cdc29b9380eb)
Week 4 (#ulink_84bc8175-fd31-5689-bbad-7d8fac1c90ea)
Crossnumber 1 (#ulink_53b0029f-c661-53f1-94f9-c0d6fa3944a4)
Week 5 (#ulink_f7ce063a-9fd6-58a0-91cc-2ceebfd0fdc8)
Week 6 (#ulink_4e6c81df-c439-5b17-a48a-f8ab60b66794)
Make a Number Challenge (#ulink_e2f74f89-0f46-5f41-b0eb-dedb10a5ce63)
Week 7 (#ulink_ca01a304-ca24-5619-abc4-0812e9d4a437)
Week 8 (#ulink_45dd2a5e-696c-5bd2-b9a4-45c8117c30e2)
Crossnumber 2 (#ulink_a04baf27-c57f-5eaa-8f1f-17a2ca27c241)
Week 9 (#ulink_37c16a7c-8255-5f4c-a64b-996caeae2711)
Week 10 (#ulink_b7311d51-8be0-5786-a520-2bb861a9fbfc)
Shuttle Challenge 1 (#ulink_8de49faa-6f29-52c5-b5c7-cc0f65138e16)
Week 11 (#ulink_c1cdb249-92be-5d0d-a63e-c081f7733f25)
Week 12 (#ulink_a46c1364-30a7-5c6e-95a1-fc0fd81ada7c)
Crossnumber 3 (#ulink_a9122b04-e7e6-51c2-bfd2-b7242a83df66)
Week 13 (#ulink_52e175ad-29ac-52c8-95e5-dca50fe38b7f)
Week 14 (#ulink_cc1db94c-154d-54b7-a2fe-dc99ca4d4e50)
Logic Challenge 2 (#ulink_f0dd4cd0-9454-5d0f-8e0b-c18d76cd23c3)
Week 15 (#ulink_25ed5334-4b49-596d-a719-499eee326fc0)
Week 16 (#ulink_ad8e14ad-5991-5a34-af0c-a60bcfc26fca)
Crossnumber 4 (#ulink_b7883565-80cf-5afb-9b12-2f20d042c1fe)
Week 17 (#ulink_aea928b7-5b87-559c-ad59-a9fafd0616a4)
Week 18 (#ulink_e6931345-6833-584d-aad2-5f4cc67619f4)
Shuttle Challenge 2 (#ulink_5a3f80bc-a13b-55a7-8304-8c1e99823445)
Week 19 (#ulink_010cfc41-2572-5a3d-a92d-8ce1835fd59b)
Week 20 (#ulink_19a98aee-cd88-54e4-bbdb-d9641a5a7cb7)
Crossnumber 5 (#ulink_e6194df0-214d-5efe-a993-ab52781073c8)
Week 21 (#ulink_8db2c308-021f-54c8-8416-a59b97770871)
Week 22 (#ulink_028880e9-7954-5ccf-9d0b-cd0006ef3411)
Logic Challenge 3 (#ulink_b6d3b590-4cd6-5277-9187-33b54e4000a5)
Week 23 (#ulink_be2a106b-aef2-57ed-bcf7-02038253c825)
Week 24 (#ulink_6bd00465-5c6d-5625-942d-0e62191b7326)
Crossnumber 6 (#ulink_84381195-b2bc-5cd0-a2f3-b7ca77afbad8)
Week 25 (#ulink_dda10365-e141-53ae-88d4-f0b72f18f7d0)
Week 26 (#ulink_47985789-f55f-51fe-b888-78322bb4c64b)
Shuttle Challenge 3 (#litres_trial_promo)
Week 27 (#litres_trial_promo)
Week 28 (#litres_trial_promo)
Crossnumber 7 (#litres_trial_promo)
Week 29 (#litres_trial_promo)
Week 30 (#litres_trial_promo)
Logic Challenge 4 (#litres_trial_promo)
Week 31 (#litres_trial_promo)
Week 32 (#litres_trial_promo)
Crossnumber 8 (#litres_trial_promo)
Week 33 (#litres_trial_promo)
Week 34 (#litres_trial_promo)
Shuttle Challenge 4 (#litres_trial_promo)
Week 35 (#litres_trial_promo)
Week 36 (#litres_trial_promo)
Crossnumber 9 (#litres_trial_promo)
Week 37 (#litres_trial_promo)
Week 38 (#litres_trial_promo)
Logic Challenge 5 (#litres_trial_promo)
Week 39 (#litres_trial_promo)
Week 40 (#litres_trial_promo)
Crossnumber 10 (#litres_trial_promo)
Week 41 (#litres_trial_promo)
Week 42 (#litres_trial_promo)
Logic Challenge 6 (#litres_trial_promo)
Week 43 (#litres_trial_promo)
Week 44 (#litres_trial_promo)
Crossnumber 11 (#litres_trial_promo)
Week 45 (#litres_trial_promo)
Week 46 (#litres_trial_promo)
Crossnumber 12 (#litres_trial_promo)
Week 47 (#litres_trial_promo)
Week 48 (#litres_trial_promo)
The Penultimate Challenge (#litres_trial_promo)
Week 49 (#litres_trial_promo)
Week 50 (#litres_trial_promo)
Crossnumber 13 (#litres_trial_promo)
Week 51 (#litres_trial_promo)
Week 52 (#litres_trial_promo)
The Ultimate Challenge (#litres_trial_promo)
Week 53 (#litres_trial_promo)
Solutions (#litres_trial_promo)
Problems 1 – 366 (#litres_trial_promo)
Make a Number Challenge (#litres_trial_promo)
Crossnumbers (#litres_trial_promo)
Logic Challenges (#litres_trial_promo)
Shuttle Challenges (#litres_trial_promo)
The Penultimate Challenge (#litres_trial_promo)
The Ultimate Challenge (#litres_trial_promo)
Glossary with Some Useful Facts (#litres_trial_promo)
Table of Squares, Primes etc. (#litres_trial_promo)
Acknowledgements (#litres_trial_promo)
About the Publisher (#litres_trial_promo)
Foreword (#ulink_f1462f54-c1e1-5308-9f65-60f349a55d54)
Marcus du Sautoy (#ulink_f1462f54-c1e1-5308-9f65-60f349a55d54)
There are two things that made me fall in love with mathematics. The first was my teacher at my comprehensive school revealing that there was more to mathematics than long division. Seeing the big stories of mathematics about prime numbers, four-dimensional geometry, symmetry and more made me realise the beauty and creativity that bubbles throughout the subject.
The second was when my teacher introduced me to the puzzles of Martin Gardner in Scientific American. It was then that I got hooked on the amazing buzz you get when you’ve struggled to crack a challenging puzzle and then suddenly you see a clever way to unlock the enigma. Be warned: cracking puzzles releases a very addictive drug.
In this book you can immerse yourself in the joys of the second reason I fell in love with mathematics. Jam-packed with problems that have formed part of the UKMT competitions over the years, these challenges range from those aimed at students taking their first steps onto the mathematical terrain to problems that are for those scaling some of the peaks of the subject.
Although the questions formed part of various competitions, I think it is important to remember that mathematics is not at its heart a competitive sport. As mathematicians we all work together to advance our understanding of the universe of numbers and geometry. Every theorem that we prove relies on all the theorems and proofs that previous generations have laid down right back to the Ancient Greeks. Of all the sciences, mathematics is perhaps unique in the way we truly stand on the shoulders of giants to see further and deeper into our subject.
So the only competitive aspect of the problems for you as a reader is the fun of competing against yourself. Enjoy the problems you can’t solve so easily. They are ultimately the ones that will give you the biggest buzz when you crack them. What gets me up in the morning to go to my desk to do mathematics is all the problems I can’t solve. And when you’ve mastered all the challenges of this book, just remember that mathematics has a whole host of big stories that are still waiting for someone to write the final chapter and reveal the mysteries that still obsess us as mathematicians.
Marcus du Sautoy is Professor of Mathematics and Simonyi Professor for the Public Understanding of Science at the University of Oxford.
Introduction
This book contains a selection of problems drawn from the various competitions and other activities organised by the United Kingdom Mathematics Trust (UKMT).
The UKMT was founded in 1996 by bringing together a number of mathematics competitions for school students organised by three different bodies: the British Mathematical Olympiad Committee, the National Committee for Mathematics Contests and the United Kingdom Mathematics Foundation. Today the UKMT organises a range of different competitions and other activities for students of all ages from 10 to 18, with the stated aim ‘to advance the education of children and young people in mathematics’.
This book consists of one short problem for each day of the year, with an interlude every fourteen days that consists of a longer challenge. The difficulty of the problems varies. We have listed at the end of the book the source of each problem. Together with the description of the competition from which it is drawn, this should give the reader some indication of its intended difficulty. The difficulty of a problem is hard to assess and will naturally depend on the reader. Our intention is that the problems should get gradually harder as each week progresses, and from the start of the year to the end. However, you will probably find many exceptions to this general rule.
Many of the problems were originally set as multiple-choice questions but have been rewritten to remove this feature. This may make some of the problems harder than they originally were, although it should be remembered that in all our competitions students are given only a limited amount of time. It is this time pressure, from which the reader of this book is free, that makes our competitions a tough challenge for school students.
In the Olympiad papers students are asked to write out full solutions and not just give the answer. In marking these papers most of the credit is given for coming up with a cogent and clearly expressed argument; a correct numerical answer by itself counts for little. This feature is missing in the Olympiad problems given in this book, as we are not able to mark your answers! For this reason the full flavour of these problems as presented here is missing. You will find it in the UKMT publications, listed below, which cover the Olympiad problems.
You should also note that the UKMT has a ‘no calculators’ rule for all its competitions. This is because our aim is to encourage good mathematical thinking. Calculators are helpful when numerical answers are required, but using them is often a substitute for thinking. You, dear reader, are, of course, free to use calculators and other electronic devices in tackling these problems, but we think they will only rarely be of any help.
How the problems have been chosen
UKMT problems are designed for students who are currently studying mathematics. In selecting problems for this book aimed at a general reader we have mainly avoided problems that require up-to-date knowledge of the current mathematics curricula. For example, although algebra will often be useful in tackling a problem, we have selected very few problems that are explicitly about algebra. We have included some geometry problems, because geometry is such a beautiful subject. However, most of the problems can be solved using no more than a little numerical knowledge, logical thinking and native wit.
For those whose knowledge is a little rusty we have included a Glossary containing some reminders of mathematical terminology and some basic geometrical facts about angles.
Solutions
To include full solutions, with detailed explanations, to all the problems would require a book four times as large as the current volume. We have therefore generally only given short solutions, but we hope that these will let you know whether you solved the problem correctly, and will be of help if you are ever baffled.
Detailed solutions to many of the problems may be found in the UKMT publications listed below. Solutions to many of the problems, especially those from recent years, may also be downloaded (for free!) from the UKMT website: www.ukmt.org.uk
The range of UKMT competitions
We list below the different UKMT competitions from which these problems have been drawn, giving an indication of the students they are aimed at and the amount of time they are given. Which competition a student is qualified for depends on their school year. Because school years are denominated differently in different parts of the UK, for simplicity we have translated them into the standard ages of the students in these school years.
Primary Team Maths Resources (PTMR)
These are a range of different team activities designed for 10- and 11-year-old students with the aim of facilitating secondary schools in running events for their feeder schools. The activities are mostly similar to those in the Team Maths Challenge but also include some logic problems that we have used for some of our interludes.
Junior Mathematical Challenge (JMC)
Intermediate Mathematical Challenge (IMC)
Senior Mathematical Challenge (SMC)
Each of these papers is made up of 25 multiple-choice questions. The JMC is aimed at students aged 12 or 13, the IMC at students aged 14, 15 or 16, and the SMC at students aged 17 or 18. These papers are often taken by students younger than the target range. For the JMC and IMC, students are given 60 minutes, and for the SMC they are given 90 minutes.
Kangaroo Competitions
The International Mathematical Kangaroo (Kangourou sans Frontières) is an 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.
The Problems (#ulink_cf88e4f6-730f-51d7-a560-f34a51ab4f93)
Week 1 (#ulink_a026831b-594c-5a00-ba44-e516811a8f15)
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?
[SOLUTION] (#litres_trial_promo)
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?
[SOLUTION] (#litres_trial_promo)
3. Granny’s meter
Yesterday, the reading on Granny’s electricity meter was 098657. She was shocked to realise that all six of these digits are different.
How many more units of electricity will she use before the next time all the digits are different?
[SOLUTION] (#litres_trial_promo)
4. Paper folding
Three shapes X, Y and Z are shown below.
A sheet of A4 paper (measuring 297 mm × 210 mm) is folded once and placed flat on the table.
Which of these shapes could be made?
[SOLUTION] (#litres_trial_promo)
5. How many triangles?
In total, how many triangles of any size are there in the diagram?
[SOLUTION] (#litres_trial_promo)
6. Four dice
Rory uses four identical standard dice to build the solid shown in the diagram.
Whenever two dice touch, the numbers on the touching faces are the same. The numbers on some of the faces of the solid are shown.
What number is written on the face marked with an asterisk?
(On a standard dice, the numbers on opposite faces add to 7.)
[SOLUTION] (#litres_trial_promo)
7. Making 73
Taran thought of a whole number and then multiplied it by either 5 or 6. Krishna added 5 or 6 to Taran’s answer. Finally Eshan subtracted either 5 or 6 from Krishna’s answer.
The final result was 73. What number did Taran choose?
[SOLUTION] (#litres_trial_promo)
Week 2 (#ulink_1016459e-b051-5406-97a6-6139c3793e01)
8. Decimal time
In the late eighteenth century, a decimal clock was proposed in which there were 100 minutes in each hour and 10 hours in each day.
Assuming that such a clock started at 0.00 at midnight, what time would it show when an ordinary clock showed 6 o’clock the following morning?
[SOLUTION] (#litres_trial_promo)
9. One size fits all
Harry’s mathematical grandmother keeps a large bag of ‘one size fits all’ socks in a dark cupboard. There are socks in red, blue, pink and green.
How many socks must she pull out to be sure of having a matching pair?
[SOLUTION] (#litres_trial_promo)
10. Cut the net
The diagram represents a rectangular fishing net made from ropes knotted together at the points shown.
The net is cut several times; each cut severs precisely one section of rope between two adjacent knots.
What is the largest number of such cuts that can be made without splitting the net into two separate pieces?
[SOLUTION] (#litres_trial_promo)
11. Times are changing
On a digital clock displaying hours, minutes and seconds, how many times in each 24-hour period do all six digits change simultaneously?
[SOLUTION] (#litres_trial_promo)
12. Making axes
In the addition sum shown, each letter represents a different non-zero digit.
What digit does each letter represent?
[SOLUTION] (#litres_trial_promo)
13. Roundabout
Four cars enter a roundabout at the same time, each one from a different direction, as shown in the diagram.
Each car drives in a clockwise direction and leaves the roundabout before making a complete circuit. No two cars leave the roundabout by the same exit.
How many different ways are there for the cars to leave the roundabout?
[SOLUTION] (#litres_trial_promo)
14. True or false?
None of these statements is true.
Exactly one of these statements is true.
Exactly two of these statements are true.
All of these statements are true.
How many of the statements in the box are true?
[SOLUTION] (#litres_trial_promo)
Logic Challenge 1 (#ulink_2d182eee-66a7-50cc-81d5-2e6005837b3b)
The team photograph
A photograph is to be taken of the school mixed five-a-side football squad, which includes three substitutes. The girls in the squad are Liz, Jenny, Sarah and Tracey. The boys are Alan, Matthew, Peter and Steve.
The team line up in two rows of four. Read the clues below to work out who is standing where and what number they are wearing (which will be one of the numbers from 1 to 8).
Place the number in the top square of the answer grid and the name in the bottom square of each row.
The clues
Tracey is in the front row in front of Jenny.
The average of the two numbers in the middle of the front row is Sarah’s number, a square.
Peter is not sitting next to a girl.
Steve is sitting between Liz and Jenny.
Players with prime numbers, which includes Alan, are sitting in the front row.
There is only one boy on the end of a row.
In both the front and back rows the two places on the right (as you look at it) are filled by a boy and a girl.
Matthew and Steve have the highest and lowest numbers a boy could wear.
Jenny’s number is three times as large as Tracey’s and twice as large as that of Peter, who is not sitting on the end of a row.
Girls have even numbers.
Back row
Front row
[SOLUTION] (#litres_trial_promo)
Week 3 (#ulink_de5bc97f-d628-5f7b-a856-f997eaea0b93)
15. A line of lamp posts
Four lamp posts are in a straight line. The distance from each post to the next is 25 metres.
What is the distance from the first post to the last?
[SOLUTION] (#litres_trial_promo)
16. Sums of digits
For how many three-digit numbers does the sum of the digits equal 25?
[SOLUTION] (#litres_trial_promo)
17. A million seconds
How many days, to the nearest day, are there in a million seconds?
[SOLUTION] (#litres_trial_promo)
18. Sum to 100
The sum of 10 distinct positive integers is 100. What is the largest possible value of any of the 10 integers?
[SOLUTION] (#litres_trial_promo)
19. x marks the spot
The numbers 2, 3, 4, 5, 6, 7, 8 are to be placed, one per square, in the diagram shown such that the four numbers in the horizontal row add up to 21 and the four numbers in the vertical column also add up to 21.
Which number should replace x?
[SOLUTION] (#litres_trial_promo)
20. The last Wednesday
One of the months in a particular year has five Wednesdays, and the third Saturday is the 19th.
Which day of the month is the last Wednesday?
[SOLUTION] (#litres_trial_promo)
21. Her brother’s age
A woman says to her brother, ‘I am four times as old as you were when I was the same age as you are now.’
The woman is 40 years old.
How old is her brother now?
[SOLUTION] (#litres_trial_promo)
Week 4 (#ulink_cdf24422-70e9-5513-8bf6-454eb7dc2098)
22. Pings and pongs
Five pings and five pongs are worth the same as two pongs and eleven pings.
How many pings is a pong worth?
[SOLUTION] (#litres_trial_promo)
23. How many sides?
A single polygon is made by joining dots in the grid with straight lines, which meet only at dots at their end points. No dot is at more than one corner. The diagram shows a five-sided polygon formed in this way.
What is the greatest possible number of sides of a polygon formed by joining the dots using these same rules?
[SOLUTION] (#litres_trial_promo)
24. A tennis club
Three-quarters of the junior members of a tennis club are boys and the rest are girls. What is the ratio of boys to girls among these members?
[SOLUTION] (#litres_trial_promo)
25. Rectangles in a square
Five equal rectangles are placed inside a square with side length 24 cm, as shown in the diagram.
What is the area in cm
of one rectangle?
[SOLUTION] (#litres_trial_promo)
26. The absent present
Four children bought a birthday present for their father. One of the children hid the present. When their mother asked them who had hidden the present, the four children made the following statements:
Alfred: ‘It was not me!’
Benjamin: ‘It was not me!’
Christian: ‘It was Daniel!’
Daniel: ‘It was Benjamin!’
It turned out that exactly one of them did not tell the truth.
Who hid the present?
[SOLUTION] (#litres_trial_promo)
27. Professor Brainstorm’s clock
Professor Brainstorm’s clock gains 16 minutes every day.
After she has set it to the correct time, how many days pass before it next tells the correct time?
[SOLUTION] (#litres_trial_promo)
28. What is n?
You are given that n is a positive integer with the property that when we add n and the sum of its digits, we obtain the number 313.
What are the possible values of n?
[SOLUTION] (#litres_trial_promo)
Crossnumber 1 (#ulink_e70c22be-7532-522f-b98f-ccea74072d81)
A crossnumber works just like a crossword except that, instead of filling each square with one of the letters from A to Z, you have to fill each square with a single digit from 0 to 9.
ACROSS
1. The square of an odd number (2)
3. 18 DOWN divided by six (3)
6. A number divisible by three (3)
7. A factor of 14 ACROSS (3)
9. An odd number that is one more than a cube (2)
10. One more than a number divisible by nine (2)
12. A number divisible by both six and a square greater than one (3)
14. A multiple of 7 ACROSS (3)
16. The sum of twice 1 ACROSS and 14 (2)
18. Seven less than a square (2)
19. One more than three times 6 ACROSS (3)
21. A Fibonacci number divisible by seven (3)
22. A palindrome that is twice a prime (3)
23. (9 ACROSS × 4) − 13 DOWN (2)
DOWN
2. A palindrome (3)
3. (3 × 9 ACROSS) − (1 ACROSS + 23 ACROSS) (3)
4. The product of two primes (2)
5. One less than a perfect number (2)
8. A cube that is also a power of two (3)
9. A factor of 6111 (3)
11. The sum of the digits of 8 DOWN and 21 ACROSS (2)
12. A number divisible by four (2)
13. A prime that is also a Fibonacci number (3)
15. Three times a prime and six greater than a square (3)
17. One less than a multiple of nine (3)
18. A number divisible by three (3)
19. (12 DOWN × 2) − 5 (2)
20. The sum of the digits of 14 ACROSS is one more than twice the sum of the digits of this number (2)
[SOLUTION] (#litres_trial_promo)
Week 5 (#ulink_edbcfc05-2c43-55b7-907c-d10e9646dd84)
29. Turbo the tortoise
Usain runs twice as fast as his mum. His mum runs five times as fast as his pet tortoise, Turbo. They all set off together for a run down the same straight path.
When Usain has run 100 metres, how far apart are his mum and Turbo the tortoise?
[SOLUTION] (#litres_trial_promo)
30. Rolling a cube
A cube is being rolled on a plane so it turns around its edges. Its bottom face passes through the positions 1, 2, 3, 4, 5, 6 and 7 in that order, as shown.
Which of these two positions were occupied by the same face of the cube?
[SOLUTION] (#litres_trial_promo)
31. Small change
My bus fare is 44p. If the driver can give me change, what is the smallest number of coins that must change hands when I pay this fare?
[The coins available are 1p, 2p, 5p, 10p, 20p, 50p, £1 and £2.]
[SOLUTION] (#litres_trial_promo)
32. Eight factors
The number 78 has exactly eight factors, including 1 and 78.
Which is the smallest integer greater than 78 that has eight factors?
[SOLUTION] (#litres_trial_promo)
33. A small sum
In the addition sum ‘TAP’ + ‘BAT’ + ‘MAN’, each letter must represent a different digit and no first digit is zero.
What is the smallest sum that can be obtained?
[SOLUTION] (#litres_trial_promo)
34. A circle on a grid
A circle is added to the grid shown.
What is the largest number of dots that the circle can pass through?
[SOLUTION] (#litres_trial_promo)
35. Numbers around a circle
Five integers are written around a circle in such a way that no two or three consecutive numbers have a sum that is a multiple of 3. Of the five numbers, how many are themselves multiples of 3?
[SOLUTION] (#litres_trial_promo)
Week 6 (#ulink_91c7769f-8820-5f25-8bf3-54a1120854da)
36. Digit sum 2001
Which is the smallest positive integer whose digits add up to 2001?
[SOLUTION] (#litres_trial_promo)
37. Seven semicircular arcs
The diagram shows a curve made from seven semicircular arcs, the radius of each of which is 1 cm, 2 cm, 4 cm or 8 cm.
What is the length of the curve?
[SOLUTION] (#litres_trial_promo)
38. Sorting dominoes
Dominoes are said to be arranged correctly if, for each pair of adjacent dominoes, the numbers of spots on the adjacent ends are equal. Paul laid six dominoes in a line, as shown in the diagram.
He can make a move either by swapping the position of any two dominoes (without rotating either domino) or by rotating one domino.
What is the smallest number of moves he needs to make to arrange all the dominoes correctly?
[SOLUTION] (#litres_trial_promo)
39. Mr Ross
Mr Ross always tells the truth on Thursdays and Fridays but always tells lies on Tuesdays. On the other days of the week he tells the truth or tells lies, at random. For seven consecutive days he was asked what his first name was, and on the first six days he gave the following answers, in order: John, Bob, John, Bob, Pit, Bob.
What was his answer on the seventh day?
[SOLUTION] (#litres_trial_promo)
40. Missing number
Ria wants to write a number in each of the seven bounded regions in the diagram.
Two regions are neighbours if they share part of their boundary. The number in each region is to be the sum of the numbers in all of its neighbours.
Ria has already written in two of the numbers, as shown.
What number must she write in the central region?
[SOLUTION] (#litres_trial_promo)
41. How many moves?
A puzzle starts with nine numbers placed in a grid, as shown.
On each move you are allowed to swap any two numbers. The aim is to arrange for the total of the numbers in each row to be a multiple of 3.
What is the smallest number of moves needed?
[SOLUTION] (#litres_trial_promo)
42. The perimeter of a square
The diagram shows a square that has been divided into five congruent rectangles. The perimeter of each rectangle is 51 cm.
What is the perimeter of the square?
[SOLUTION] (#litres_trial_promo)
Make a Number Challenge (#ulink_3659be5e-8057-52d9-914e-99908f79df38)
In each case you are given some numbers and are challenged to use them to make the target number.
You can use the basic mathematical operations + − × ÷ and brackets, but no other mathematical symbols.
You can use each of the given numbers just once, but you don’t have to use all the numbers. You can’t put digits together to make larger numbers and you can’t use exponents.
Example: Use 1, 2, 3 and 8 to make 27.
Answer: (1 + 8) × 3 and 8 × 3 + 1 + 2 are both correct. However, 81 ÷ 3 and 3
are not acceptable answers.
The following challenges are taken from the Primary Team Maths Resources for 2015.
Question 1
Use 1, 4, 4, 6, 6 and 75 to make 324.
Question 2
Use 1, 2, 4, 6, 7 and 50 to make 405.
Question 3
Use 1, 2, 3, 3, 6 and 100 to make 154.
Question 4
Use 1, 4, 5, 8, 9 and 75 to make 760.
Question 5
Use 5, 6, 8, 8, 9 and 25 to make 426.
Question 6
Use 1, 2, 4, 6, 7 and 75 to make 441.
Question 7
Use 1, 2, 3, 5, 7 and 25 to make 851.
Question 8
Use 2, 4, 5, 7, 8 and 25 to make 594.
Question 9
Use 1, 2, 6, 7, 8 and 25 to make 483.
Question 10
Use 1, 5, 6, 6, 7 and 100 to make 521.
[SOLUTION] (#litres_trial_promo)
Week 7 (#ulink_03a34ff2-82cf-5c81-b47f-f6923aa152a3)
43. Easter eggs
Mary has three brothers and four sisters.
If they, and Mary, all buy each other an Easter egg, how many eggs will be bought?
[SOLUTION] (#litres_trial_promo)
44. A shape sum
In the sum shown, different shapes represent different digits.
What digit does the square represent?
[SOLUTION] (#litres_trial_promo)
45. The pages of a newspaper
A newspaper has thirty-six pages.
Which other pages are on the same sheet as page 10?
[SOLUTION] (#litres_trial_promo)
46. The sum of two primes
The number 12 345 can be expressed as the sum of two primes in exactly one way.
What is the larger of the two primes?
[SOLUTION] (#litres_trial_promo)
47. A perimeter
The diagram shows three touching circles, each of radius 5 cm, and a line touching two of them.
What is the total length of the perimeter of the shaded region?
[SOLUTION] (#litres_trial_promo)
48. The oldest tree
Today the combined age of three oak trees is exactly 900 years. When the youngest tree has reached the present age of the middle tree, the middle tree will be the present age of the oldest tree and four times the present age of the youngest tree.
What is the present age of the oldest tree?
[SOLUTION] (#litres_trial_promo)
49. Who’s done their homework?
Miss Spelling, the English teacher, asked five of her students how many of the five of them had done their homework the day before. Daniel said none, Ellen said only one, Cara said exactly two, Zain said exactly three and Marcus said exactly four. Miss Spelling knew that the students who had not done their homework were not telling the truth but those who had done their homework were telling the truth.
How many of these students had done their homework the day before?
[SOLUTION] (#litres_trial_promo)
Week 8 (#ulink_44197a9e-2266-5731-b2b7-7bb67f9fb6ca)
50. A stack of cubes
Katie writes a different positive integer on the top face of each of the fourteen cubes in the pyramid shown.
The sum of the nine integers written on the cubes in the bottom layer is 50. The integer written on each of the cubes in the middle and top layers of the pyramid is equal to the sum of the integers on the four cubes underneath it.
What is the greatest possible integer that she can write on the top cube?
[SOLUTION] (#litres_trial_promo)
51. The largest remainder
Gregor divides 2015 successively by 1, 2, 3, and so on up to and including 1000. He writes down the remainder for each division.
What is the largest remainder he writes down?
[SOLUTION] (#litres_trial_promo)
52. Go on and on and on and on
In this addition, G, N and O represent different digits, none of which is zero.
What are the numbers in this sum?
[SOLUTION] (#litres_trial_promo)
53. A list of primes
Alice writes down a list of prime numbers less than 100, using each of the digits 1, 2, 3, 4 and 5 only once and using no other digits.
Which prime number must be in her list?
[SOLUTION] (#litres_trial_promo)
54. Continue the pattern
The diagram shows the first three patterns in a sequence in which each pattern has a square hole in the middle.
How many small shaded squares are needed to build the tenth pattern in the sequence?
[SOLUTION] (#litres_trial_promo)
55. How many codes?
Peter has a lock with a three-digit code. He knows that all the digits of his code are different, and that if he divides the second digit by the third and then squares his answer he will get the first digit.
What is the difference between the largest and smallest possible codes?
[SOLUTION] (#litres_trial_promo)
56. A word product
What is the value of P + Q + R in the multiplication shown?
[SOLUTION] (#litres_trial_promo)
Crossnumber 2 (#ulink_ae3d6f66-07db-51f9-a427-e4d0df2050e7)
ACROSS
1. The highest common factor of 5 DOWN and 8 DOWN (2)
3. A prime factor of 2007 (3)
5. 3 DOWN plus the square root of 4 DOWN (3)
6. The product of three consecutive integers, two of which are prime (3)
7. One less than a multiple of 2 DOWN (3)
9. Five less than 14 ACROSS (4)
11. Seven more than the product of the digits of 22 ACROSS (2)
13. Three more than a triangular number (2)
14. 9 ACROSS plus five (4)
16. A square whose digit sum is three more than its square root (3)
18. Three times the product of two consecutive prime numbers (3)
21. The mean of 11 ACROSS and 21 ACROSS is 16 ACROSS (3)
22. Twice a prime number (3)
23. Two less than a square (2)
DOWN
1. Eight less than a multiple of nine (3)
2. A prime factor of 12 DOWN (2)
3. A Fibonacci number that is also a prime (3)
4. A square (2)
5. One less than twice a triangular number (3)
7. 10 DOWN minus three (4)
8. One third the product of three consecutive numbers, two of which are prime (3)
10. 7 DOWN plus three (4)
12. A number whose digit sum is equal to one of its factors (3)
15. The product of two consecutive prime numbers (3)
17. p
+ 1, where p is prime (3)
19. The sum of 16 ACROSS and 3 ACROSS (3)
20. Six less than twice 13 ACROSS (2)
21. Fifteen plus the mean of 1 ACROSS and 11 ACROSS (2)
[SOLUTION] (#litres_trial_promo)
Week 9 (#ulink_cb4e776c-6058-5380-8ff9-9b6fd258652f)
57. Three Tuesdays
Three Tuesdays of a month fall on even-numbered dates.
Which day of the week was the twenty-first day of the month?
[SOLUTION] (#litres_trial_promo)
58. Crack the code
In a seven-digit numerical code, each group of four adjacent digits adds to 16 and each group of five adjacent digits adds to 19.
What is the code?
[SOLUTION] (#litres_trial_promo)
59. Mr Bean’s fruit
Despite his name, Mr Bean likes to eat lots of fruit. He finds that four apples and two oranges cost £1.54 and that two oranges and four bananas cost £1.70.
How much would he have to pay if he bought one apple, one orange and one banana?
[SOLUTION] (#litres_trial_promo)
60. Ali’s bookshelves
Ali is arranging the books on his bookshelves. He puts half his books on the bottom shelf and two-thirds of what remains on the second shelf. Finally, he splits the rest of his books over the other two shelves so that the third shelf contains four more books than the top shelf. There are three books on the top shelf.
How many books are on the bottom shelf?
[SOLUTION] (#litres_trial_promo)
61. An unfair dice
I have an unfair dice that has probability
of landing on a six, with all the other numbers equally likely. If the dice is thrown twice, what is the probability of obtaining a total score of ten?
[SOLUTION] (#litres_trial_promo)
62. A room in Ginkrail
The town of Ginkrail is inhabited entirely by knights and liars. Every sentence spoken by a knight is true, and every sentence spoken by a liar is false. One day some inhabitants of Ginkrail were alone in a room and three of them spoke.
The first one said: ‘There are no more than three of us in the room. All of us are liars.’
The second said: ‘There are no more than four of us in the room. Not all of us are liars.’
The third said: ‘There are five of us in the room. Three of us are liars.’
How many people were in the room and how many liars were among them?
[SOLUTION] (#litres_trial_promo)
63. Curious integers
In the following puzzle, each different capital letter represents a different digit. Thus ‘SEVEN’ represents a five-digit decimal number.
‘SEVEN’ is prime and, as one would expect, ‘SEVEN’ minus ‘THREE’ equals ‘FOUR’.
Curiously, ‘FOUR’ is prime (as is ‘RUOF’) but ‘THREE’ is not prime. Another oddity is that ‘TEN’ is a square.
Find the values of ‘FOUR’ and ‘TEN’.
[SOLUTION] (#litres_trial_promo)
Week 10 (#ulink_4c5f022e-4027-57ea-a354-2985ddf7784b)
64. Eight factors
A certain number has exactly eight factors including 1 and itself. Two of its factors are 21 and 35.
What is the number?
[SOLUTION] (#litres_trial_promo)
65. A nonagon problem
The diagram shows a regular nine-sided polygon (a nonagon or an enneagon) with two of the sides extended to meet at the point X.
What is the size of the acute angle at X?
[SOLUTION] (#litres_trial_promo)
66. How many primes?
Peter wrote a list of all the numbers that could be produced by changing one digit of the number 200.
How many of the numbers on Peter’s list are prime?
[SOLUTION] (#litres_trial_promo)
67. Fill in the blanks
Sam wants to complete the diagram so that each of the nine circles contains one of the digits from 1 to 9 inclusive and each contains a different digit.
Also, the digits in each of the three lines of four circles must have the same total. What is this total?
[SOLUTION] (#litres_trial_promo)
68. The school netball league
In our school netball league, a team gains a certain whole number of points if it wins a game, a lower whole number of points if it draws a game and no points if it loses a game.
After 10 games my team has won 7 games, drawn 3 and gained 44 points. My sister’s team has won 5 games, drawn 2 and lost 3.
How many points has her team gained?
[SOLUTION] (#litres_trial_promo)
69. How many zogs?
The currency used on the planet Zog consists of bank notes of a fixed size differing only in colour. Three green notes and eight blue notes are worth 46 zogs; eight green notes and three blue notes are worth 31 zogs.
How many zogs are two green notes and three blue notes worth?
[SOLUTION] (#litres_trial_promo)
70. How many V-numbers?
A three-digit integer is called a ‘V-number’ if the digits go ‘high-low-high’ – that is, if the tens digit is smaller than both the hundreds digit and the units (or ‘ones’) digit.
How many three-digit ‘V-numbers’ are there?
[SOLUTION] (#litres_trial_promo)
Shuttle Challenge 1 (#ulink_129e2f4c-45c9-5218-b3e6-5e6f33850e8b)
In the Shuttle rounds of the Team and Senior Team Maths Challenges, each team of four students is divided into two pairs who sit at opposite ends of a table. One pair tackles questions 1 and 3; the other pair attempts questions 2 and 4. The numerical answer to question 1 is passed across the table to the other pair who need it to answer question 2, and so on. The answer that is passed on is called A in the subsequent question.
The teams have eight minutes to answer all four questions. They get bonus marks if they answer all the questions correctly within six minutes.
How long will it take you?
Question 1
What is the value of (4
+ 5
) × 7
?
Question 2
[A is the answer to Question 1.]
At which number will the minute hand of a clock be pointing to (A + 1) minutes after midnight?
Question 3
[A is the answer to Question 2.]
John has three sticks that he has formed into a triangle. The length of each stick is a whole number of centimetres.
The length of one of the sticks is (A + 1) cm, and the length of another of the sticks is (A − 1) cm.
How many different possibilities are there for the length of John’s third stick?
Question 4
[A is the answer to Question 3.]
A pyramid with a polygonal base has A faces.
How many edges does the pyramid have?
[SOLUTION] (#litres_trial_promo)
Week 11 (#ulink_2d3fb083-63a2-5504-a0b5-29982fb14514)
71. A magic square
In a magic square, each row, each column and both main diagonals have the same total.
In the partially completed magic square shown, what number should replace N?
[SOLUTION] (#litres_trial_promo)
72. Fly, fly, fly away
In this addition sum, each letter represents a different non-zero digit.
What are the numbers in this sum?
[SOLUTION] (#litres_trial_promo)
73. What is the units digit?
Catherine’s computer correctly calculates
What is the units digit of its answer?
[SOLUTION] (#litres_trial_promo)
74. Minnie’s training
After a year’s training, Minnie Midriffe increases her average speed in the London Marathon by 25%.
By what percentage did her time decrease?
[SOLUTION] (#litres_trial_promo)
75. Telling the truth
The Queen of Hearts always lies for the whole day or always tells the truth for the whole day.
Which of these statements can she never say?
A. ‘Yesterday, I told the truth.’
B. ‘Yesterday, I lied.’
C. ‘Today, I tell the truth.’
D. ‘Today, I lie.’
E. ‘Tomorrow, I shall tell the truth.’
[SOLUTION] (#litres_trial_promo)
76. What is the unshaded area?
Eight congruent semicircles are drawn inside a square of side length 4.
Each semicircle begins at a vertex of the square and ends at a midpoint of an edge of the square.
What is the area of the unshaded part of the square?
[SOLUTION] (#litres_trial_promo)
77. Aimee goes to work
Every day, Aimee goes up an escalator on her journey to work. If she stands still, it takes her 60 seconds to travel from the bottom to the top. One day the escalator was broken so she had to walk up it. This took her 90 seconds.
How many seconds would it take her to travel up the escalator if she walked up at the same speed as before while it was working?
[SOLUTION] (#litres_trial_promo)
Week 12 (#ulink_bbb3881f-c648-5399-b549-b7bd973ec51b)
78. The pages of a book
The pages of a book are numbered 1, 2, 3, and so on. In total, it takes 852 digits to number all the pages of the book. What is the number of the last page?
[SOLUTION] (#litres_trial_promo)
79. A letter sum
Each letter in the sum shown represents a different digit.
The letter A represents an odd digit.
What are the numbers in this sum?
[SOLUTION] (#litres_trial_promo)
80. Timi’s ears
Three inhabitants of the planet Zog met in a crater and counted each other’s ears. Imi said, ‘I can see exactly 8 ears’; Dimi said, ‘I can see exactly 7 ears’; Timi said, ‘I can see exactly 5 ears.’ None of them could see their own ears.
How many ears does Timi have?
[SOLUTION] (#litres_trial_promo)
81. Unusual noughts and crosses
In this unusual game of noughts and crosses, the first player to form a line of three Os or three Xs loses.
It is X’s turn. Where should she place her cross to make sure that she does not lose?
[SOLUTION] (#litres_trial_promo)
82. An average
The average of 16 different positive integers is 16.
What is the greatest possible value that any of these integers could have?
[SOLUTION] (#litres_trial_promo)
83. Painting a cube
Each face of a cube is painted with a different colour from a selection of six colours.
How many different-looking cubes can be made in this way?
[SOLUTION] (#litres_trial_promo)
84. A Suko puzzle
In the puzzle Suko, the numbers from 1 to 9 are to be placed in the spaces (one number in each) so that the number in each circle is equal to the sum of the numbers in the four surrounding spaces.
How many solutions are there to the Suko puzzle shown?
[SOLUTION] (#litres_trial_promo)
Crossnumber 3 (#ulink_b4b21f71-ddf6-546e-a5cf-0fa3cff12984)
ACROSS
2. The sum of a square and a cube (3)
4. Nine less than half 26 ACROSS (2)
6. 13 DOWN plus 5 DOWN minus 2 ACROSS minus 10 DOWN (3)
7. A prime factor of (6 ACROSS plus 15) (2)
8. The square root of 4 ACROSS cubed (2)
9. One more than a multiple of 8 (3)
12. Fifteen less than a cube (2)
14. A multiple of fourteen (3)
17. A prime greater than 13 and whose digits are different (2)
18. The mean of 5 DOWN, 21 DOWN and 28 ACROSS (2)
19. Three more than a square (3)
22. An even number that is less than 24 DOWN (2)
24. The sum of 9 ACROSS and a multiple of five (3)
26. The difference of two two-digit triangular numbers and also one more than an odd square (2)
28. The first two digits of the square of 17 ACROSS (2)
29. The hypotenuse of a triangle whose shorter sides are 21 DOWN and 20 DOWN (3)
30. The lowest common multiple of 10 DOWN and 15 DOWN (2)
31. A factor of 6789 (3)
DOWN
1. The square of (one more than a multiple of 29) (4)
2. A prime factor of 2008 (3)
3. A Fibonacci number that is one more than 2 ACROSS (3)
5. Two less than 11 DOWN minus 4 ACROSS (2)
10. Eight more than half 4 ACROSS (2)
11. The sum of a Fibonacci number and a triangular number in three distinct ways (2)
13. The number whose digits are those of 31 ACROSS reversed (3)
14. 3 DOWN plus 11 DOWN (3)
15. A multiple of 5 that is less than 16 DOWN (2)
16. A power of 2 that is greater than 4 ACROSS and less than 28 ACROSS (2)
20. The exterior angle, in degrees, of a regular polygon (2)
21. 70 per cent of 30 ACROSS (2)
23. A prime whose digits are increasing consecutive numbers (4)
24. A power of 2 multiplied by a power of 3 (3)
25. Three less than a multiple of seven (3)
27. 11 DOWN plus 15 plus a quarter of 4 ACROSS (2)
[SOLUTION] (#litres_trial_promo)
Week 13 (#ulink_e62ac059-2da8-5d33-aff5-2f1c4c98ab87)
85. The Beans’ beans
The Bean family are very particular about beans. At every meal all Beans eat some beans. Pa Bean always eats more beans than Ma Bean but never eats more than half the beans. Ma Bean always eats the same number of beans as both of their children together and the two children always eat the same number of beans as each other. At their last meal they ate 23 beans.
How many beans did Pa Bean eat?
[SOLUTION] (#litres_trial_promo)
86. Palindromic years
The number 2002 is a palindrome, since it reads the same forwards and backwards.
For how many other years this century will the number of the year be a palindrome?
[SOLUTION] (#litres_trial_promo)
87. Swallowing spiders
It was reported recently that, in an average lifetime of 70 years, each human is likely to swallow around 8 spiders while sleeping.
Supposing that the population of the UK is around 60 million, what is the best estimate of the number of unfortunate spiders consumed in this way in the UK each year?
[SOLUTION] (#litres_trial_promo)
88. Multiple missing digits
The two-digit by two-digit multiplication shown has lots of digits missing.
What are the missing digits?
[SOLUTION] (#litres_trial_promo)
89. Pippa’s visit
Pippa is visiting her grandparents. She spends half the time playing, a third sleeping and the remaining 35 minutes eating.
How long is her visit?
[SOLUTION] (#litres_trial_promo)
90. Possible ps
The eight-digit number ‘ppppqqqq’, where p and q are digits, is a multiple of 45.
What are the possible values of p?
[SOLUTION] (#litres_trial_promo)
91. A magic square
A 3 × 3 grid contains nine numbers, not necessarily integers, one in each cell. Each number is doubled to obtain the number on its immediate right and trebled to obtain the number immediately below it.
The sum of the nine numbers is 13. What is the number in the central cell?
[SOLUTION] (#litres_trial_promo)
Week 14 (#ulink_6543a76f-befc-57e5-baee-b333ccfcc3c0)
92. A line of coins
Sixty 20p coins are lined up side by side. Every second 20p coin is then replaced by a 10p coin. Then every third coin is replaced by a 5p coin. Finally, every fourth coin in the row is replaced by a 2p coin.
What is the final value of all the coins in the line?
[SOLUTION] (#litres_trial_promo)
93. What is the area?
The figure shows two shapes that fit together exactly.
Each shape is formed by four semicircles of radius 1. What is the total shaded area?
[SOLUTION] (#litres_trial_promo)
94. My children’s ages
The product of my children’s ages is 1664. The youngest is half as old as the eldest.
How many children do I have?
[SOLUTION] (#litres_trial_promo)
95. Tickets for a school play
Tickets for a school play cost £3 for adults and £1 for children. The total amount collected from ticket sales was £1320. The play was staged in a hall seating 600, but the hall was not completely full.
What was the smallest possible number of adults at the play?
[SOLUTION] (#litres_trial_promo)
96. A mini crossnumber
The solution to each clue of this crossnumber is a two-digit number. None of these numbers begins with zero.
Complete the crossnumber.
Across
1. Multiple of 3
3. Three times a prime
Down
1. Multiple of 25
2. Square
[SOLUTION] (#litres_trial_promo)
97. What is ‘abc’?
The letters a, b and c stand for non-zero digits. The integer ‘abc’ is a multiple of 3; the integer ‘cbabc’ is a multiple of 15; and the integer ‘abcba’ is a multiple of 8.
What is the integer ‘abc’?
[SOLUTION] (#litres_trial_promo)
98. The ninth term
In a sequence of positive integers, each term is larger than the previous term. Also, after the first two terms, each term is the sum of the previous two terms.
The eighth term of the sequence is 390. What is the ninth term?
[SOLUTION] (#litres_trial_promo)
Logic Challenge 2 (#ulink_13836ae1-02b9-5754-be71-4e892b48e0e3)
Five clowns are standing in a line. They are being judged as to who is the most colourful clown.
Read the clues below to work out where each clown is standing in the line, and what colour hat, hair, nose and shoes they are wearing. Here ‘left’ and ‘right’ refer to the position in which the clowns appear to someone who is standing facing them.
The blue hat is worn by Jessie.
The red hat is worn by the clown with the blue nose.
Amy has green hair.
Jessie is on the left, at the end of the row, wearing yellow shoes.
Mitch, with yellow hair, has Jessie and Amy next to him.
Kenny is immediately to the right of the clown with red shoes.
Alby has red hair.
The middle clown in the line-up facing the class has a yellow nose.
Both Jessie and Kenny are wearing yellow shoes.
The clown with yellow hair is standing in between clowns wearing yellow and red shoes.
A clown with a blue nose is next to a clown wearing a blue hat.
Neither Mitch nor Jessie are wearing anything green.
For each clown except Amy, their hat, hair, nose and shoes are different colours.
Kenny’s hat, Jessie’s hair, Mitch’s shoes and Alby’s nose are all the same colour.
The hats worn by the five clowns are all different colours. The same is also true for their hair and noses.
Alby’s hair is the same colour as Amy’s shoes, and Amy’s hair is the same colour as Alby’s shoes.
Everyone except Amy is wearing something that is orange.
[SOLUTION] (#litres_trial_promo)
Week 15 (#ulink_4a42fc09-392a-5dba-9a1b-c03273c9185c)
99. Folded shapes
A sheet of A4 paper (297 mm × 210 mm) is folded once and then laid flat on the table.
Which of these shapes could not be made?
[SOLUTION] (#litres_trial_promo)
100. Einstein’s clocks
Albert Einstein is experimenting with two unusual clocks that both have 24-hour displays. One clock goes at twice the normal speed. The other clock goes backwards, but at the normal speed. Both clocks show the correct time at 13:00.
What is the correct time when the displays on the clocks next agree?
[SOLUTION] (#litres_trial_promo)
101. The total area
The diagram shows three semicircles, each of radius 1.
What is the size of the total shaded area?
[SOLUTION] (#litres_trial_promo)
102. How many weeks?
How many weeks are there in 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 minutes?
[SOLUTION] (#litres_trial_promo)
103. A platinum question
Platinum is a very rare metal, even rarer than gold. Its density is 21.45 g/cm
. Assuming that the world production has been about 110 tonnes for each of the past 50 years, and negligible before that, which of the following has a comparable volume to that of the total amount of platinum ever produced?
(a) a shoe box;
(b) a cupboard;
(c) a house;
(d) Buckingham Palace;
(e) the Grand Canyon
[SOLUTION] (#litres_trial_promo)
104. Underlining numbers
Ten different numbers (not necessarily integers) are written down. Any number that is equal to the product of the other nine numbers is then underlined.
At most, how many numbers can be underlined?
[SOLUTION] (#litres_trial_promo)
105. Placing draughts
Barbara wants to place draughts on a board in such a way that the number of draughts in each row is equal to the number shown at the end of the row, and the number of draughts in each column is equal to the number shown at the bottom of the column. No more than one draught is to be placed in any cell.
In how many ways can this be done?
[SOLUTION] (#litres_trial_promo)
Week 16 (#ulink_d43b1c40-5cfb-56ba-9b3c-33f4cad8768d)
106. Square roots
How many of the numbers
are greater than 10?
[SOLUTION] (#litres_trial_promo)
107. How many lines?
The picture shows seven points and the connections between them.
What is the least number of connecting lines that could be added to the picture so that each of the seven points has the same number of connections with other points?
(Connecting lines are allowed to cross each other.)
[SOLUTION] (#litres_trial_promo)
108. Where in the list?
There are 120 different ways of arranging the letters U, K, M, I and C. All of these arrangements are listed in dictionary order, starting with CIKMU.
Which position in the list does UKIMC occupy?
[SOLUTION] (#litres_trial_promo)
109. Eva’s sport
Two sportsmen (Ben and Filip) and two sportswomen (Eva and Andrea) − a speed skater, a skier, a hockey player and a snowboarder − had dinner at a square table, with one person on each edge of the square.
The skier sat at Andrea’s left hand.
The speed skater sat opposite Ben.
Eva and Filip sat next to each other.
A woman sat at the hockey player’s left hand.
Which sport did Eva do?
[SOLUTION] (#litres_trial_promo)
110. Pedro’s numbers
Pedro writes down a list of six different positive integers, the largest of which is N. There is exactly one pair of these numbers for which the smaller number does not divide the larger.
What is the smallest possible value of N?
[SOLUTION] (#litres_trial_promo)
111. The speed of the train
A train travelling at constant speed takes five seconds to pass completely through a tunnel that is 85 m long, and eight seconds to pass completely through a second tunnel that is 160 m long.
What is the speed of the train?
[SOLUTION] (#litres_trial_promo)
112. What is ‘pqrst’?
The digits p, q, r, s and t are all different.
What is the smallest five-digit integer ‘pqrst’ that is divisible by 1, 2, 3, 4 and 5?
[SOLUTION] (#litres_trial_promo)
Crossnumber 4 (#ulink_65a487f9-458b-5223-977a-e327917da255)
ACROSS
2. A power of two (4)
5. A prime factor of 12345 (3)
6. Six more than a multiple of 13 ACROSS (3)
8. A cube (2)
10. The product of the digits of 25 ACROSS and also less than half of 23 ACROSS (2)
11. The mean of 4 DOWN, 8 ACROSS, 10 ACROSS, 13 ACROSS and 20 ACROSS and more than 3 DOWN (2)
13. A Fibonacci number (2)
14. A multiple of seven (3)
17. Eight less than a square (3)
19. Seven less than 26 DOWN (2)
20. A number that is greater than 3 DOWN and less than 27 DOWN (2)
22. An even number that is the sum of a square and a triangular number in two different ways (2)
23. A prime whose digits add up to five (2)
25. A square and a multiple of five (3)
28. A multiple of 14 that includes a two and an eight among its digits (3)
29. Nine more than a power of 20 ACROSS (4)
DOWN
1. One hundred and ninety-five less than a square (4)
2. One less than a Fibonacci number (3)
3. The highest common factor of 9 DOWN and 15 DOWN (2)
4. The sum of two powers of two (2)
6. (25 ACROSS) per cent of 24 DOWN (3)
7. The shortest side of a right-angled triangle whose longer sides are 24 DOWN and 25 ACROSS (3)
9. The square of a triangular number; also one less than a multiple of five (3)
12. A factor of 732, each of whose digits is a power of two (3)
15. Five multiplied by 3 DOWN (3)
16. An even square; also a multiple of 8 ACROSS (3)
17. A multiple of 17, the product of whose digits is a square multiplied by seven (3)
18. A multiple of nine (3)
21. A power of 21 (4)
24. A factor of 360 (3)
26. Seven more than 19 ACROSS (2)
27. A cube (2)
[SOLUTION] (#litres_trial_promo)
Week 17 (#ulink_390b9ecf-46c7-59fd-b7cc-f66a2cecdf58)
113. How many routes?
How many different routes are there from S to T that do not go through either of the points U and V more than once?
[SOLUTION] (#litres_trial_promo)
114. What Rachel drinks
A bottle contains 750 ml of mineral water. Rachel drinks 50% more than Ross, and these two friends finish the bottle between them.
How much does Rachel drink?
[SOLUTION] (#litres_trial_promo)
115. A magic product square
Place the numbers
in the squares of the grid, with one number in each square, so that the products of the numbers in the three rows, the three columns and the two diagonals are all equal to 1.
[SOLUTION] (#litres_trial_promo)
116. What is the angle?
The diagram shows a regular hexagon PQRSTU, a square PUWX and an equilateral triangle UVW.
What is the size of angle TVU?
[SOLUTION] (#litres_trial_promo)
117. A sum of numbers
Consider the list of all four-digit numbers that can be formed using only the digits 1, 2, 3 and 4, with no repetitions.
What is the sum of all the numbers in this list?
[SOLUTION] (#litres_trial_promo)
118. How many knights?
A group of 25 people consists of knights, serfs and damsels.
Each knight always tells the truth, each serf always lies, and each damsel alternates between telling the truth and lying.
When each of them was asked: ‘Are you a knight?’, 17 of them said ‘Yes’. When each of them was then asked: ‘Are you a damsel?’, 12 of them said ‘Yes’. When each of them was then asked: ‘Are you a serf?’, 8 of them said ‘Yes’.
How many knights are in the group?
[SOLUTION] (#litres_trial_promo)
119. Crossing the river
Two adults and two children wish to cross a river. They make a raft, but it will carry only the weight of one adult or two children.
What is the minimum number of times the raft must cross the river to get all four people to the other side?
(Note: The raft may not cross the river without at least one person on board.)
[SOLUTION] (#litres_trial_promo)
Week 18 (#ulink_52ffdfc3-4e3b-5380-97dc-fb065c1474c4)
120. Gluing cubes
A cube is made by gluing together a number of unit cubes face-to-face. The number of unit cubes that are glued to exactly four other unit cubes is 96.
How many unit cubes are glued to exactly five other unit cubes?
[SOLUTION] (#litres_trial_promo)
121. Mr Gallop’s ponies
Mr Gallop has two stables that each initially housed three ponies. His prize pony, Rein Beau, is worth £250 000. Rein Beau usually spends his day in the small stable, but when he wandered across into the large stable, Mr Gallop was surprised to find that the average value of the ponies in each stable rose by £10 000.
What is the total value of all six ponies?
[SOLUTION] (#litres_trial_promo)
122. Making a square
I have two types of square tile. One type has a side length of 1 cm and the other has a side length of 2 cm.
What is the smallest square that can be made with equal numbers of each type of tile?
[SOLUTION] (#litres_trial_promo)
123. How many pairs?
How many pairs of digits (p, q) are there so that the five-digit integer ‘p869q’ is a multiple of 15?
[SOLUTION] (#litres_trial_promo)
124. What is the product?
Lucy wants to put the numbers 2, 3, 4, 5, 6 and 10 into the circles so that the products of the three numbers along each edge are the same, and as large as possible.
In how many ways can this be done?
[SOLUTION] (#litres_trial_promo)
125. A five-team league
Five teams played in a competition and every team played once against each of the other four teams. Each team received three points for a match it won, one point for a match it drew and no points for a match it lost.
At the end of the competition the points were as follows:
How many of the matches resulted in a draw?
What were the results of the Greens’ matches against the other teams?
[SOLUTION] (#litres_trial_promo)
126. A mini crossnumber
The solution to each clue of this crossnumber is a two-digit number, not beginning with zero.
Across
1. A triangular number
3. A triangular number
Down
1. A square
2. A multiple of 5
In how many different ways can the crossnumber be completed correctly?
[SOLUTION] (#litres_trial_promo)
Shuttle Challenge 2 (#ulink_85f08238-e567-578f-bc49-24130ecd3c2e)
See Shuttle Challenge 1 (#ulink_8de49faa-6f29-52c5-b5c7-cc0f65138e16) for how the Shuttle works.
Question 1
What is the value of
?
Question 2
[A is the answer to Question 1.]
The number A is an example of a palindromic integer – one that is unchanged when the order of its digits is reversed.
How many palindromic integers are there from 300 to A inclusive?
Question 3
[A is the answer to Question 2.]
The diagram shows a triangle drawn on a square grid made up of nine smaller squares.
The area of the shaded triangle is A cm
.
What is the area, in cm
, of one of the smaller squares?
Question 4
[A is the answer to Question 3.]
Write the number A as a word in the gap shown in the following sentence.
Out of the first __________ letters in this sentence, what fraction is vowels?
Now answer the question.
[SOLUTION] (#litres_trial_promo)
Week 19 (#ulink_4f22030e-3560-5924-86ee-69853a96e52f)
127. Coins in a frame
The diagram shows 10 identical coins that fit exactly inside a wooden frame. As a result, each coin is prevented from sliding.
What is the largest number of coins that may be removed so that each remaining coin is still unable to slide?
[SOLUTION] (#litres_trial_promo)
128. Cheetah v. snail
In a sponsored ‘Animal Streak’, the cheetah ran at 90 kilometres per hour, while the snail slimed along at 20 hours per kilometre. The cheetah kept going for 18 seconds.
Roughly how long would the snail take to cover the same distance as the cheetah?
[SOLUTION] (#litres_trial_promo)
129. Memorable phone numbers
A new taxi firm needs a memorable phone number. They want a number that has a maximum of two different digits. Their phone number must start with the digit 3 and be six digits long.
How many such numbers are possible?
[SOLUTION] (#litres_trial_promo)
130. How many games?
Cleo played 40 games of chess and scored 25 points.
A win counts as one point, a draw counts as half a point, and a loss counts as zero points.
How many more games did she win than lose?
[SOLUTION] (#litres_trial_promo)
131. How many draughts?
Barbara wants to place draughts on a 4 × 4 board in such a way that the number of draughts in each row and in each column are all different. (She may place more than one draught in a square, and a square may be empty.)
What is the smallest number of draughts that she would need?
[SOLUTION] (#litres_trial_promo)
132. To and from Jena
In a certain region these are five towns: Freiburg, Göttingen, Hamburg, Ingolstadt and Jena.
One day, 40 trains each made a journey, leaving one of these towns and arriving at another.
10 trains travelled either to or from Freiburg.
10 trains travelled either to or from Göttingen.
10 trains travelled either to or from Hamburg.
10 trains travelled either to or from Ingolstadt.
How many trains travelled either to or from Jena?
[SOLUTION] (#litres_trial_promo)
133. Largest possible remainder
What is the largest possible remainder that is obtained when a two-digit number is divided by the sum of its digits?
[SOLUTION] (#litres_trial_promo)
Week 20 (#ulink_c45feba2-ca27-5f18-aed1-b89c3ac93dff)
134. nth termn
The first term of a sequence of positive integers is 6. The other terms in the sequence follow these rules:
if a term is even then divide it by 2 to obtain the next term;
if a term is odd then multiply it by 5 and subtract 1 to obtain the next term.
For which values of n is the nth term equal to n?
[SOLUTION] (#litres_trial_promo)
135. How many numbers?
Rafael writes down a five-digit number whose digits are all distinct, and whose first digit is equal to the sum of the other four digits.
How many five-digit numbers with this property are there?
[SOLUTION] (#litres_trial_promo)
136. A square area
A regular octagon is placed inside a square, as shown.
The shaded square connects the midpoints of four sides of the octagon.
What fraction of the outer square is shaded?
[SOLUTION] (#litres_trial_promo)
137. ODD plus ODD is EVEN
Find all possible solutions to the ‘word sum’ shown.
Each letter stands for one of the digits 0−9 and has the same meaning each time it occurs. Different letters stand for different digits. No number starts with a zero.
[SOLUTION] (#litres_trial_promo)
138. Only odd digits
How many three-digit multiples of 9 consist only of odd digits?
[SOLUTION] (#litres_trial_promo)
139. How many tests?
Before the last of a series of tests, Sam calculated that a mark of 17 would enable her to average 80 over the series, but that a mark of 92 would raise her average mark over the series to 85.
How many tests were in the series?
[SOLUTION] (#litres_trial_promo)
140. Three primes
Find all positive integers p such that p, p + 8 and p + 16 are all prime.
[SOLUTION] (#litres_trial_promo)
Crossnumber 5 (#ulink_6f4f13ad-e843-5f09-a109-9e98fe449e11)
ACROSS
1. The cube of a square (5)
4. Eight less than 5 DOWN (3)
6. One less than a multiple of seven (3)
7. A prime factor of 20 902 (4)
10. A number whose digits successively decrease by one (3)
12. Sixty per cent of 20 DOWN (3)
14. A multiple of seven (3)
15. A multiple of three whose digits have an even sum (3)
16. The square of a square (3)
17. A prime that is one less than a multiple of six (3)
19. Eleven more than a cube (4)
22. A number all of whose digits are the same (3)
24. A number that leaves a remainder of eleven when divided by thirteen (3)
25. The square of a prime; the sum of the digits of this square is ten (5)
DOWN
1. A number with an odd number of factors (3)
2. Four less than a triangular number (3)
3. The square root of 9 DOWN (2)
4. A factor of 12 ACROSS (3)
5. The longest side of a right-angled triangle whose shorter sides are 3 DOWN and 4 ACROSS (3)
8. A Fibonacci number (5)
9. The square of 3 DOWN (4)
11. Three more than an even cube (5)
13. A prime factor of 34567 (4)
17. The mean of 10 ACROSS, 16 ACROSS, 18 DOWN, 20 DOWN and 21 DOWN (3)
18. A power of eighteen (3)
20. Two less than 22 ACROSS (3)
21. A number whose digits are those of 12 ACROSS reversed (3)
23. A multiple of twenty-three (2)
[SOLUTION] (#litres_trial_promo)
Week 21 (#ulink_7a647005-8dd7-5e6f-be47-4578f5bcdf2e)
141. Joey’s and Zoë’s sums
Joey calculated the sum of the largest and smallest two-digit numbers that are multiples of three. Zoë calculated the sum of the largest and smallest two-digit numbers that are not multiples of three.
What is the difference between their answers?
[SOLUTION] (#litres_trial_promo)
142. When is the party?
Six friends are having dinner together in their local restaurant. The first eats there every day, the second eats there every other day, the third eats there every third day, the fourth eats there every fourth day, the fifth every fifth day and the sixth eats there every sixth day. They agree to have a party the next time they all eat together there. In how many days’ time is the party?
[SOLUTION] (#litres_trial_promo)
143. A multiple of 11
The eight-digit number ‘1234d678’ is a multiple of 11.
Which digit is d?
[SOLUTION] (#litres_trial_promo)
144. Two squares
ABCD is a square. P and Q are squares drawn in the triangles ADC and ABC, as shown.
What is the ratio of the area of the square P to the area of the square Q?
[SOLUTION] (#litres_trial_promo)
145. Proper divisors
Excluding 1 and 24 itself, the positive whole numbers that divide into 24 are 2, 3, 4, 6, 8 and 12. These six numbers are called the proper divisors of 24.
Suppose that you wanted to list in increasing order all those positive integers greater than 1 that are equal to the product of their proper divisors. Which would be the first six numbers in your list?
[SOLUTION] (#litres_trial_promo)
146. Kangaroo game
In the expression
the same letter stands for the same non-zero digit and different letters stand for different digits.
What is the smallest possible positive integer value of the expression?
[SOLUTION] (#litres_trial_promo)
147. A game with sweets
There are 20 sweets on the table. Two players take turns to eat as many sweets as they choose, but they must eat at least one, and never more than half of what remains. The loser is the player who has no valid move.
Is it possible for one of the two players to force the other to lose? If so, how?
[SOLUTION] (#litres_trial_promo)
Week 22 (#ulink_c7102e18-9563-54bc-9859-d92be81711b7)
148. A 1000-digit number
What is the largest number of digits that can be erased from the 1000-digit number 201820182018 … 2018 so that the sum of the remaining digits is 2018?
[SOLUTION] (#litres_trial_promo)
149. Gardeners at work
It takes four gardeners four hours to dig four circular flower beds, each of diameter four metres.
How long will it take six gardeners to dig six circular flower beds each of diameter six metres?
[SOLUTION] (#litres_trial_promo)
150. Overlapping squares
The diagram shows four overlapping squares that have sides of lengths 5 cm, 7 cm, 9 cm and 11 cm.
What is the difference between the total area shaded grey and the total hatched area?
[SOLUTION] (#litres_trial_promo)
151. What can T be?
Each of the numbers from 1 to 10 is to be placed in the circles so that the sum of each line of three numbers is equal to T. Four numbers have already been entered.
Find all the possible values of T.
[SOLUTION] (#litres_trial_promo)
152. Increases of 75%
Find all the two-digit numbers and three-digit numbers that are increased by 75% when their digits are reversed.
[SOLUTION] (#litres_trial_promo)
153. Three groups
For which values of the positive integer n is it possible to divide the first 3n positive integers into three groups each of which has the same sum?
[SOLUTION] (#litres_trial_promo)
154. A board game
Two players, X and Y, play a game on a board that consists of a narrow strip that is one square wide and n squares long. They take turns in placing counters that are one square wide and two squares long on unoccupied squares on the board.
The first player who cannot place a counter on the board loses. X always plays first, and both players always make the best available move.
Who wins the game in the cases where n = 2, 3, 4, 5, 6, 7 and 8?
[SOLUTION] (#litres_trial_promo)
Logic Challenge 3 (#ulink_3030ba14-43de-5b9b-a5a8-04e29f6c1d24)
Five teachers work in a school. By using the clues in the statements below, you need to work out what subject the teacher teaches and what sport they like, and information about their classroom, including the room name and number, and the colour of the classroom door.
Fill in the answer grid with information about each teacher.
Mr Smith teaches Art.
The Science teacher is in the classroom called ‘Square’.
History is taught by Miss Jones.
The favourite sport of Mr Henry does not involve a ball.
Mrs Talbot’s classroom door is coloured yellow and is next to the classroom that has the largest single-digit prime number as its number.
The Maths teacher has the largest classroom door number.
The favourite sport of the teacher in the classroom called ‘Triangle’ is netball.
Miss Jones’s door is coloured orange.
The teacher in the classroom called ‘Circle’ is next to the Maths teacher.
English is taught in the classroom called ‘Sphere’.
The classroom door numbered 3 is next to the teacher who is next to the teacher whose favourite sport is jogging.
Football is the favourite sport of the teacher in the classroom called ‘Cylinder’.
The teacher in the classroom called ‘Square’ is next to the classrooms with the green and orange doors.
The classroom called ‘Cylinder’ is next to the classroom of the Science teacher.
English is taught in the classroom with the ‘unlucky’ prime number.
The classroom door of the Maths teacher is green.
The classroom numbers of Mrs Talbot and Mrs Richard add up to 30.
All the classroom door numbers are prime numbers.
The English teacher is next to a classroom that is next to the classroom of the Science teacher.
The teacher in the classroom with the red door likes rugby and is next to the Maths teacher.
The sum of all the classroom door numbers is 51.
The favourite sport of the teacher in the classroom named ‘Sphere’ is cricket.
The classroom called ‘Triangle’ is next door to the classrooms of the Art and Science teachers.
One of the classrooms has a white door.
[SOLUTION] (#litres_trial_promo)
Week 23 (#ulink_924edb11-6ead-5fdc-98e1-dccaa447f978)
155. Angles around a triangle
What is the value of a + b + c + d + e + f?
[SOLUTION] (#litres_trial_promo)
156. Einstein sees two clocks
Albert Einstein was standing on the station platform thinking about relativity when he noticed he could see two station clocks. Each clock was digital, showing only hours and minutes. He observed that the display on one clock changed to the next minute 10 seconds before the correct time, whereas the display on the other clock changed to the next minute 10 seconds after the correct time.
For what fraction of the time did both clocks show the same time?
[SOLUTION] (#litres_trial_promo)
157. Halving an annulus
The shaded region in the diagram, bounded by two concentric circles, is called an annulus.
The circles have radii 2 cm and 14 cm. The dashed circle divides the area of this annulus into two equal areas.
What is its radius?
[SOLUTION] (#litres_trial_promo)
158. How many pairs?
How many pairs of numbers (a, b) exist such that the sum a + b, the product ab and the quotient
of these two numbers are all equal?
[SOLUTION] (#litres_trial_promo)
159. Two ages
Abi and Becky were comparing their ages and found that Becky is as old as Abi was when Becky was as old as Abi had been when Becky was half as old as Abi is. The sum of their present ages is 44.
How old is Abi?
[SOLUTION] (#litres_trial_promo)
160. At McBride Academy
At McBride Academy there are 300 children, each of whom represents the school in both summer and winter sports. In summer, 60% of these play tennis and the other 40% play badminton. In winter, they play hockey or swim, but not both. 56% of the hockey players play tennis in the summer and 30% of the tennis players swim in the winter.
How many both swim and play badminton?
[SOLUTION] (#litres_trial_promo)
161. Maths, maths, Cayley
How many different solutions are there to this word sum, where each letter stands for a different non-zero digit?
[SOLUTION] (#litres_trial_promo)
Week 24 (#ulink_2ab90312-70ad-589c-966c-621f3c723d72)
162. An angle in a square
The diagram shows a square ABCD and an equilateral triangle ABE.
The point F lies on BC so that EC = EF.
Calculate the angle BEF.
[SOLUTION] (#litres_trial_promo)
163. Areas in a quarter circle
The diagram shows a quarter circle with centre O and two semicircular arcs with diameters OA and OB.
Calculate the ratio of the area of the region shaded grey to the area of the region shaded black.
[SOLUTION] (#litres_trial_promo)
164. How many extensions?
In a large office, each person has their own telephone extension consisting of three digits, but not all possible extensions are in use. To try to prevent wrong numbers, no used number can be converted to another just by swapping two of its digits.
What is the largest possible number of extensions in use in the office?
[SOLUTION] (#litres_trial_promo)
165. The top ball
Six pool balls numbered 1 to 6 are to be arranged in a triangle, as shown.
After three balls are placed in the bottom row, each of the remaining balls is placed so that its number is the difference of the two below it.
Which balls can land up at the top of the triangle?
[SOLUTION] (#litres_trial_promo)
166. Two squares
A square has four digits. When each digit is increased by 1, another square is formed.
What are the two squares?
[SOLUTION] (#litres_trial_promo)
167. Four vehicles
Four vehicles travelled along a road with constant speeds. The car overtook the scooter at 12:00 noon, then met the bike at 14:00 and the motorcycle at 16:00. The motorcycle met the scooter at 17:00 and overtook the bike at 18:00.
At what time did the bike and the scooter meet?
[SOLUTION] (#litres_trial_promo)
168. A marching band
A marching band is having difficulty lining up for a parade. When they line up in rows of 3, one person is left over. When they line up in rows of 4, two people are left over. When they line up in rows of 5, three people are left over. When they line up in rows of 6, four people are left over.
However, the band is able to line up in rows of 7 with nobody left over. What is the smallest possible number of marchers in the band?
[SOLUTION] (#litres_trial_promo)
Crossnumber 6 (#ulink_c1f643ff-f08c-5e86-908b-47a8cc0ae2ea)
ACROSS
1. A prime factor of 8765 (4)
4. The interior angle, in degrees, of a regular polygon; its digits have a product of 12 (3)
6. A Fibonacci number whose digits add up to twenty-four (3)
7. A prime that is one greater than a square (2)
8. Ninety-nine greater than the number formed by reversing the order of its digits (3)
10. A multiple of 25 ACROSS (3)
12. 9 DOWN multiplied by seven (3)
15. A number with seven factors (2)
17. The third side of a right-angled triangle with hypotenuse 9 DOWN and other side 25 ACROSS (2)
19. (10 ACROSS) per cent of 8 ACROSS (3)
21. An odd multiple of nine (3)
23. The product of the seventh prime and the eleventh prime (3)
25. An even number (2)
26. The lowest common multiple of 25 ACROSS and 11 DOWN (3)
28. A multiple of eleven, and also the mean of 12 ACROSS, 16 DOWN, 19 ACROSS, 24 DOWN and 27 DOWN (3)
29. A power of nineteen (4)
DOWN
2. An even square (3)
3. One third of 21 ACROSS (2)
4. The interior angle, in degrees, of the regular polygon that has twice as many sides as the regular polygon whose interior angle, in degrees, is 4 ACROSS (3)
5. One less than a multiple of eleven (3)
8. A power of nine (4)
9. The mean of 3 DOWN, 7 ACROSS, 13 DOWN, 17 ACROSS and 27 DOWN (2)
11. Ninety-one less than 10 ACROSS (2)
13. The sum of the squares of the digits of 23 ACROSS (2)
14. A factor of 4567 (4)
16. The difference between 11 DOWN and 3 DOWN (2)
18. The square root of 8 DOWN (2)
20. The highest common factor of 10 ACROSS and 24 DOWN (2)
21. The total number of days in a year in the months whose names do not contain the letter A (3)
22. A prime that is ten less than a cube (3)
24. A square multiplied by five; the product of its digits is 40 (3)
27. A triangular number that is the sum of two prime numbers that differ by eight (2)
[SOLUTION] (#litres_trial_promo)
Week 25 (#ulink_e0e86d87-357d-528d-ad4b-82ed7f1ff11a)
169. Non-factors of 720
Sam starts to list in ascending order every positive integer that is not a factor of 720.
Which is the tenth number in her list?
[SOLUTION] (#litres_trial_promo)
170. What size is JMO?
In the diagram, JK and ML are parallel. JK = KO = OJ = OM and LM = LO = LK.
Find the size of the angle JMO.
[SOLUTION] (#litres_trial_promo)
171. Amrita’s numbers
Amrita has written down four whole numbers. If she chooses three of her numbers at a time and adds up each triple, she obtains totals of 115, 153, 169 and 181.
What is the largest of Amrita’s numbers?
[SOLUTION] (#litres_trial_promo)
172. Where do the children come from?
Five children, boys Vince, Will and Zac, and girls Xenia and Yvonne, sit at a round table. They come from five different cities: Aberdeen, Belfast, Cardiff, Durham and Edinburgh. The child from Aberdeen sits between Zac and the child from Edinburgh. Neither of the two girls is sitting next to Will. Vince sits between Yvonne and the child from Durham. Zac writes to the child from Cardiff.
Where does each child come from?
[SOLUTION] (#litres_trial_promo)
173. Find four integers
Find four integers whose sum is 400 and such that the first integer is equal to twice the second integer, three times the third integer and four times the fourth integer.
[SOLUTION] (#litres_trial_promo)
174. Bradley’s Bicycle Bazaar
In the window of Bradley’s Bicycle Bazaar there are some unicycles, some bicycles and some tricycles. Laura sees that there are seven saddles in total, thirteen wheels in total and more bicycles than tricycles.
How many unicycles are in the window?
[SOLUTION] (#litres_trial_promo)
175. Liz and Mary’s problems
Liz and Mary compete in solving problems. Each of them is given the same list of 100 problems. For any problem, the first of them to solve it gets 4 points, while the second to solve it gets 1 point. Liz solved 60 problems, and Mary also solved 60 problems. Together, they got 312 points.
How many problems were solved by both of them?
[SOLUTION] (#litres_trial_promo)
Week 26 (#ulink_4ca51d59-2d51-5fec-b069-f57ff887c55b)
176. Edinburgh to London
Travelling by train from Edinburgh to London, I passed a sign saying ‘London 150 miles’. After seven more miles, I passed another sign saying ‘Edinburgh 250 miles’.
How far is it by train from Edinburgh to London?
[SOLUTION] (#litres_trial_promo)
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Литагент HarperCollins
’Be warned: cracking puzzles releases a very addictive drug.’ – Marcus du SautoyHave you ever wanted to be a puzzle pro or logical luminary? Well, look no further!The perfect way to liven up your day, The Ultimate Mathematical Challenge has over 365 puzzles to test your wits and excite your mind. From starter puzzles to perplexing Olympiad problems designed to stretch even the strongest mathematicians, this book is the ideal forum to get your brain into gear and feed it with the challenges it craves.Specially curated from the UK Mathematics Trust’s catalogue of puzzles, most of these problems can be tackled using no more than a little numerical knowledge, logical thinking and native wit. Including interludes of crossnumber conundrums and shuttle challenges, space for your working out and a handy glossary for those obscure mathematical terms, this book has everything you need to solve captivating problems all year round.Do you have what it takes to conquer The Ultimate Mathematical Challenge?
Copyright (#ulink_76ca4698-1304-5af5-9590-269a5d3f3ad1)
HarperCollinsPublishers
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London SE1 9GF
www.harpercollins.co.uk (http://www.harpercollins.co.uk)
First published by HarperCollinsPublishers 2018
FIRST EDITION
© The UK Mathematics Trust 2018
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A catalogue record of this book is available from the British Library
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Source ISBN: 9780008316402
Ebook Edition © November 2018 ISBN: 9780008316419
Version 2018-11-22
Contents
Cover (#ua10b24fa-5137-5ed0-8232-74a38c96554e)
Title Page (#u45d0f140-b767-5816-955e-d4a3b407da0e)
Copyright (#ulink_c7989f6c-17e5-58e5-a3f7-45f7b2a2faa4)
Foreword by Marcus du Sautoy (#ulink_37cfa5b0-e885-5236-bf6f-0d291edc0e42)
Introduction (#ulink_ba90edcf-bd78-5597-9d9d-50d7e39e48ec)
The Problems (#ulink_1f63c861-a8c5-5e82-b5b3-59819c39051f)
Week 1 (#ulink_8b129db6-15e4-5249-94b8-96ea37e6c4c7)
Week 2 (#ulink_d75a4250-adf1-5973-bb56-19aaa81827bc)
Logic Challenge 1 (#ulink_5adef3ac-3021-5a73-a1ac-d2f7467f1b14)
Week 3 (#ulink_6d8c0dcf-79de-51ab-9552-cdc29b9380eb)
Week 4 (#ulink_84bc8175-fd31-5689-bbad-7d8fac1c90ea)
Crossnumber 1 (#ulink_53b0029f-c661-53f1-94f9-c0d6fa3944a4)
Week 5 (#ulink_f7ce063a-9fd6-58a0-91cc-2ceebfd0fdc8)
Week 6 (#ulink_4e6c81df-c439-5b17-a48a-f8ab60b66794)
Make a Number Challenge (#ulink_e2f74f89-0f46-5f41-b0eb-dedb10a5ce63)
Week 7 (#ulink_ca01a304-ca24-5619-abc4-0812e9d4a437)
Week 8 (#ulink_45dd2a5e-696c-5bd2-b9a4-45c8117c30e2)
Crossnumber 2 (#ulink_a04baf27-c57f-5eaa-8f1f-17a2ca27c241)
Week 9 (#ulink_37c16a7c-8255-5f4c-a64b-996caeae2711)
Week 10 (#ulink_b7311d51-8be0-5786-a520-2bb861a9fbfc)
Shuttle Challenge 1 (#ulink_8de49faa-6f29-52c5-b5c7-cc0f65138e16)
Week 11 (#ulink_c1cdb249-92be-5d0d-a63e-c081f7733f25)
Week 12 (#ulink_a46c1364-30a7-5c6e-95a1-fc0fd81ada7c)
Crossnumber 3 (#ulink_a9122b04-e7e6-51c2-bfd2-b7242a83df66)
Week 13 (#ulink_52e175ad-29ac-52c8-95e5-dca50fe38b7f)
Week 14 (#ulink_cc1db94c-154d-54b7-a2fe-dc99ca4d4e50)
Logic Challenge 2 (#ulink_f0dd4cd0-9454-5d0f-8e0b-c18d76cd23c3)
Week 15 (#ulink_25ed5334-4b49-596d-a719-499eee326fc0)
Week 16 (#ulink_ad8e14ad-5991-5a34-af0c-a60bcfc26fca)
Crossnumber 4 (#ulink_b7883565-80cf-5afb-9b12-2f20d042c1fe)
Week 17 (#ulink_aea928b7-5b87-559c-ad59-a9fafd0616a4)
Week 18 (#ulink_e6931345-6833-584d-aad2-5f4cc67619f4)
Shuttle Challenge 2 (#ulink_5a3f80bc-a13b-55a7-8304-8c1e99823445)
Week 19 (#ulink_010cfc41-2572-5a3d-a92d-8ce1835fd59b)
Week 20 (#ulink_19a98aee-cd88-54e4-bbdb-d9641a5a7cb7)
Crossnumber 5 (#ulink_e6194df0-214d-5efe-a993-ab52781073c8)
Week 21 (#ulink_8db2c308-021f-54c8-8416-a59b97770871)
Week 22 (#ulink_028880e9-7954-5ccf-9d0b-cd0006ef3411)
Logic Challenge 3 (#ulink_b6d3b590-4cd6-5277-9187-33b54e4000a5)
Week 23 (#ulink_be2a106b-aef2-57ed-bcf7-02038253c825)
Week 24 (#ulink_6bd00465-5c6d-5625-942d-0e62191b7326)
Crossnumber 6 (#ulink_84381195-b2bc-5cd0-a2f3-b7ca77afbad8)
Week 25 (#ulink_dda10365-e141-53ae-88d4-f0b72f18f7d0)
Week 26 (#ulink_47985789-f55f-51fe-b888-78322bb4c64b)
Shuttle Challenge 3 (#litres_trial_promo)
Week 27 (#litres_trial_promo)
Week 28 (#litres_trial_promo)
Crossnumber 7 (#litres_trial_promo)
Week 29 (#litres_trial_promo)
Week 30 (#litres_trial_promo)
Logic Challenge 4 (#litres_trial_promo)
Week 31 (#litres_trial_promo)
Week 32 (#litres_trial_promo)
Crossnumber 8 (#litres_trial_promo)
Week 33 (#litres_trial_promo)
Week 34 (#litres_trial_promo)
Shuttle Challenge 4 (#litres_trial_promo)
Week 35 (#litres_trial_promo)
Week 36 (#litres_trial_promo)
Crossnumber 9 (#litres_trial_promo)
Week 37 (#litres_trial_promo)
Week 38 (#litres_trial_promo)
Logic Challenge 5 (#litres_trial_promo)
Week 39 (#litres_trial_promo)
Week 40 (#litres_trial_promo)
Crossnumber 10 (#litres_trial_promo)
Week 41 (#litres_trial_promo)
Week 42 (#litres_trial_promo)
Logic Challenge 6 (#litres_trial_promo)
Week 43 (#litres_trial_promo)
Week 44 (#litres_trial_promo)
Crossnumber 11 (#litres_trial_promo)
Week 45 (#litres_trial_promo)
Week 46 (#litres_trial_promo)
Crossnumber 12 (#litres_trial_promo)
Week 47 (#litres_trial_promo)
Week 48 (#litres_trial_promo)
The Penultimate Challenge (#litres_trial_promo)
Week 49 (#litres_trial_promo)
Week 50 (#litres_trial_promo)
Crossnumber 13 (#litres_trial_promo)
Week 51 (#litres_trial_promo)
Week 52 (#litres_trial_promo)
The Ultimate Challenge (#litres_trial_promo)
Week 53 (#litres_trial_promo)
Solutions (#litres_trial_promo)
Problems 1 – 366 (#litres_trial_promo)
Make a Number Challenge (#litres_trial_promo)
Crossnumbers (#litres_trial_promo)
Logic Challenges (#litres_trial_promo)
Shuttle Challenges (#litres_trial_promo)
The Penultimate Challenge (#litres_trial_promo)
The Ultimate Challenge (#litres_trial_promo)
Glossary with Some Useful Facts (#litres_trial_promo)
Table of Squares, Primes etc. (#litres_trial_promo)
Acknowledgements (#litres_trial_promo)
About the Publisher (#litres_trial_promo)
Foreword (#ulink_f1462f54-c1e1-5308-9f65-60f349a55d54)
Marcus du Sautoy (#ulink_f1462f54-c1e1-5308-9f65-60f349a55d54)
There are two things that made me fall in love with mathematics. The first was my teacher at my comprehensive school revealing that there was more to mathematics than long division. Seeing the big stories of mathematics about prime numbers, four-dimensional geometry, symmetry and more made me realise the beauty and creativity that bubbles throughout the subject.
The second was when my teacher introduced me to the puzzles of Martin Gardner in Scientific American. It was then that I got hooked on the amazing buzz you get when you’ve struggled to crack a challenging puzzle and then suddenly you see a clever way to unlock the enigma. Be warned: cracking puzzles releases a very addictive drug.
In this book you can immerse yourself in the joys of the second reason I fell in love with mathematics. Jam-packed with problems that have formed part of the UKMT competitions over the years, these challenges range from those aimed at students taking their first steps onto the mathematical terrain to problems that are for those scaling some of the peaks of the subject.
Although the questions formed part of various competitions, I think it is important to remember that mathematics is not at its heart a competitive sport. As mathematicians we all work together to advance our understanding of the universe of numbers and geometry. Every theorem that we prove relies on all the theorems and proofs that previous generations have laid down right back to the Ancient Greeks. Of all the sciences, mathematics is perhaps unique in the way we truly stand on the shoulders of giants to see further and deeper into our subject.
So the only competitive aspect of the problems for you as a reader is the fun of competing against yourself. Enjoy the problems you can’t solve so easily. They are ultimately the ones that will give you the biggest buzz when you crack them. What gets me up in the morning to go to my desk to do mathematics is all the problems I can’t solve. And when you’ve mastered all the challenges of this book, just remember that mathematics has a whole host of big stories that are still waiting for someone to write the final chapter and reveal the mysteries that still obsess us as mathematicians.
Marcus du Sautoy is Professor of Mathematics and Simonyi Professor for the Public Understanding of Science at the University of Oxford.
Introduction
This book contains a selection of problems drawn from the various competitions and other activities organised by the United Kingdom Mathematics Trust (UKMT).
The UKMT was founded in 1996 by bringing together a number of mathematics competitions for school students organised by three different bodies: the British Mathematical Olympiad Committee, the National Committee for Mathematics Contests and the United Kingdom Mathematics Foundation. Today the UKMT organises a range of different competitions and other activities for students of all ages from 10 to 18, with the stated aim ‘to advance the education of children and young people in mathematics’.
This book consists of one short problem for each day of the year, with an interlude every fourteen days that consists of a longer challenge. The difficulty of the problems varies. We have listed at the end of the book the source of each problem. Together with the description of the competition from which it is drawn, this should give the reader some indication of its intended difficulty. The difficulty of a problem is hard to assess and will naturally depend on the reader. Our intention is that the problems should get gradually harder as each week progresses, and from the start of the year to the end. However, you will probably find many exceptions to this general rule.
Many of the problems were originally set as multiple-choice questions but have been rewritten to remove this feature. This may make some of the problems harder than they originally were, although it should be remembered that in all our competitions students are given only a limited amount of time. It is this time pressure, from which the reader of this book is free, that makes our competitions a tough challenge for school students.
In the Olympiad papers students are asked to write out full solutions and not just give the answer. In marking these papers most of the credit is given for coming up with a cogent and clearly expressed argument; a correct numerical answer by itself counts for little. This feature is missing in the Olympiad problems given in this book, as we are not able to mark your answers! For this reason the full flavour of these problems as presented here is missing. You will find it in the UKMT publications, listed below, which cover the Olympiad problems.
You should also note that the UKMT has a ‘no calculators’ rule for all its competitions. This is because our aim is to encourage good mathematical thinking. Calculators are helpful when numerical answers are required, but using them is often a substitute for thinking. You, dear reader, are, of course, free to use calculators and other electronic devices in tackling these problems, but we think they will only rarely be of any help.
How the problems have been chosen
UKMT problems are designed for students who are currently studying mathematics. In selecting problems for this book aimed at a general reader we have mainly avoided problems that require up-to-date knowledge of the current mathematics curricula. For example, although algebra will often be useful in tackling a problem, we have selected very few problems that are explicitly about algebra. We have included some geometry problems, because geometry is such a beautiful subject. However, most of the problems can be solved using no more than a little numerical knowledge, logical thinking and native wit.
For those whose knowledge is a little rusty we have included a Glossary containing some reminders of mathematical terminology and some basic geometrical facts about angles.
Solutions
To include full solutions, with detailed explanations, to all the problems would require a book four times as large as the current volume. We have therefore generally only given short solutions, but we hope that these will let you know whether you solved the problem correctly, and will be of help if you are ever baffled.
Detailed solutions to many of the problems may be found in the UKMT publications listed below. Solutions to many of the problems, especially those from recent years, may also be downloaded (for free!) from the UKMT website: www.ukmt.org.uk
The range of UKMT competitions
We list below the different UKMT competitions from which these problems have been drawn, giving an indication of the students they are aimed at and the amount of time they are given. Which competition a student is qualified for depends on their school year. Because school years are denominated differently in different parts of the UK, for simplicity we have translated them into the standard ages of the students in these school years.
Primary Team Maths Resources (PTMR)
These are a range of different team activities designed for 10- and 11-year-old students with the aim of facilitating secondary schools in running events for their feeder schools. The activities are mostly similar to those in the Team Maths Challenge but also include some logic problems that we have used for some of our interludes.
Junior Mathematical Challenge (JMC)
Intermediate Mathematical Challenge (IMC)
Senior Mathematical Challenge (SMC)
Each of these papers is made up of 25 multiple-choice questions. The JMC is aimed at students aged 12 or 13, the IMC at students aged 14, 15 or 16, and the SMC at students aged 17 or 18. These papers are often taken by students younger than the target range. For the JMC and IMC, students are given 60 minutes, and for the SMC they are given 90 minutes.
Kangaroo Competitions
The International Mathematical Kangaroo (Kangourou sans Frontières) is an 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.
The Problems (#ulink_cf88e4f6-730f-51d7-a560-f34a51ab4f93)
Week 1 (#ulink_a026831b-594c-5a00-ba44-e516811a8f15)
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?
[SOLUTION] (#litres_trial_promo)
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?
[SOLUTION] (#litres_trial_promo)
3. Granny’s meter
Yesterday, the reading on Granny’s electricity meter was 098657. She was shocked to realise that all six of these digits are different.
How many more units of electricity will she use before the next time all the digits are different?
[SOLUTION] (#litres_trial_promo)
4. Paper folding
Three shapes X, Y and Z are shown below.
A sheet of A4 paper (measuring 297 mm × 210 mm) is folded once and placed flat on the table.
Which of these shapes could be made?
[SOLUTION] (#litres_trial_promo)
5. How many triangles?
In total, how many triangles of any size are there in the diagram?
[SOLUTION] (#litres_trial_promo)
6. Four dice
Rory uses four identical standard dice to build the solid shown in the diagram.
Whenever two dice touch, the numbers on the touching faces are the same. The numbers on some of the faces of the solid are shown.
What number is written on the face marked with an asterisk?
(On a standard dice, the numbers on opposite faces add to 7.)
[SOLUTION] (#litres_trial_promo)
7. Making 73
Taran thought of a whole number and then multiplied it by either 5 or 6. Krishna added 5 or 6 to Taran’s answer. Finally Eshan subtracted either 5 or 6 from Krishna’s answer.
The final result was 73. What number did Taran choose?
[SOLUTION] (#litres_trial_promo)
Week 2 (#ulink_1016459e-b051-5406-97a6-6139c3793e01)
8. Decimal time
In the late eighteenth century, a decimal clock was proposed in which there were 100 minutes in each hour and 10 hours in each day.
Assuming that such a clock started at 0.00 at midnight, what time would it show when an ordinary clock showed 6 o’clock the following morning?
[SOLUTION] (#litres_trial_promo)
9. One size fits all
Harry’s mathematical grandmother keeps a large bag of ‘one size fits all’ socks in a dark cupboard. There are socks in red, blue, pink and green.
How many socks must she pull out to be sure of having a matching pair?
[SOLUTION] (#litres_trial_promo)
10. Cut the net
The diagram represents a rectangular fishing net made from ropes knotted together at the points shown.
The net is cut several times; each cut severs precisely one section of rope between two adjacent knots.
What is the largest number of such cuts that can be made without splitting the net into two separate pieces?
[SOLUTION] (#litres_trial_promo)
11. Times are changing
On a digital clock displaying hours, minutes and seconds, how many times in each 24-hour period do all six digits change simultaneously?
[SOLUTION] (#litres_trial_promo)
12. Making axes
In the addition sum shown, each letter represents a different non-zero digit.
What digit does each letter represent?
[SOLUTION] (#litres_trial_promo)
13. Roundabout
Four cars enter a roundabout at the same time, each one from a different direction, as shown in the diagram.
Each car drives in a clockwise direction and leaves the roundabout before making a complete circuit. No two cars leave the roundabout by the same exit.
How many different ways are there for the cars to leave the roundabout?
[SOLUTION] (#litres_trial_promo)
14. True or false?
None of these statements is true.
Exactly one of these statements is true.
Exactly two of these statements are true.
All of these statements are true.
How many of the statements in the box are true?
[SOLUTION] (#litres_trial_promo)
Logic Challenge 1 (#ulink_2d182eee-66a7-50cc-81d5-2e6005837b3b)
The team photograph
A photograph is to be taken of the school mixed five-a-side football squad, which includes three substitutes. The girls in the squad are Liz, Jenny, Sarah and Tracey. The boys are Alan, Matthew, Peter and Steve.
The team line up in two rows of four. Read the clues below to work out who is standing where and what number they are wearing (which will be one of the numbers from 1 to 8).
Place the number in the top square of the answer grid and the name in the bottom square of each row.
The clues
Tracey is in the front row in front of Jenny.
The average of the two numbers in the middle of the front row is Sarah’s number, a square.
Peter is not sitting next to a girl.
Steve is sitting between Liz and Jenny.
Players with prime numbers, which includes Alan, are sitting in the front row.
There is only one boy on the end of a row.
In both the front and back rows the two places on the right (as you look at it) are filled by a boy and a girl.
Matthew and Steve have the highest and lowest numbers a boy could wear.
Jenny’s number is three times as large as Tracey’s and twice as large as that of Peter, who is not sitting on the end of a row.
Girls have even numbers.
Back row
Front row
[SOLUTION] (#litres_trial_promo)
Week 3 (#ulink_de5bc97f-d628-5f7b-a856-f997eaea0b93)
15. A line of lamp posts
Four lamp posts are in a straight line. The distance from each post to the next is 25 metres.
What is the distance from the first post to the last?
[SOLUTION] (#litres_trial_promo)
16. Sums of digits
For how many three-digit numbers does the sum of the digits equal 25?
[SOLUTION] (#litres_trial_promo)
17. A million seconds
How many days, to the nearest day, are there in a million seconds?
[SOLUTION] (#litres_trial_promo)
18. Sum to 100
The sum of 10 distinct positive integers is 100. What is the largest possible value of any of the 10 integers?
[SOLUTION] (#litres_trial_promo)
19. x marks the spot
The numbers 2, 3, 4, 5, 6, 7, 8 are to be placed, one per square, in the diagram shown such that the four numbers in the horizontal row add up to 21 and the four numbers in the vertical column also add up to 21.
Which number should replace x?
[SOLUTION] (#litres_trial_promo)
20. The last Wednesday
One of the months in a particular year has five Wednesdays, and the third Saturday is the 19th.
Which day of the month is the last Wednesday?
[SOLUTION] (#litres_trial_promo)
21. Her brother’s age
A woman says to her brother, ‘I am four times as old as you were when I was the same age as you are now.’
The woman is 40 years old.
How old is her brother now?
[SOLUTION] (#litres_trial_promo)
Week 4 (#ulink_cdf24422-70e9-5513-8bf6-454eb7dc2098)
22. Pings and pongs
Five pings and five pongs are worth the same as two pongs and eleven pings.
How many pings is a pong worth?
[SOLUTION] (#litres_trial_promo)
23. How many sides?
A single polygon is made by joining dots in the grid with straight lines, which meet only at dots at their end points. No dot is at more than one corner. The diagram shows a five-sided polygon formed in this way.
What is the greatest possible number of sides of a polygon formed by joining the dots using these same rules?
[SOLUTION] (#litres_trial_promo)
24. A tennis club
Three-quarters of the junior members of a tennis club are boys and the rest are girls. What is the ratio of boys to girls among these members?
[SOLUTION] (#litres_trial_promo)
25. Rectangles in a square
Five equal rectangles are placed inside a square with side length 24 cm, as shown in the diagram.
What is the area in cm
of one rectangle?
[SOLUTION] (#litres_trial_promo)
26. The absent present
Four children bought a birthday present for their father. One of the children hid the present. When their mother asked them who had hidden the present, the four children made the following statements:
Alfred: ‘It was not me!’
Benjamin: ‘It was not me!’
Christian: ‘It was Daniel!’
Daniel: ‘It was Benjamin!’
It turned out that exactly one of them did not tell the truth.
Who hid the present?
[SOLUTION] (#litres_trial_promo)
27. Professor Brainstorm’s clock
Professor Brainstorm’s clock gains 16 minutes every day.
After she has set it to the correct time, how many days pass before it next tells the correct time?
[SOLUTION] (#litres_trial_promo)
28. What is n?
You are given that n is a positive integer with the property that when we add n and the sum of its digits, we obtain the number 313.
What are the possible values of n?
[SOLUTION] (#litres_trial_promo)
Crossnumber 1 (#ulink_e70c22be-7532-522f-b98f-ccea74072d81)
A crossnumber works just like a crossword except that, instead of filling each square with one of the letters from A to Z, you have to fill each square with a single digit from 0 to 9.
ACROSS
1. The square of an odd number (2)
3. 18 DOWN divided by six (3)
6. A number divisible by three (3)
7. A factor of 14 ACROSS (3)
9. An odd number that is one more than a cube (2)
10. One more than a number divisible by nine (2)
12. A number divisible by both six and a square greater than one (3)
14. A multiple of 7 ACROSS (3)
16. The sum of twice 1 ACROSS and 14 (2)
18. Seven less than a square (2)
19. One more than three times 6 ACROSS (3)
21. A Fibonacci number divisible by seven (3)
22. A palindrome that is twice a prime (3)
23. (9 ACROSS × 4) − 13 DOWN (2)
DOWN
2. A palindrome (3)
3. (3 × 9 ACROSS) − (1 ACROSS + 23 ACROSS) (3)
4. The product of two primes (2)
5. One less than a perfect number (2)
8. A cube that is also a power of two (3)
9. A factor of 6111 (3)
11. The sum of the digits of 8 DOWN and 21 ACROSS (2)
12. A number divisible by four (2)
13. A prime that is also a Fibonacci number (3)
15. Three times a prime and six greater than a square (3)
17. One less than a multiple of nine (3)
18. A number divisible by three (3)
19. (12 DOWN × 2) − 5 (2)
20. The sum of the digits of 14 ACROSS is one more than twice the sum of the digits of this number (2)
[SOLUTION] (#litres_trial_promo)
Week 5 (#ulink_edbcfc05-2c43-55b7-907c-d10e9646dd84)
29. Turbo the tortoise
Usain runs twice as fast as his mum. His mum runs five times as fast as his pet tortoise, Turbo. They all set off together for a run down the same straight path.
When Usain has run 100 metres, how far apart are his mum and Turbo the tortoise?
[SOLUTION] (#litres_trial_promo)
30. Rolling a cube
A cube is being rolled on a plane so it turns around its edges. Its bottom face passes through the positions 1, 2, 3, 4, 5, 6 and 7 in that order, as shown.
Which of these two positions were occupied by the same face of the cube?
[SOLUTION] (#litres_trial_promo)
31. Small change
My bus fare is 44p. If the driver can give me change, what is the smallest number of coins that must change hands when I pay this fare?
[The coins available are 1p, 2p, 5p, 10p, 20p, 50p, £1 and £2.]
[SOLUTION] (#litres_trial_promo)
32. Eight factors
The number 78 has exactly eight factors, including 1 and 78.
Which is the smallest integer greater than 78 that has eight factors?
[SOLUTION] (#litres_trial_promo)
33. A small sum
In the addition sum ‘TAP’ + ‘BAT’ + ‘MAN’, each letter must represent a different digit and no first digit is zero.
What is the smallest sum that can be obtained?
[SOLUTION] (#litres_trial_promo)
34. A circle on a grid
A circle is added to the grid shown.
What is the largest number of dots that the circle can pass through?
[SOLUTION] (#litres_trial_promo)
35. Numbers around a circle
Five integers are written around a circle in such a way that no two or three consecutive numbers have a sum that is a multiple of 3. Of the five numbers, how many are themselves multiples of 3?
[SOLUTION] (#litres_trial_promo)
Week 6 (#ulink_91c7769f-8820-5f25-8bf3-54a1120854da)
36. Digit sum 2001
Which is the smallest positive integer whose digits add up to 2001?
[SOLUTION] (#litres_trial_promo)
37. Seven semicircular arcs
The diagram shows a curve made from seven semicircular arcs, the radius of each of which is 1 cm, 2 cm, 4 cm or 8 cm.
What is the length of the curve?
[SOLUTION] (#litres_trial_promo)
38. Sorting dominoes
Dominoes are said to be arranged correctly if, for each pair of adjacent dominoes, the numbers of spots on the adjacent ends are equal. Paul laid six dominoes in a line, as shown in the diagram.
He can make a move either by swapping the position of any two dominoes (without rotating either domino) or by rotating one domino.
What is the smallest number of moves he needs to make to arrange all the dominoes correctly?
[SOLUTION] (#litres_trial_promo)
39. Mr Ross
Mr Ross always tells the truth on Thursdays and Fridays but always tells lies on Tuesdays. On the other days of the week he tells the truth or tells lies, at random. For seven consecutive days he was asked what his first name was, and on the first six days he gave the following answers, in order: John, Bob, John, Bob, Pit, Bob.
What was his answer on the seventh day?
[SOLUTION] (#litres_trial_promo)
40. Missing number
Ria wants to write a number in each of the seven bounded regions in the diagram.
Two regions are neighbours if they share part of their boundary. The number in each region is to be the sum of the numbers in all of its neighbours.
Ria has already written in two of the numbers, as shown.
What number must she write in the central region?
[SOLUTION] (#litres_trial_promo)
41. How many moves?
A puzzle starts with nine numbers placed in a grid, as shown.
On each move you are allowed to swap any two numbers. The aim is to arrange for the total of the numbers in each row to be a multiple of 3.
What is the smallest number of moves needed?
[SOLUTION] (#litres_trial_promo)
42. The perimeter of a square
The diagram shows a square that has been divided into five congruent rectangles. The perimeter of each rectangle is 51 cm.
What is the perimeter of the square?
[SOLUTION] (#litres_trial_promo)
Make a Number Challenge (#ulink_3659be5e-8057-52d9-914e-99908f79df38)
In each case you are given some numbers and are challenged to use them to make the target number.
You can use the basic mathematical operations + − × ÷ and brackets, but no other mathematical symbols.
You can use each of the given numbers just once, but you don’t have to use all the numbers. You can’t put digits together to make larger numbers and you can’t use exponents.
Example: Use 1, 2, 3 and 8 to make 27.
Answer: (1 + 8) × 3 and 8 × 3 + 1 + 2 are both correct. However, 81 ÷ 3 and 3
are not acceptable answers.
The following challenges are taken from the Primary Team Maths Resources for 2015.
Question 1
Use 1, 4, 4, 6, 6 and 75 to make 324.
Question 2
Use 1, 2, 4, 6, 7 and 50 to make 405.
Question 3
Use 1, 2, 3, 3, 6 and 100 to make 154.
Question 4
Use 1, 4, 5, 8, 9 and 75 to make 760.
Question 5
Use 5, 6, 8, 8, 9 and 25 to make 426.
Question 6
Use 1, 2, 4, 6, 7 and 75 to make 441.
Question 7
Use 1, 2, 3, 5, 7 and 25 to make 851.
Question 8
Use 2, 4, 5, 7, 8 and 25 to make 594.
Question 9
Use 1, 2, 6, 7, 8 and 25 to make 483.
Question 10
Use 1, 5, 6, 6, 7 and 100 to make 521.
[SOLUTION] (#litres_trial_promo)
Week 7 (#ulink_03a34ff2-82cf-5c81-b47f-f6923aa152a3)
43. Easter eggs
Mary has three brothers and four sisters.
If they, and Mary, all buy each other an Easter egg, how many eggs will be bought?
[SOLUTION] (#litres_trial_promo)
44. A shape sum
In the sum shown, different shapes represent different digits.
What digit does the square represent?
[SOLUTION] (#litres_trial_promo)
45. The pages of a newspaper
A newspaper has thirty-six pages.
Which other pages are on the same sheet as page 10?
[SOLUTION] (#litres_trial_promo)
46. The sum of two primes
The number 12 345 can be expressed as the sum of two primes in exactly one way.
What is the larger of the two primes?
[SOLUTION] (#litres_trial_promo)
47. A perimeter
The diagram shows three touching circles, each of radius 5 cm, and a line touching two of them.
What is the total length of the perimeter of the shaded region?
[SOLUTION] (#litres_trial_promo)
48. The oldest tree
Today the combined age of three oak trees is exactly 900 years. When the youngest tree has reached the present age of the middle tree, the middle tree will be the present age of the oldest tree and four times the present age of the youngest tree.
What is the present age of the oldest tree?
[SOLUTION] (#litres_trial_promo)
49. Who’s done their homework?
Miss Spelling, the English teacher, asked five of her students how many of the five of them had done their homework the day before. Daniel said none, Ellen said only one, Cara said exactly two, Zain said exactly three and Marcus said exactly four. Miss Spelling knew that the students who had not done their homework were not telling the truth but those who had done their homework were telling the truth.
How many of these students had done their homework the day before?
[SOLUTION] (#litres_trial_promo)
Week 8 (#ulink_44197a9e-2266-5731-b2b7-7bb67f9fb6ca)
50. A stack of cubes
Katie writes a different positive integer on the top face of each of the fourteen cubes in the pyramid shown.
The sum of the nine integers written on the cubes in the bottom layer is 50. The integer written on each of the cubes in the middle and top layers of the pyramid is equal to the sum of the integers on the four cubes underneath it.
What is the greatest possible integer that she can write on the top cube?
[SOLUTION] (#litres_trial_promo)
51. The largest remainder
Gregor divides 2015 successively by 1, 2, 3, and so on up to and including 1000. He writes down the remainder for each division.
What is the largest remainder he writes down?
[SOLUTION] (#litres_trial_promo)
52. Go on and on and on and on
In this addition, G, N and O represent different digits, none of which is zero.
What are the numbers in this sum?
[SOLUTION] (#litres_trial_promo)
53. A list of primes
Alice writes down a list of prime numbers less than 100, using each of the digits 1, 2, 3, 4 and 5 only once and using no other digits.
Which prime number must be in her list?
[SOLUTION] (#litres_trial_promo)
54. Continue the pattern
The diagram shows the first three patterns in a sequence in which each pattern has a square hole in the middle.
How many small shaded squares are needed to build the tenth pattern in the sequence?
[SOLUTION] (#litres_trial_promo)
55. How many codes?
Peter has a lock with a three-digit code. He knows that all the digits of his code are different, and that if he divides the second digit by the third and then squares his answer he will get the first digit.
What is the difference between the largest and smallest possible codes?
[SOLUTION] (#litres_trial_promo)
56. A word product
What is the value of P + Q + R in the multiplication shown?
[SOLUTION] (#litres_trial_promo)
Crossnumber 2 (#ulink_ae3d6f66-07db-51f9-a427-e4d0df2050e7)
ACROSS
1. The highest common factor of 5 DOWN and 8 DOWN (2)
3. A prime factor of 2007 (3)
5. 3 DOWN plus the square root of 4 DOWN (3)
6. The product of three consecutive integers, two of which are prime (3)
7. One less than a multiple of 2 DOWN (3)
9. Five less than 14 ACROSS (4)
11. Seven more than the product of the digits of 22 ACROSS (2)
13. Three more than a triangular number (2)
14. 9 ACROSS plus five (4)
16. A square whose digit sum is three more than its square root (3)
18. Three times the product of two consecutive prime numbers (3)
21. The mean of 11 ACROSS and 21 ACROSS is 16 ACROSS (3)
22. Twice a prime number (3)
23. Two less than a square (2)
DOWN
1. Eight less than a multiple of nine (3)
2. A prime factor of 12 DOWN (2)
3. A Fibonacci number that is also a prime (3)
4. A square (2)
5. One less than twice a triangular number (3)
7. 10 DOWN minus three (4)
8. One third the product of three consecutive numbers, two of which are prime (3)
10. 7 DOWN plus three (4)
12. A number whose digit sum is equal to one of its factors (3)
15. The product of two consecutive prime numbers (3)
17. p
+ 1, where p is prime (3)
19. The sum of 16 ACROSS and 3 ACROSS (3)
20. Six less than twice 13 ACROSS (2)
21. Fifteen plus the mean of 1 ACROSS and 11 ACROSS (2)
[SOLUTION] (#litres_trial_promo)
Week 9 (#ulink_cb4e776c-6058-5380-8ff9-9b6fd258652f)
57. Three Tuesdays
Three Tuesdays of a month fall on even-numbered dates.
Which day of the week was the twenty-first day of the month?
[SOLUTION] (#litres_trial_promo)
58. Crack the code
In a seven-digit numerical code, each group of four adjacent digits adds to 16 and each group of five adjacent digits adds to 19.
What is the code?
[SOLUTION] (#litres_trial_promo)
59. Mr Bean’s fruit
Despite his name, Mr Bean likes to eat lots of fruit. He finds that four apples and two oranges cost £1.54 and that two oranges and four bananas cost £1.70.
How much would he have to pay if he bought one apple, one orange and one banana?
[SOLUTION] (#litres_trial_promo)
60. Ali’s bookshelves
Ali is arranging the books on his bookshelves. He puts half his books on the bottom shelf and two-thirds of what remains on the second shelf. Finally, he splits the rest of his books over the other two shelves so that the third shelf contains four more books than the top shelf. There are three books on the top shelf.
How many books are on the bottom shelf?
[SOLUTION] (#litres_trial_promo)
61. An unfair dice
I have an unfair dice that has probability
of landing on a six, with all the other numbers equally likely. If the dice is thrown twice, what is the probability of obtaining a total score of ten?
[SOLUTION] (#litres_trial_promo)
62. A room in Ginkrail
The town of Ginkrail is inhabited entirely by knights and liars. Every sentence spoken by a knight is true, and every sentence spoken by a liar is false. One day some inhabitants of Ginkrail were alone in a room and three of them spoke.
The first one said: ‘There are no more than three of us in the room. All of us are liars.’
The second said: ‘There are no more than four of us in the room. Not all of us are liars.’
The third said: ‘There are five of us in the room. Three of us are liars.’
How many people were in the room and how many liars were among them?
[SOLUTION] (#litres_trial_promo)
63. Curious integers
In the following puzzle, each different capital letter represents a different digit. Thus ‘SEVEN’ represents a five-digit decimal number.
‘SEVEN’ is prime and, as one would expect, ‘SEVEN’ minus ‘THREE’ equals ‘FOUR’.
Curiously, ‘FOUR’ is prime (as is ‘RUOF’) but ‘THREE’ is not prime. Another oddity is that ‘TEN’ is a square.
Find the values of ‘FOUR’ and ‘TEN’.
[SOLUTION] (#litres_trial_promo)
Week 10 (#ulink_4c5f022e-4027-57ea-a354-2985ddf7784b)
64. Eight factors
A certain number has exactly eight factors including 1 and itself. Two of its factors are 21 and 35.
What is the number?
[SOLUTION] (#litres_trial_promo)
65. A nonagon problem
The diagram shows a regular nine-sided polygon (a nonagon or an enneagon) with two of the sides extended to meet at the point X.
What is the size of the acute angle at X?
[SOLUTION] (#litres_trial_promo)
66. How many primes?
Peter wrote a list of all the numbers that could be produced by changing one digit of the number 200.
How many of the numbers on Peter’s list are prime?
[SOLUTION] (#litres_trial_promo)
67. Fill in the blanks
Sam wants to complete the diagram so that each of the nine circles contains one of the digits from 1 to 9 inclusive and each contains a different digit.
Also, the digits in each of the three lines of four circles must have the same total. What is this total?
[SOLUTION] (#litres_trial_promo)
68. The school netball league
In our school netball league, a team gains a certain whole number of points if it wins a game, a lower whole number of points if it draws a game and no points if it loses a game.
After 10 games my team has won 7 games, drawn 3 and gained 44 points. My sister’s team has won 5 games, drawn 2 and lost 3.
How many points has her team gained?
[SOLUTION] (#litres_trial_promo)
69. How many zogs?
The currency used on the planet Zog consists of bank notes of a fixed size differing only in colour. Three green notes and eight blue notes are worth 46 zogs; eight green notes and three blue notes are worth 31 zogs.
How many zogs are two green notes and three blue notes worth?
[SOLUTION] (#litres_trial_promo)
70. How many V-numbers?
A three-digit integer is called a ‘V-number’ if the digits go ‘high-low-high’ – that is, if the tens digit is smaller than both the hundreds digit and the units (or ‘ones’) digit.
How many three-digit ‘V-numbers’ are there?
[SOLUTION] (#litres_trial_promo)
Shuttle Challenge 1 (#ulink_129e2f4c-45c9-5218-b3e6-5e6f33850e8b)
In the Shuttle rounds of the Team and Senior Team Maths Challenges, each team of four students is divided into two pairs who sit at opposite ends of a table. One pair tackles questions 1 and 3; the other pair attempts questions 2 and 4. The numerical answer to question 1 is passed across the table to the other pair who need it to answer question 2, and so on. The answer that is passed on is called A in the subsequent question.
The teams have eight minutes to answer all four questions. They get bonus marks if they answer all the questions correctly within six minutes.
How long will it take you?
Question 1
What is the value of (4
+ 5
) × 7
?
Question 2
[A is the answer to Question 1.]
At which number will the minute hand of a clock be pointing to (A + 1) minutes after midnight?
Question 3
[A is the answer to Question 2.]
John has three sticks that he has formed into a triangle. The length of each stick is a whole number of centimetres.
The length of one of the sticks is (A + 1) cm, and the length of another of the sticks is (A − 1) cm.
How many different possibilities are there for the length of John’s third stick?
Question 4
[A is the answer to Question 3.]
A pyramid with a polygonal base has A faces.
How many edges does the pyramid have?
[SOLUTION] (#litres_trial_promo)
Week 11 (#ulink_2d3fb083-63a2-5504-a0b5-29982fb14514)
71. A magic square
In a magic square, each row, each column and both main diagonals have the same total.
In the partially completed magic square shown, what number should replace N?
[SOLUTION] (#litres_trial_promo)
72. Fly, fly, fly away
In this addition sum, each letter represents a different non-zero digit.
What are the numbers in this sum?
[SOLUTION] (#litres_trial_promo)
73. What is the units digit?
Catherine’s computer correctly calculates
What is the units digit of its answer?
[SOLUTION] (#litres_trial_promo)
74. Minnie’s training
After a year’s training, Minnie Midriffe increases her average speed in the London Marathon by 25%.
By what percentage did her time decrease?
[SOLUTION] (#litres_trial_promo)
75. Telling the truth
The Queen of Hearts always lies for the whole day or always tells the truth for the whole day.
Which of these statements can she never say?
A. ‘Yesterday, I told the truth.’
B. ‘Yesterday, I lied.’
C. ‘Today, I tell the truth.’
D. ‘Today, I lie.’
E. ‘Tomorrow, I shall tell the truth.’
[SOLUTION] (#litres_trial_promo)
76. What is the unshaded area?
Eight congruent semicircles are drawn inside a square of side length 4.
Each semicircle begins at a vertex of the square and ends at a midpoint of an edge of the square.
What is the area of the unshaded part of the square?
[SOLUTION] (#litres_trial_promo)
77. Aimee goes to work
Every day, Aimee goes up an escalator on her journey to work. If she stands still, it takes her 60 seconds to travel from the bottom to the top. One day the escalator was broken so she had to walk up it. This took her 90 seconds.
How many seconds would it take her to travel up the escalator if she walked up at the same speed as before while it was working?
[SOLUTION] (#litres_trial_promo)
Week 12 (#ulink_bbb3881f-c648-5399-b549-b7bd973ec51b)
78. The pages of a book
The pages of a book are numbered 1, 2, 3, and so on. In total, it takes 852 digits to number all the pages of the book. What is the number of the last page?
[SOLUTION] (#litres_trial_promo)
79. A letter sum
Each letter in the sum shown represents a different digit.
The letter A represents an odd digit.
What are the numbers in this sum?
[SOLUTION] (#litres_trial_promo)
80. Timi’s ears
Three inhabitants of the planet Zog met in a crater and counted each other’s ears. Imi said, ‘I can see exactly 8 ears’; Dimi said, ‘I can see exactly 7 ears’; Timi said, ‘I can see exactly 5 ears.’ None of them could see their own ears.
How many ears does Timi have?
[SOLUTION] (#litres_trial_promo)
81. Unusual noughts and crosses
In this unusual game of noughts and crosses, the first player to form a line of three Os or three Xs loses.
It is X’s turn. Where should she place her cross to make sure that she does not lose?
[SOLUTION] (#litres_trial_promo)
82. An average
The average of 16 different positive integers is 16.
What is the greatest possible value that any of these integers could have?
[SOLUTION] (#litres_trial_promo)
83. Painting a cube
Each face of a cube is painted with a different colour from a selection of six colours.
How many different-looking cubes can be made in this way?
[SOLUTION] (#litres_trial_promo)
84. A Suko puzzle
In the puzzle Suko, the numbers from 1 to 9 are to be placed in the spaces (one number in each) so that the number in each circle is equal to the sum of the numbers in the four surrounding spaces.
How many solutions are there to the Suko puzzle shown?
[SOLUTION] (#litres_trial_promo)
Crossnumber 3 (#ulink_b4b21f71-ddf6-546e-a5cf-0fa3cff12984)
ACROSS
2. The sum of a square and a cube (3)
4. Nine less than half 26 ACROSS (2)
6. 13 DOWN plus 5 DOWN minus 2 ACROSS minus 10 DOWN (3)
7. A prime factor of (6 ACROSS plus 15) (2)
8. The square root of 4 ACROSS cubed (2)
9. One more than a multiple of 8 (3)
12. Fifteen less than a cube (2)
14. A multiple of fourteen (3)
17. A prime greater than 13 and whose digits are different (2)
18. The mean of 5 DOWN, 21 DOWN and 28 ACROSS (2)
19. Three more than a square (3)
22. An even number that is less than 24 DOWN (2)
24. The sum of 9 ACROSS and a multiple of five (3)
26. The difference of two two-digit triangular numbers and also one more than an odd square (2)
28. The first two digits of the square of 17 ACROSS (2)
29. The hypotenuse of a triangle whose shorter sides are 21 DOWN and 20 DOWN (3)
30. The lowest common multiple of 10 DOWN and 15 DOWN (2)
31. A factor of 6789 (3)
DOWN
1. The square of (one more than a multiple of 29) (4)
2. A prime factor of 2008 (3)
3. A Fibonacci number that is one more than 2 ACROSS (3)
5. Two less than 11 DOWN minus 4 ACROSS (2)
10. Eight more than half 4 ACROSS (2)
11. The sum of a Fibonacci number and a triangular number in three distinct ways (2)
13. The number whose digits are those of 31 ACROSS reversed (3)
14. 3 DOWN plus 11 DOWN (3)
15. A multiple of 5 that is less than 16 DOWN (2)
16. A power of 2 that is greater than 4 ACROSS and less than 28 ACROSS (2)
20. The exterior angle, in degrees, of a regular polygon (2)
21. 70 per cent of 30 ACROSS (2)
23. A prime whose digits are increasing consecutive numbers (4)
24. A power of 2 multiplied by a power of 3 (3)
25. Three less than a multiple of seven (3)
27. 11 DOWN plus 15 plus a quarter of 4 ACROSS (2)
[SOLUTION] (#litres_trial_promo)
Week 13 (#ulink_e62ac059-2da8-5d33-aff5-2f1c4c98ab87)
85. The Beans’ beans
The Bean family are very particular about beans. At every meal all Beans eat some beans. Pa Bean always eats more beans than Ma Bean but never eats more than half the beans. Ma Bean always eats the same number of beans as both of their children together and the two children always eat the same number of beans as each other. At their last meal they ate 23 beans.
How many beans did Pa Bean eat?
[SOLUTION] (#litres_trial_promo)
86. Palindromic years
The number 2002 is a palindrome, since it reads the same forwards and backwards.
For how many other years this century will the number of the year be a palindrome?
[SOLUTION] (#litres_trial_promo)
87. Swallowing spiders
It was reported recently that, in an average lifetime of 70 years, each human is likely to swallow around 8 spiders while sleeping.
Supposing that the population of the UK is around 60 million, what is the best estimate of the number of unfortunate spiders consumed in this way in the UK each year?
[SOLUTION] (#litres_trial_promo)
88. Multiple missing digits
The two-digit by two-digit multiplication shown has lots of digits missing.
What are the missing digits?
[SOLUTION] (#litres_trial_promo)
89. Pippa’s visit
Pippa is visiting her grandparents. She spends half the time playing, a third sleeping and the remaining 35 minutes eating.
How long is her visit?
[SOLUTION] (#litres_trial_promo)
90. Possible ps
The eight-digit number ‘ppppqqqq’, where p and q are digits, is a multiple of 45.
What are the possible values of p?
[SOLUTION] (#litres_trial_promo)
91. A magic square
A 3 × 3 grid contains nine numbers, not necessarily integers, one in each cell. Each number is doubled to obtain the number on its immediate right and trebled to obtain the number immediately below it.
The sum of the nine numbers is 13. What is the number in the central cell?
[SOLUTION] (#litres_trial_promo)
Week 14 (#ulink_6543a76f-befc-57e5-baee-b333ccfcc3c0)
92. A line of coins
Sixty 20p coins are lined up side by side. Every second 20p coin is then replaced by a 10p coin. Then every third coin is replaced by a 5p coin. Finally, every fourth coin in the row is replaced by a 2p coin.
What is the final value of all the coins in the line?
[SOLUTION] (#litres_trial_promo)
93. What is the area?
The figure shows two shapes that fit together exactly.
Each shape is formed by four semicircles of radius 1. What is the total shaded area?
[SOLUTION] (#litres_trial_promo)
94. My children’s ages
The product of my children’s ages is 1664. The youngest is half as old as the eldest.
How many children do I have?
[SOLUTION] (#litres_trial_promo)
95. Tickets for a school play
Tickets for a school play cost £3 for adults and £1 for children. The total amount collected from ticket sales was £1320. The play was staged in a hall seating 600, but the hall was not completely full.
What was the smallest possible number of adults at the play?
[SOLUTION] (#litres_trial_promo)
96. A mini crossnumber
The solution to each clue of this crossnumber is a two-digit number. None of these numbers begins with zero.
Complete the crossnumber.
Across
1. Multiple of 3
3. Three times a prime
Down
1. Multiple of 25
2. Square
[SOLUTION] (#litres_trial_promo)
97. What is ‘abc’?
The letters a, b and c stand for non-zero digits. The integer ‘abc’ is a multiple of 3; the integer ‘cbabc’ is a multiple of 15; and the integer ‘abcba’ is a multiple of 8.
What is the integer ‘abc’?
[SOLUTION] (#litres_trial_promo)
98. The ninth term
In a sequence of positive integers, each term is larger than the previous term. Also, after the first two terms, each term is the sum of the previous two terms.
The eighth term of the sequence is 390. What is the ninth term?
[SOLUTION] (#litres_trial_promo)
Logic Challenge 2 (#ulink_13836ae1-02b9-5754-be71-4e892b48e0e3)
Five clowns are standing in a line. They are being judged as to who is the most colourful clown.
Read the clues below to work out where each clown is standing in the line, and what colour hat, hair, nose and shoes they are wearing. Here ‘left’ and ‘right’ refer to the position in which the clowns appear to someone who is standing facing them.
The blue hat is worn by Jessie.
The red hat is worn by the clown with the blue nose.
Amy has green hair.
Jessie is on the left, at the end of the row, wearing yellow shoes.
Mitch, with yellow hair, has Jessie and Amy next to him.
Kenny is immediately to the right of the clown with red shoes.
Alby has red hair.
The middle clown in the line-up facing the class has a yellow nose.
Both Jessie and Kenny are wearing yellow shoes.
The clown with yellow hair is standing in between clowns wearing yellow and red shoes.
A clown with a blue nose is next to a clown wearing a blue hat.
Neither Mitch nor Jessie are wearing anything green.
For each clown except Amy, their hat, hair, nose and shoes are different colours.
Kenny’s hat, Jessie’s hair, Mitch’s shoes and Alby’s nose are all the same colour.
The hats worn by the five clowns are all different colours. The same is also true for their hair and noses.
Alby’s hair is the same colour as Amy’s shoes, and Amy’s hair is the same colour as Alby’s shoes.
Everyone except Amy is wearing something that is orange.
[SOLUTION] (#litres_trial_promo)
Week 15 (#ulink_4a42fc09-392a-5dba-9a1b-c03273c9185c)
99. Folded shapes
A sheet of A4 paper (297 mm × 210 mm) is folded once and then laid flat on the table.
Which of these shapes could not be made?
[SOLUTION] (#litres_trial_promo)
100. Einstein’s clocks
Albert Einstein is experimenting with two unusual clocks that both have 24-hour displays. One clock goes at twice the normal speed. The other clock goes backwards, but at the normal speed. Both clocks show the correct time at 13:00.
What is the correct time when the displays on the clocks next agree?
[SOLUTION] (#litres_trial_promo)
101. The total area
The diagram shows three semicircles, each of radius 1.
What is the size of the total shaded area?
[SOLUTION] (#litres_trial_promo)
102. How many weeks?
How many weeks are there in 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 minutes?
[SOLUTION] (#litres_trial_promo)
103. A platinum question
Platinum is a very rare metal, even rarer than gold. Its density is 21.45 g/cm
. Assuming that the world production has been about 110 tonnes for each of the past 50 years, and negligible before that, which of the following has a comparable volume to that of the total amount of platinum ever produced?
(a) a shoe box;
(b) a cupboard;
(c) a house;
(d) Buckingham Palace;
(e) the Grand Canyon
[SOLUTION] (#litres_trial_promo)
104. Underlining numbers
Ten different numbers (not necessarily integers) are written down. Any number that is equal to the product of the other nine numbers is then underlined.
At most, how many numbers can be underlined?
[SOLUTION] (#litres_trial_promo)
105. Placing draughts
Barbara wants to place draughts on a board in such a way that the number of draughts in each row is equal to the number shown at the end of the row, and the number of draughts in each column is equal to the number shown at the bottom of the column. No more than one draught is to be placed in any cell.
In how many ways can this be done?
[SOLUTION] (#litres_trial_promo)
Week 16 (#ulink_d43b1c40-5cfb-56ba-9b3c-33f4cad8768d)
106. Square roots
How many of the numbers
are greater than 10?
[SOLUTION] (#litres_trial_promo)
107. How many lines?
The picture shows seven points and the connections between them.
What is the least number of connecting lines that could be added to the picture so that each of the seven points has the same number of connections with other points?
(Connecting lines are allowed to cross each other.)
[SOLUTION] (#litres_trial_promo)
108. Where in the list?
There are 120 different ways of arranging the letters U, K, M, I and C. All of these arrangements are listed in dictionary order, starting with CIKMU.
Which position in the list does UKIMC occupy?
[SOLUTION] (#litres_trial_promo)
109. Eva’s sport
Two sportsmen (Ben and Filip) and two sportswomen (Eva and Andrea) − a speed skater, a skier, a hockey player and a snowboarder − had dinner at a square table, with one person on each edge of the square.
The skier sat at Andrea’s left hand.
The speed skater sat opposite Ben.
Eva and Filip sat next to each other.
A woman sat at the hockey player’s left hand.
Which sport did Eva do?
[SOLUTION] (#litres_trial_promo)
110. Pedro’s numbers
Pedro writes down a list of six different positive integers, the largest of which is N. There is exactly one pair of these numbers for which the smaller number does not divide the larger.
What is the smallest possible value of N?
[SOLUTION] (#litres_trial_promo)
111. The speed of the train
A train travelling at constant speed takes five seconds to pass completely through a tunnel that is 85 m long, and eight seconds to pass completely through a second tunnel that is 160 m long.
What is the speed of the train?
[SOLUTION] (#litres_trial_promo)
112. What is ‘pqrst’?
The digits p, q, r, s and t are all different.
What is the smallest five-digit integer ‘pqrst’ that is divisible by 1, 2, 3, 4 and 5?
[SOLUTION] (#litres_trial_promo)
Crossnumber 4 (#ulink_65a487f9-458b-5223-977a-e327917da255)
ACROSS
2. A power of two (4)
5. A prime factor of 12345 (3)
6. Six more than a multiple of 13 ACROSS (3)
8. A cube (2)
10. The product of the digits of 25 ACROSS and also less than half of 23 ACROSS (2)
11. The mean of 4 DOWN, 8 ACROSS, 10 ACROSS, 13 ACROSS and 20 ACROSS and more than 3 DOWN (2)
13. A Fibonacci number (2)
14. A multiple of seven (3)
17. Eight less than a square (3)
19. Seven less than 26 DOWN (2)
20. A number that is greater than 3 DOWN and less than 27 DOWN (2)
22. An even number that is the sum of a square and a triangular number in two different ways (2)
23. A prime whose digits add up to five (2)
25. A square and a multiple of five (3)
28. A multiple of 14 that includes a two and an eight among its digits (3)
29. Nine more than a power of 20 ACROSS (4)
DOWN
1. One hundred and ninety-five less than a square (4)
2. One less than a Fibonacci number (3)
3. The highest common factor of 9 DOWN and 15 DOWN (2)
4. The sum of two powers of two (2)
6. (25 ACROSS) per cent of 24 DOWN (3)
7. The shortest side of a right-angled triangle whose longer sides are 24 DOWN and 25 ACROSS (3)
9. The square of a triangular number; also one less than a multiple of five (3)
12. A factor of 732, each of whose digits is a power of two (3)
15. Five multiplied by 3 DOWN (3)
16. An even square; also a multiple of 8 ACROSS (3)
17. A multiple of 17, the product of whose digits is a square multiplied by seven (3)
18. A multiple of nine (3)
21. A power of 21 (4)
24. A factor of 360 (3)
26. Seven more than 19 ACROSS (2)
27. A cube (2)
[SOLUTION] (#litres_trial_promo)
Week 17 (#ulink_390b9ecf-46c7-59fd-b7cc-f66a2cecdf58)
113. How many routes?
How many different routes are there from S to T that do not go through either of the points U and V more than once?
[SOLUTION] (#litres_trial_promo)
114. What Rachel drinks
A bottle contains 750 ml of mineral water. Rachel drinks 50% more than Ross, and these two friends finish the bottle between them.
How much does Rachel drink?
[SOLUTION] (#litres_trial_promo)
115. A magic product square
Place the numbers
in the squares of the grid, with one number in each square, so that the products of the numbers in the three rows, the three columns and the two diagonals are all equal to 1.
[SOLUTION] (#litres_trial_promo)
116. What is the angle?
The diagram shows a regular hexagon PQRSTU, a square PUWX and an equilateral triangle UVW.
What is the size of angle TVU?
[SOLUTION] (#litres_trial_promo)
117. A sum of numbers
Consider the list of all four-digit numbers that can be formed using only the digits 1, 2, 3 and 4, with no repetitions.
What is the sum of all the numbers in this list?
[SOLUTION] (#litres_trial_promo)
118. How many knights?
A group of 25 people consists of knights, serfs and damsels.
Each knight always tells the truth, each serf always lies, and each damsel alternates between telling the truth and lying.
When each of them was asked: ‘Are you a knight?’, 17 of them said ‘Yes’. When each of them was then asked: ‘Are you a damsel?’, 12 of them said ‘Yes’. When each of them was then asked: ‘Are you a serf?’, 8 of them said ‘Yes’.
How many knights are in the group?
[SOLUTION] (#litres_trial_promo)
119. Crossing the river
Two adults and two children wish to cross a river. They make a raft, but it will carry only the weight of one adult or two children.
What is the minimum number of times the raft must cross the river to get all four people to the other side?
(Note: The raft may not cross the river without at least one person on board.)
[SOLUTION] (#litres_trial_promo)
Week 18 (#ulink_52ffdfc3-4e3b-5380-97dc-fb065c1474c4)
120. Gluing cubes
A cube is made by gluing together a number of unit cubes face-to-face. The number of unit cubes that are glued to exactly four other unit cubes is 96.
How many unit cubes are glued to exactly five other unit cubes?
[SOLUTION] (#litres_trial_promo)
121. Mr Gallop’s ponies
Mr Gallop has two stables that each initially housed three ponies. His prize pony, Rein Beau, is worth £250 000. Rein Beau usually spends his day in the small stable, but when he wandered across into the large stable, Mr Gallop was surprised to find that the average value of the ponies in each stable rose by £10 000.
What is the total value of all six ponies?
[SOLUTION] (#litres_trial_promo)
122. Making a square
I have two types of square tile. One type has a side length of 1 cm and the other has a side length of 2 cm.
What is the smallest square that can be made with equal numbers of each type of tile?
[SOLUTION] (#litres_trial_promo)
123. How many pairs?
How many pairs of digits (p, q) are there so that the five-digit integer ‘p869q’ is a multiple of 15?
[SOLUTION] (#litres_trial_promo)
124. What is the product?
Lucy wants to put the numbers 2, 3, 4, 5, 6 and 10 into the circles so that the products of the three numbers along each edge are the same, and as large as possible.
In how many ways can this be done?
[SOLUTION] (#litres_trial_promo)
125. A five-team league
Five teams played in a competition and every team played once against each of the other four teams. Each team received three points for a match it won, one point for a match it drew and no points for a match it lost.
At the end of the competition the points were as follows:
How many of the matches resulted in a draw?
What were the results of the Greens’ matches against the other teams?
[SOLUTION] (#litres_trial_promo)
126. A mini crossnumber
The solution to each clue of this crossnumber is a two-digit number, not beginning with zero.
Across
1. A triangular number
3. A triangular number
Down
1. A square
2. A multiple of 5
In how many different ways can the crossnumber be completed correctly?
[SOLUTION] (#litres_trial_promo)
Shuttle Challenge 2 (#ulink_85f08238-e567-578f-bc49-24130ecd3c2e)
See Shuttle Challenge 1 (#ulink_8de49faa-6f29-52c5-b5c7-cc0f65138e16) for how the Shuttle works.
Question 1
What is the value of
?
Question 2
[A is the answer to Question 1.]
The number A is an example of a palindromic integer – one that is unchanged when the order of its digits is reversed.
How many palindromic integers are there from 300 to A inclusive?
Question 3
[A is the answer to Question 2.]
The diagram shows a triangle drawn on a square grid made up of nine smaller squares.
The area of the shaded triangle is A cm
.
What is the area, in cm
, of one of the smaller squares?
Question 4
[A is the answer to Question 3.]
Write the number A as a word in the gap shown in the following sentence.
Out of the first __________ letters in this sentence, what fraction is vowels?
Now answer the question.
[SOLUTION] (#litres_trial_promo)
Week 19 (#ulink_4f22030e-3560-5924-86ee-69853a96e52f)
127. Coins in a frame
The diagram shows 10 identical coins that fit exactly inside a wooden frame. As a result, each coin is prevented from sliding.
What is the largest number of coins that may be removed so that each remaining coin is still unable to slide?
[SOLUTION] (#litres_trial_promo)
128. Cheetah v. snail
In a sponsored ‘Animal Streak’, the cheetah ran at 90 kilometres per hour, while the snail slimed along at 20 hours per kilometre. The cheetah kept going for 18 seconds.
Roughly how long would the snail take to cover the same distance as the cheetah?
[SOLUTION] (#litres_trial_promo)
129. Memorable phone numbers
A new taxi firm needs a memorable phone number. They want a number that has a maximum of two different digits. Their phone number must start with the digit 3 and be six digits long.
How many such numbers are possible?
[SOLUTION] (#litres_trial_promo)
130. How many games?
Cleo played 40 games of chess and scored 25 points.
A win counts as one point, a draw counts as half a point, and a loss counts as zero points.
How many more games did she win than lose?
[SOLUTION] (#litres_trial_promo)
131. How many draughts?
Barbara wants to place draughts on a 4 × 4 board in such a way that the number of draughts in each row and in each column are all different. (She may place more than one draught in a square, and a square may be empty.)
What is the smallest number of draughts that she would need?
[SOLUTION] (#litres_trial_promo)
132. To and from Jena
In a certain region these are five towns: Freiburg, Göttingen, Hamburg, Ingolstadt and Jena.
One day, 40 trains each made a journey, leaving one of these towns and arriving at another.
10 trains travelled either to or from Freiburg.
10 trains travelled either to or from Göttingen.
10 trains travelled either to or from Hamburg.
10 trains travelled either to or from Ingolstadt.
How many trains travelled either to or from Jena?
[SOLUTION] (#litres_trial_promo)
133. Largest possible remainder
What is the largest possible remainder that is obtained when a two-digit number is divided by the sum of its digits?
[SOLUTION] (#litres_trial_promo)
Week 20 (#ulink_c45feba2-ca27-5f18-aed1-b89c3ac93dff)
134. nth termn
The first term of a sequence of positive integers is 6. The other terms in the sequence follow these rules:
if a term is even then divide it by 2 to obtain the next term;
if a term is odd then multiply it by 5 and subtract 1 to obtain the next term.
For which values of n is the nth term equal to n?
[SOLUTION] (#litres_trial_promo)
135. How many numbers?
Rafael writes down a five-digit number whose digits are all distinct, and whose first digit is equal to the sum of the other four digits.
How many five-digit numbers with this property are there?
[SOLUTION] (#litres_trial_promo)
136. A square area
A regular octagon is placed inside a square, as shown.
The shaded square connects the midpoints of four sides of the octagon.
What fraction of the outer square is shaded?
[SOLUTION] (#litres_trial_promo)
137. ODD plus ODD is EVEN
Find all possible solutions to the ‘word sum’ shown.
Each letter stands for one of the digits 0−9 and has the same meaning each time it occurs. Different letters stand for different digits. No number starts with a zero.
[SOLUTION] (#litres_trial_promo)
138. Only odd digits
How many three-digit multiples of 9 consist only of odd digits?
[SOLUTION] (#litres_trial_promo)
139. How many tests?
Before the last of a series of tests, Sam calculated that a mark of 17 would enable her to average 80 over the series, but that a mark of 92 would raise her average mark over the series to 85.
How many tests were in the series?
[SOLUTION] (#litres_trial_promo)
140. Three primes
Find all positive integers p such that p, p + 8 and p + 16 are all prime.
[SOLUTION] (#litres_trial_promo)
Crossnumber 5 (#ulink_6f4f13ad-e843-5f09-a109-9e98fe449e11)
ACROSS
1. The cube of a square (5)
4. Eight less than 5 DOWN (3)
6. One less than a multiple of seven (3)
7. A prime factor of 20 902 (4)
10. A number whose digits successively decrease by one (3)
12. Sixty per cent of 20 DOWN (3)
14. A multiple of seven (3)
15. A multiple of three whose digits have an even sum (3)
16. The square of a square (3)
17. A prime that is one less than a multiple of six (3)
19. Eleven more than a cube (4)
22. A number all of whose digits are the same (3)
24. A number that leaves a remainder of eleven when divided by thirteen (3)
25. The square of a prime; the sum of the digits of this square is ten (5)
DOWN
1. A number with an odd number of factors (3)
2. Four less than a triangular number (3)
3. The square root of 9 DOWN (2)
4. A factor of 12 ACROSS (3)
5. The longest side of a right-angled triangle whose shorter sides are 3 DOWN and 4 ACROSS (3)
8. A Fibonacci number (5)
9. The square of 3 DOWN (4)
11. Three more than an even cube (5)
13. A prime factor of 34567 (4)
17. The mean of 10 ACROSS, 16 ACROSS, 18 DOWN, 20 DOWN and 21 DOWN (3)
18. A power of eighteen (3)
20. Two less than 22 ACROSS (3)
21. A number whose digits are those of 12 ACROSS reversed (3)
23. A multiple of twenty-three (2)
[SOLUTION] (#litres_trial_promo)
Week 21 (#ulink_7a647005-8dd7-5e6f-be47-4578f5bcdf2e)
141. Joey’s and Zoë’s sums
Joey calculated the sum of the largest and smallest two-digit numbers that are multiples of three. Zoë calculated the sum of the largest and smallest two-digit numbers that are not multiples of three.
What is the difference between their answers?
[SOLUTION] (#litres_trial_promo)
142. When is the party?
Six friends are having dinner together in their local restaurant. The first eats there every day, the second eats there every other day, the third eats there every third day, the fourth eats there every fourth day, the fifth every fifth day and the sixth eats there every sixth day. They agree to have a party the next time they all eat together there. In how many days’ time is the party?
[SOLUTION] (#litres_trial_promo)
143. A multiple of 11
The eight-digit number ‘1234d678’ is a multiple of 11.
Which digit is d?
[SOLUTION] (#litres_trial_promo)
144. Two squares
ABCD is a square. P and Q are squares drawn in the triangles ADC and ABC, as shown.
What is the ratio of the area of the square P to the area of the square Q?
[SOLUTION] (#litres_trial_promo)
145. Proper divisors
Excluding 1 and 24 itself, the positive whole numbers that divide into 24 are 2, 3, 4, 6, 8 and 12. These six numbers are called the proper divisors of 24.
Suppose that you wanted to list in increasing order all those positive integers greater than 1 that are equal to the product of their proper divisors. Which would be the first six numbers in your list?
[SOLUTION] (#litres_trial_promo)
146. Kangaroo game
In the expression
the same letter stands for the same non-zero digit and different letters stand for different digits.
What is the smallest possible positive integer value of the expression?
[SOLUTION] (#litres_trial_promo)
147. A game with sweets
There are 20 sweets on the table. Two players take turns to eat as many sweets as they choose, but they must eat at least one, and never more than half of what remains. The loser is the player who has no valid move.
Is it possible for one of the two players to force the other to lose? If so, how?
[SOLUTION] (#litres_trial_promo)
Week 22 (#ulink_c7102e18-9563-54bc-9859-d92be81711b7)
148. A 1000-digit number
What is the largest number of digits that can be erased from the 1000-digit number 201820182018 … 2018 so that the sum of the remaining digits is 2018?
[SOLUTION] (#litres_trial_promo)
149. Gardeners at work
It takes four gardeners four hours to dig four circular flower beds, each of diameter four metres.
How long will it take six gardeners to dig six circular flower beds each of diameter six metres?
[SOLUTION] (#litres_trial_promo)
150. Overlapping squares
The diagram shows four overlapping squares that have sides of lengths 5 cm, 7 cm, 9 cm and 11 cm.
What is the difference between the total area shaded grey and the total hatched area?
[SOLUTION] (#litres_trial_promo)
151. What can T be?
Each of the numbers from 1 to 10 is to be placed in the circles so that the sum of each line of three numbers is equal to T. Four numbers have already been entered.
Find all the possible values of T.
[SOLUTION] (#litres_trial_promo)
152. Increases of 75%
Find all the two-digit numbers and three-digit numbers that are increased by 75% when their digits are reversed.
[SOLUTION] (#litres_trial_promo)
153. Three groups
For which values of the positive integer n is it possible to divide the first 3n positive integers into three groups each of which has the same sum?
[SOLUTION] (#litres_trial_promo)
154. A board game
Two players, X and Y, play a game on a board that consists of a narrow strip that is one square wide and n squares long. They take turns in placing counters that are one square wide and two squares long on unoccupied squares on the board.
The first player who cannot place a counter on the board loses. X always plays first, and both players always make the best available move.
Who wins the game in the cases where n = 2, 3, 4, 5, 6, 7 and 8?
[SOLUTION] (#litres_trial_promo)
Logic Challenge 3 (#ulink_3030ba14-43de-5b9b-a5a8-04e29f6c1d24)
Five teachers work in a school. By using the clues in the statements below, you need to work out what subject the teacher teaches and what sport they like, and information about their classroom, including the room name and number, and the colour of the classroom door.
Fill in the answer grid with information about each teacher.
Mr Smith teaches Art.
The Science teacher is in the classroom called ‘Square’.
History is taught by Miss Jones.
The favourite sport of Mr Henry does not involve a ball.
Mrs Talbot’s classroom door is coloured yellow and is next to the classroom that has the largest single-digit prime number as its number.
The Maths teacher has the largest classroom door number.
The favourite sport of the teacher in the classroom called ‘Triangle’ is netball.
Miss Jones’s door is coloured orange.
The teacher in the classroom called ‘Circle’ is next to the Maths teacher.
English is taught in the classroom called ‘Sphere’.
The classroom door numbered 3 is next to the teacher who is next to the teacher whose favourite sport is jogging.
Football is the favourite sport of the teacher in the classroom called ‘Cylinder’.
The teacher in the classroom called ‘Square’ is next to the classrooms with the green and orange doors.
The classroom called ‘Cylinder’ is next to the classroom of the Science teacher.
English is taught in the classroom with the ‘unlucky’ prime number.
The classroom door of the Maths teacher is green.
The classroom numbers of Mrs Talbot and Mrs Richard add up to 30.
All the classroom door numbers are prime numbers.
The English teacher is next to a classroom that is next to the classroom of the Science teacher.
The teacher in the classroom with the red door likes rugby and is next to the Maths teacher.
The sum of all the classroom door numbers is 51.
The favourite sport of the teacher in the classroom named ‘Sphere’ is cricket.
The classroom called ‘Triangle’ is next door to the classrooms of the Art and Science teachers.
One of the classrooms has a white door.
[SOLUTION] (#litres_trial_promo)
Week 23 (#ulink_924edb11-6ead-5fdc-98e1-dccaa447f978)
155. Angles around a triangle
What is the value of a + b + c + d + e + f?
[SOLUTION] (#litres_trial_promo)
156. Einstein sees two clocks
Albert Einstein was standing on the station platform thinking about relativity when he noticed he could see two station clocks. Each clock was digital, showing only hours and minutes. He observed that the display on one clock changed to the next minute 10 seconds before the correct time, whereas the display on the other clock changed to the next minute 10 seconds after the correct time.
For what fraction of the time did both clocks show the same time?
[SOLUTION] (#litres_trial_promo)
157. Halving an annulus
The shaded region in the diagram, bounded by two concentric circles, is called an annulus.
The circles have radii 2 cm and 14 cm. The dashed circle divides the area of this annulus into two equal areas.
What is its radius?
[SOLUTION] (#litres_trial_promo)
158. How many pairs?
How many pairs of numbers (a, b) exist such that the sum a + b, the product ab and the quotient
of these two numbers are all equal?
[SOLUTION] (#litres_trial_promo)
159. Two ages
Abi and Becky were comparing their ages and found that Becky is as old as Abi was when Becky was as old as Abi had been when Becky was half as old as Abi is. The sum of their present ages is 44.
How old is Abi?
[SOLUTION] (#litres_trial_promo)
160. At McBride Academy
At McBride Academy there are 300 children, each of whom represents the school in both summer and winter sports. In summer, 60% of these play tennis and the other 40% play badminton. In winter, they play hockey or swim, but not both. 56% of the hockey players play tennis in the summer and 30% of the tennis players swim in the winter.
How many both swim and play badminton?
[SOLUTION] (#litres_trial_promo)
161. Maths, maths, Cayley
How many different solutions are there to this word sum, where each letter stands for a different non-zero digit?
[SOLUTION] (#litres_trial_promo)
Week 24 (#ulink_2ab90312-70ad-589c-966c-621f3c723d72)
162. An angle in a square
The diagram shows a square ABCD and an equilateral triangle ABE.
The point F lies on BC so that EC = EF.
Calculate the angle BEF.
[SOLUTION] (#litres_trial_promo)
163. Areas in a quarter circle
The diagram shows a quarter circle with centre O and two semicircular arcs with diameters OA and OB.
Calculate the ratio of the area of the region shaded grey to the area of the region shaded black.
[SOLUTION] (#litres_trial_promo)
164. How many extensions?
In a large office, each person has their own telephone extension consisting of three digits, but not all possible extensions are in use. To try to prevent wrong numbers, no used number can be converted to another just by swapping two of its digits.
What is the largest possible number of extensions in use in the office?
[SOLUTION] (#litres_trial_promo)
165. The top ball
Six pool balls numbered 1 to 6 are to be arranged in a triangle, as shown.
After three balls are placed in the bottom row, each of the remaining balls is placed so that its number is the difference of the two below it.
Which balls can land up at the top of the triangle?
[SOLUTION] (#litres_trial_promo)
166. Two squares
A square has four digits. When each digit is increased by 1, another square is formed.
What are the two squares?
[SOLUTION] (#litres_trial_promo)
167. Four vehicles
Four vehicles travelled along a road with constant speeds. The car overtook the scooter at 12:00 noon, then met the bike at 14:00 and the motorcycle at 16:00. The motorcycle met the scooter at 17:00 and overtook the bike at 18:00.
At what time did the bike and the scooter meet?
[SOLUTION] (#litres_trial_promo)
168. A marching band
A marching band is having difficulty lining up for a parade. When they line up in rows of 3, one person is left over. When they line up in rows of 4, two people are left over. When they line up in rows of 5, three people are left over. When they line up in rows of 6, four people are left over.
However, the band is able to line up in rows of 7 with nobody left over. What is the smallest possible number of marchers in the band?
[SOLUTION] (#litres_trial_promo)
Crossnumber 6 (#ulink_c1f643ff-f08c-5e86-908b-47a8cc0ae2ea)
ACROSS
1. A prime factor of 8765 (4)
4. The interior angle, in degrees, of a regular polygon; its digits have a product of 12 (3)
6. A Fibonacci number whose digits add up to twenty-four (3)
7. A prime that is one greater than a square (2)
8. Ninety-nine greater than the number formed by reversing the order of its digits (3)
10. A multiple of 25 ACROSS (3)
12. 9 DOWN multiplied by seven (3)
15. A number with seven factors (2)
17. The third side of a right-angled triangle with hypotenuse 9 DOWN and other side 25 ACROSS (2)
19. (10 ACROSS) per cent of 8 ACROSS (3)
21. An odd multiple of nine (3)
23. The product of the seventh prime and the eleventh prime (3)
25. An even number (2)
26. The lowest common multiple of 25 ACROSS and 11 DOWN (3)
28. A multiple of eleven, and also the mean of 12 ACROSS, 16 DOWN, 19 ACROSS, 24 DOWN and 27 DOWN (3)
29. A power of nineteen (4)
DOWN
2. An even square (3)
3. One third of 21 ACROSS (2)
4. The interior angle, in degrees, of the regular polygon that has twice as many sides as the regular polygon whose interior angle, in degrees, is 4 ACROSS (3)
5. One less than a multiple of eleven (3)
8. A power of nine (4)
9. The mean of 3 DOWN, 7 ACROSS, 13 DOWN, 17 ACROSS and 27 DOWN (2)
11. Ninety-one less than 10 ACROSS (2)
13. The sum of the squares of the digits of 23 ACROSS (2)
14. A factor of 4567 (4)
16. The difference between 11 DOWN and 3 DOWN (2)
18. The square root of 8 DOWN (2)
20. The highest common factor of 10 ACROSS and 24 DOWN (2)
21. The total number of days in a year in the months whose names do not contain the letter A (3)
22. A prime that is ten less than a cube (3)
24. A square multiplied by five; the product of its digits is 40 (3)
27. A triangular number that is the sum of two prime numbers that differ by eight (2)
[SOLUTION] (#litres_trial_promo)
Week 25 (#ulink_e0e86d87-357d-528d-ad4b-82ed7f1ff11a)
169. Non-factors of 720
Sam starts to list in ascending order every positive integer that is not a factor of 720.
Which is the tenth number in her list?
[SOLUTION] (#litres_trial_promo)
170. What size is JMO?
In the diagram, JK and ML are parallel. JK = KO = OJ = OM and LM = LO = LK.
Find the size of the angle JMO.
[SOLUTION] (#litres_trial_promo)
171. Amrita’s numbers
Amrita has written down four whole numbers. If she chooses three of her numbers at a time and adds up each triple, she obtains totals of 115, 153, 169 and 181.
What is the largest of Amrita’s numbers?
[SOLUTION] (#litres_trial_promo)
172. Where do the children come from?
Five children, boys Vince, Will and Zac, and girls Xenia and Yvonne, sit at a round table. They come from five different cities: Aberdeen, Belfast, Cardiff, Durham and Edinburgh. The child from Aberdeen sits between Zac and the child from Edinburgh. Neither of the two girls is sitting next to Will. Vince sits between Yvonne and the child from Durham. Zac writes to the child from Cardiff.
Where does each child come from?
[SOLUTION] (#litres_trial_promo)
173. Find four integers
Find four integers whose sum is 400 and such that the first integer is equal to twice the second integer, three times the third integer and four times the fourth integer.
[SOLUTION] (#litres_trial_promo)
174. Bradley’s Bicycle Bazaar
In the window of Bradley’s Bicycle Bazaar there are some unicycles, some bicycles and some tricycles. Laura sees that there are seven saddles in total, thirteen wheels in total and more bicycles than tricycles.
How many unicycles are in the window?
[SOLUTION] (#litres_trial_promo)
175. Liz and Mary’s problems
Liz and Mary compete in solving problems. Each of them is given the same list of 100 problems. For any problem, the first of them to solve it gets 4 points, while the second to solve it gets 1 point. Liz solved 60 problems, and Mary also solved 60 problems. Together, they got 312 points.
How many problems were solved by both of them?
[SOLUTION] (#litres_trial_promo)
Week 26 (#ulink_4ca51d59-2d51-5fec-b069-f57ff887c55b)
176. Edinburgh to London
Travelling by train from Edinburgh to London, I passed a sign saying ‘London 150 miles’. After seven more miles, I passed another sign saying ‘Edinburgh 250 miles’.
How far is it by train from Edinburgh to London?
[SOLUTION] (#litres_trial_promo)
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Литагент HarperCollins
Тип: электронная книга
Жанр: Развлечения
Язык: на английском языке
Издательство: HarperCollins
Дата публикации: 16.04.2024
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О книге: ’Be warned: cracking puzzles releases a very addictive drug.’ – Marcus du SautoyHave you ever wanted to be a puzzle pro or logical luminary? Well, look no further!The perfect way to liven up your day, The Ultimate Mathematical Challenge has over 365 puzzles to test your wits and excite your mind. From starter puzzles to perplexing Olympiad problems designed to stretch even the strongest mathematicians, this book is the ideal forum to get your brain into gear and feed it with the challenges it craves.Specially curated from the UK Mathematics Trust’s catalogue of puzzles, most of these problems can be tackled using no more than a little numerical knowledge, logical thinking and native wit. Including interludes of crossnumber conundrums and shuttle challenges, space for your working out and a handy glossary for those obscure mathematical terms, this book has everything you need to solve captivating problems all year round.Do you have what it takes to conquer The Ultimate Mathematical Challenge?