Secondary Mathematical Challenges
Welcome to the 2024-2025 Scottish Secondary Mathematical Challenges. This package contains
This Welcome Page (including Section Information)
The Secondary Guidelines
Round 1 Questions
Information About Payments
Data Protection and Privacy Policy
How to Enter your School and Pupil Names on the Marks Website
The Annual Poster
Archives and Book Order Form
Once again, the name of the Section Organiser is not on the question paper. Their details are on the
website but are repeated here for convenience:
Section 1
Aberdeen City; Aberdeenshire; Highland; Moray; Orkney Islands; Shetland Islands; Western Isles
Dr Richard Hepworth (r.hepworth@abdn.ac.uk)
Mathematical Challenge
Department of Mathematical Sciences, University of Aberdeen,
Aberdeen AB24 3UE
Section 2
Angus; Clackmannanshire; Dundee City; Falkirk; Fife; Perth & Kinross; Stirling
Dr Jean Reinaud (jnr1@st-andrews.ac.uk)
Mathematical Institute, University of St Andrews,
St Andrews, Fife KY16 9SS
Section 3
East Lothian; Edinburgh City; Midlothian; Scottish Borders; West Lothian
Andrew Gallacher (A.Gallacher@napier.ac.uk)
Head of Teacher Education, Edinburgh Napier University, School of Applied Sciences,
Room 2.B.37, Sighthill Court, Edinburgh EH11 4BN
Section 4
Argyll & Bute; Dumfries & Galloway; East Ayrshire; East Dunbartonshire;
East Renfrewshire; Glasgow City; Inverclyde; North Ayrshire; North Lanarkshire;
Renfrewshire; South Ayrshire; South Lanarkshire; West Dunbartonshire
Scottish Mathematical Challenge Organiser (wpr3145@gmail.com),
Department of Mathematics and Statistics, University of Strathclyde,
26 Richmond Street, Glasgow G1 1XH
The competition timetable for 2024-2025 is as follows:
Last date for receipt Last date for receipt
Set No. of questions by schools of solutions from pupils
I Friday 23 August 2024 Friday 1 November 2024
II Friday 22 November 2024 Friday 21 February 2025
If there are organisational difficulties you may contact me, Bill Richardson, (wpr3145@gmail.com).
The Scottish Mathematical Council
MC homepage: www.scot-maths.co.uk/
MATHEMATICAL CHALLENGE 2024−2025
A national problem solving competition for schools in Scotland
SECONDARY DIVISIONS
GUIDELINES FOR TEACHERS
1. Mathematical Challenge is a problem-solving competition which goes back to 1976. The Challenge
is open to all students educated in Scotland. Its aim is to promote mathematics as a source of interest
and pleasurable achievement through challenging problems which require only elementary
techniques and simple logic.
Please ensure that all teachers involved in the competition see these Guidelines.
How Mathematical Challenge operates
2. There are four divisions: JUNIOR for Sl and S2, MIDDLE for S3 and S4, SENIOR for S5 and S6,
and PRIMARY (for which a separate circular is available).
Pupils may enter only one division and must specify that division on their first entry.
Please contact your local organiser, whose name and address are on the proforma on page 2 of each
problem sheet and in the Contacts section of the Web pages, if there is any doubt about divisions, or
if further information is required.
3. There are no written examinations. For the Junior, Middle and Senior Divisions, two sets of five
problems each will be available for schools to download according to a timetable outlined in §13 below.
Problems for different divisions will be on separate sheets. Some problems may be common to different
divisions. The problems will also be available from the Mathematical Challenge Web pages (see above
for address).
4. A registration fee is required from participating schools. For a secondary school the fee is £20
for the first 10 entrants and £15 for each subsequent batch of 10 entrants or part thereof. A fee
form is included with the downloadable pack of materials.
For individual participants NOT entering through a school, the fee is £8.
Entries and Marking
5. Entries must be the unaided efforts of individual pupils. Group working is not appropriate in
Mathematical Challenge. Participants may consult books or the internet for information on facts or
on how to tackle problems. Whilst teachers or parents may give guidance on interpretation of
wording, they should not be involved in the solution of a problem. Furthermore, the work should
not interfere with normal teaching and in no circumstances should it be a class assignment.
6. All Sections must use the software package to assist in the processing of the results. A Record of
Entries must be made electronically by the school, or it will not be possible to process the results.
• Go to the marks website: https://www.scottishmathschallenge.org.uk/
• Choose “School Login” and enter your login details or “Register here” to set up a new account.
• When you have logged in, go to "Add/Edit Entrants" _ enter the names and school year of
each entrant.
• The marks will eventually appear on the “Marks page”.
• Messages from the organiser may also appear there on the first page from time to time.
Use a paper copy of the ‘Printable version of details and entrants’ from the marks website as a cover
sheet for the school's entries. This contains the school details and the alphabetical list of entrants in
each section, as entered on the website. All entries submitted will be marked even if earlier problem
sets are missed.
7. Entries will not be returned. Entrants should keep a copy of their solutions. The Scottish
Mathematical Council reserves the right to publish good solutions in its Journal.
8. Participants should explain their solutions as fully as they can. Marks will be given for
explanations of answers not just for the answers themselves. We should be most grateful if
teachers would stress this point. Incomplete or incorrect answers may gain some credit.
In outline, the marking scheme for each problem is as follows:
4 : a completely correct solution, with full explanation.
3 : a solution, with explanation, which is correct apart from a minor slip or
omission of a special case.
2 : a solution with explanation which contains a serious error or omission, but
which nevertheless involves good ideas.
1 : there is an indication of an interesting idea or method, but not necessarily one
which could lead to a correct solution.
A bonus mark may be given for a completely correct solution, with full explanation, which contains
additional good ideas, such as a successful generalisation of the problem.
A solution in which an answer is given without any explanation will normally be awarded no
marks, even if the answer is correct. However, correct working may be accepted as providing an
explanation, so long as the various steps are clear.
9. No problems set in Mathematical Challenge require the use of a computer package (e.g. a
spreadsheet) to obtain a correct solution. If computer software is used, then a proper mathematical
explanation of its use is essential.
Awards
10. There are three classes of award: Gold, Silver and Bronze. Award winners will be selected primarily
on the basis of the total number of marks obtained over both sets of Problems. Special circumstances
for individual entrants may be taken into account.
11. All award winners will qualify for certificates. Where an award ceremony can be arranged, the most
successful entrants will be invited to attend to receive their certificates and Mathematical Challenge
mugs. Certificates not presented at a ceremony will be sent by post.
Important notes
12. Large numbers of entries can impose a considerable strain on markers and on organisers. Local
organisers may have to set limits on the total numbers of entries per school. Schools submitting large
numbers of entries may be asked to provide additional markers. Any such markers would not mark
entries from their own schools.
13. The timetable for 2024-2025 is as follows:
Last date for receipt Last date for receipt
Set No. of questions by schools of solutions from pupils
I Friday 23 August 2024 Friday 1 November 2024
II Friday 22 November 2024 Friday 21 February 2025
14. The problems of earlier sessions form an excellent resource. Those for the years 1991-92 to 2005-
2006, including solutions, are available in the books Mathematical Challenges III, Mathematical
Challenges IV, Mathematical Challenges V and Mathematical Challenges VI which are published by
The Scottish Mathematical Council. Copies can be obtained from Bill Richardson, Kintail,
Longmorn, Elgin IV30 8RJ, prices £7.50, £8, £8, £8 respectively.
In addition, it seems unlikely that any further books will be printed so questions and solutions for
2006-2021 can be accessed at: www.wpr3.co.uk/MC-archive/
Comments on the usefulness of these to wpr3145@gmail.com would be welcome.
15. For other information, please contact your local organiser, whose name and address are given in the
Contacts section of the Mathematical Challenge Web pages
www.scot-maths.co.uk
as well as on the materials download menu page
www.wpr3.co.uk/MC/materials
The Scottish Mathematical Council
www.scot-maths.co.uk
MATHEMATICAL CHALLENGE 2024−2025
Entries must be the unaided efforts of individual pupils.
Solutions must include explanations and answers without explanation will be given no credit.
Do not feel that you must hand in answers to all the questions.
CURRENT AND RECENT SPONSORS OF MATHEMATICAL CHALLENGE ARE
The Edinburgh Mathematical Society, The Maxwell Foundation,
The London Mathematical Society and The Scottish International Education Trust.
The Scottish Mathematical Council is indebted to the above for their generous support and gratefully acknowledges financial and
other assistance from schools, universities and education authorities.
Particular thanks are due to the Universities of Aberdeen, Edinburgh Napier, Moray House, St Andrews, Stirling,
Strathclyde and to George Heriot's School, Gryffe High School and Kelvinside Academy.
Junior Division: Problems 1
J1. Over coffee the other day my friend MacAngus and I were mulling over the passing of time. ‘Four years
ago’, MacAngus said, ‘I was four times as old as young Callum and now I'm only three times as old.’
‘If you go on like that’ I said, ‘you’ll soon both be the same age.’
When we finished laughing over this fallacy, it occurred to me that from what MacAngus said it
would be possible to figure out his age. How old is my friend MacAngus?
J2. Four explorers wish to get one of their number as far as possible into the wilderness from their base.
Each explorer can carry supplies for up to 10 days. At any time supplies can be transferred between
explorers and individual explorers can return to base, provided they have sufficient supplies for the
return journey. Supplies cannot be left unattended in the wilderness.
What is the greatest number of days the lead explorer can travel from the base so that all return safely?
J3. An integer n, between 100 and 999 inclusive, is chosen at random. What is the probability that the
sum of the digits of n is 23?
J4. A van delivering maths textbooks travels to a school at an average speed of 60mph. Due to a
mechanical problem the van could only do the return journey at an average speed of 40mph.
What was the van's average speed over the entire journey?
J5. R
A B P Q C D
In the diagram, points B, P, Q and C lie on line segment AD. The semi-circle with diameter AC has
centre P and the semi-circle with diameter BD has centre Q. The two semi-circles intersect at R. If
∠PRQ = 30 degrees, determine the size of ∠ARD.
END OF PROBLEM SET 1
CLOSING DATE FOR RECEIPT OF SOLUTIONS :
1 November 2024
Look out for Problems 2 in late November!
SEE OVER FOR LINKS TO THE MATHS CHALLENGES ARCHIVES
Mathematical Challenge Problems 1
JUNIOR DIVISION 2024-2025
SMC PLEASE USE CAPITALS TO COMPLETE
FOR OFFICIAL USE
SURNAME
Marker
OTHER NAME(S)
(underline the one
you prefer) Marks
SCHOOL 1 2 3 4 5
Total
AGE YEAR OF STUDY S
CUT ALONG HERE
Please write your solutions on A4 paper and staple the above form to them.
PLEASE WRITE YOUR NAME ON EVERY PAGE.
Send your entry through your school to the section organiser.
For further information on the competition, please see the School Materials which have been distributed to
schools. A copy of these Materials can be obtained from
http://www.wpr3.co.uk/MC/materials/index.html
There are separate links for primary and secondary schools. This page also includes a list of authorities in
each section and names and addresses of section organisers.
For information about Mathematical Challenge, look on the MC web site:
www.scot-maths.co.uk
MATHS CHALLENGES ARCHIVES
There are archives of previous questions on: www.wpr3.co.uk/MC-archive/J/index-J.html
Here is a shortcut for your smartphone or tablet
The Scottish Mathematical Council
www.scot-maths.co.uk
MATHEMATICAL CHALLENGE 2024−2025
Entries must be the unaided efforts of individual pupils.
Solutions must include explanations and answers without explanation will be given no credit.
Do not feel that you must hand in answers to all the questions.
CURRENT AND RECENT SPONSORS OF MATHEMATICAL CHALLENGE ARE
The Edinburgh Mathematical Society, The Maxwell Foundation,
The London Mathematical Society and The Scottish International Education Trust.
The Scottish Mathematical Council is indebted to the above for their generous support and gratefully acknowledges financial and
other assistance from schools, universities and education authorities.
Particular thanks are due to the Universities of Aberdeen, Edinburgh Napier, Moray House, St Andrews, Stirling,
Strathclyde and to George Heriot's School, Gryffe High School and Kelvinside Academy.
Middle Division: Problems 1
M1. A van delivering maths textbooks travels to a school at an average speed of 60mph. Due to a
mechanical problem the van could only do the return journey at an average speed of 40mph.
What was the van's average speed over the entire journey?
M2. R
A B P Q C D
In the diagram, points B, P, Q and C lie on line segment AD. The semi-circle with diameter AC has
centre P and the semi-circle with diameter BD has centre Q. The two semi-circles intersect at R. If
∠PRQ = 30 degrees, determine the size of ∠ARD.
M3. A cross country track is in the form of a capital letter L.
The short side, which comes at the beginning of the race, is 2 miles long. When the runners have
gone 1½ miles along the other side they are as far from the starting point (as the crow flies) as they
are from the winning post.
How far (as the crow flies) is the distance from the starting point to the winning post?
M4. In the addition sum below, only one out of the five decimal points is in the correct position.
47.5
38.627
125.4
1583.3
4508.57
Find all the possible ways to alter the four incorrect decimal places and make the sum add up
correctly.
SEE OVER FOR QUESTION M5.
Mathematical Challenge Problems 1
MIDDLE DIVISION 2024-2025
SMC PLEASE USE CAPITALS TO COMPLETE
SURNAME FOR OFFICIAL USE
Marker
OTHER NAME(S)
(underline the one
you prefer) Marks
SCHOOL 1 2 3 4 5
Total
AGE YEAR OF STUDY S
CUT ALONG HERE
Please write your solutions on A4 paper and staple the above form to them.
PLEASE WRITE YOUR NAME ON EVERY PAGE.
Send your entry through your school to the section organiser.
For further information on the competition, please see the School Materials which have been distributed to
schools. A copy of these Materials can be obtained from
http://www.wpr3.co.uk/MC/materials/index.html
There are separate links for primary and secondary schools. This page also includes a list of authorities in
each section and names and addresses of section organisers.
M5. When Oliver walks briskly down a downward moving escalator he takes 60 steps of the escalator to
reach the bottom. When Oliver walks slowly down the escalator at half his previous speed he takes
42 steps of the escalator to reach the bottom. Assuming constant speeds for walking briskly, walking
slowly and the movement of the escalator, find how many steps the escalator shows when it is
stationary.
END OF PROBLEM SET 1
CLOSING DATE FOR RECEIPT OF SOLUTIONS :
1 November 2024 Look out for Problems 2 in late November!
For information about Mathematical Challenge, look on the MC web site:
www.scot-maths.co.uk
LINKS TO THE MATHS CHALLENGES ARCHIVES
There are archives of previous questions on: www.wpr3.co.uk/MC-archive/M/index-M.html
Here is a shortcut for your smartphone or tablet
The Scottish Mathematical Council
www.scot-maths.co.uk
MATHEMATICAL CHALLENGE 2024−2025
Entries must be the unaided efforts of individual pupils.
Solutions must include explanations and answers without explanation will be given no credit.
Do not feel that you must hand in answers to all the questions.
CURRENT AND RECENT SPONSORS OF MATHEMATICAL CHALLENGE ARE
The Edinburgh Mathematical Society, The Maxwell Foundation,
The London Mathematical Society and The Scottish International Education Trust.
The Scottish Mathematical Council is indebted to the above for their generous support and gratefully acknowledges financial and
other assistance from schools, universities and education authorities.
Particular thanks are due to the Universities of Aberdeen, Edinburgh Napier, Moray House, St Andrews, Stirling,
Strathclyde and to George Heriot's School, Gryffe High School and Kelvinside Academy.
Senior Division: Problems 1
S1. In the addition sum below, only one out of the five decimal points is in the correct position.
47.5
38.627
125.4
1583.3
4508.57
Find all the possible ways to alter the four incorrect decimal places and make the sum add up correctly.
S2. When Oliver walks briskly down a downward moving escalator he takes 60 steps of the escalator to
reach the bottom. When Oliver walks slowly down the escalator at half his previous speed he takes
42 steps of the escalator to reach the bottom. Assuming constant speeds for walking briskly, walking
slowly and the movement of the escalator, find how many steps the escalator shows when it is
stationary.
S3. Four cards with integers on are placed face down on a table. Five people in succession each take two
cards and state the total value of the two cards, but do not reveal the individual values. These totals are
8, 13, 14, 17 and 11.
Determine the numbers on each of the four individual cards.
S4. Cyclic quadrilateral ABCD has AB = AD = 1, CD = cos ∠ABC and cos ∠BAD = − 13 .
Prove that BC is a diameter of the circumscribed circle.
S5. Choose 6 different non-zero digits.
(a) How many different 6 digit numbers can be formed from these 6 digits?
(b) Find the largest prime factor of the sum of all of these different 6 digit numbers.
END OF PROBLEM SET 1
CLOSING DATE FOR RECEIPT OF SOLUTIONS :
1 November 2024 Look out for Problems 2 in late November!
SEE OVER FOR LINKS TO THE MATHS CHALLENGES ARCHIVES
Mathematical Challenge Problems 1
SENIOR DIVISION 2024-2025
SMC PLEASE USE CAPITALS TO COMPLETE
SURNAME FOR OFFICIAL USE
Marker
OTHER NAME(S)
(underline the one
you prefer) Marks
SCHOOL 1 2 3 4 5
Total
AGE YEAR OF STUDY S
CUT ALONG HERE
Please write your solutions on A4 paper and staple the above form to them.
PLEASE WRITE YOUR NAME ON EVERY PAGE.
Send your entry through your school to the section organiser.
For further information on the competition, please see the School Materials which have been distributed to
schools. A copy of these Materials can be obtained from
http://www.wpr3.co.uk/MC/materials/index.html
There are separate links for primary and secondary schools. This page also includes a list of authorities in
each section and names and addresses of section organisers.
For information about Mathematical Challenge, look on the MC web site:
www.scot-maths.co.uk
MATHS CHALLENGES ARCHIVES
There are archives of previous questions on: www.wpr3.co.uk/MC-archive/S/index-S.html
Here is a shortcut for your smartphone or tablet
The Scottish Mathematical Council
MATHEMATICAL CHALLENGE (SECONDARY)
Registration Fee 2024−2025
Participating schools are required to pay a registration fee. The fee covers both sets of
problems. The secondary school fee is £20 for 1-10 entrants in Round 1, £15 for the next 10 (or
part thereof) and so on. A cheque made payable to
SMC Mathematical Challenge
should be attached to this form (but see below) and
returned with the first set of problems.
School details Name .................................
Telephone .................................
Address .................................
.................................
.................................
.................................
Education Authority
(where appropriate) .................................
Contact teacher Name .................................
email address .................................
(if used)
Payments: we prefer to be paid by cheque but if this is not possible we will accept payment by
BACS. Our account is:
SMC MATHEMATICAL CHALLENGE
The details are: 83-06-08 1054 8147.
The reference you provide must include the school postcode followed by the school name.
Data Protection
Our policy is attached. By completing and returning this form we will assume that you have
read it and accept it.
Submission of scripts
You must submit on time and use correct postage. As a guide, first class large letters rates are:
Weight below 100g 250g 500g 750g
Postage £2.10 £2.90 £3.50 £3.50
Data Protection and Privacy Policy for the SMC Maths Challenge
In the following, the data referred to is stored on the computer used for entering pupil and
school details and for recording pupil performance. Access to all such information is password
protected.
For each entrant the data stored is:
• surname, other name, school, year of study, and (after marking) the marks for each
question. Permission to record this data is given by completing the tear-off form
attached to the solutions.
For each school the data stored is:
• school name, address, phone, Primary or Secondary, LEA (Local Education Authority
area), whether independent, contact teacher name and e-mail, and other information
you want to add (if any).
Teachers have access, by logging in with their username and password, only to the data for
their own school and its entrants, and are responsible for communicating results and awards to
their entrants.
Section Organisers have access, by logging in with their username and password, only to data
about schools and entrants within their own Section.
Designated members of the Maths Challenge National Committee have access, by logging in
with their username and password, to all the data in the database to facilitate running the
challenge.
No data will be kept for more than 4 years. This would allow for production of comparative
reports over 3 years for the Scottish Mathematical Council Journal.
The data is stored on a secure server.
Contact: scottishmathschallenge@gmail.com
Last update: 16/8/24
Using the marks website
A Record of Entrants must be made on the marks website by the school or it will not be possible for
the section organiser to later add the marks for each entrant.
Go to the website:
https://www.scottishmathschallenge.org.uk/
If you have used the website for your school in a previous year and can
remember the username and password log in as usual.
Otherwise choose
"Register here"
which enables you to enter your details and create a password. Your
username will be your email address so please make sure that you enter
this correctly. The system will send you an email with an activation link:
click on this link to verify your email address and then log in.
You will need your username and password each time you log in.
Once logged in click on the link
“Edit School Details"
Check that all the contact details are correct, making changes as
necessary. In particular, make sure that you select primary or secondary
and the relevant local authority whenever you make changes.
Later, when you have a pile of entries, or a list of entrants, log in again
and go to
"Add/Edit Entrants"
and enter
• the division (only for secondary pupils),
• the names, and
• the school year of each entrant.
When all your entrants have been added, to get a printout to send with your entries go to
"Printable version of details and entrants"
When the marks for each round are released, to view the marks for your school go to
"Marks page"
On the main page after you log in you may, from time to time, see messages from the section
organisers.
NOTE: If you experience difficulties with the website please contact either
your section organiser or
Helen Martin h.martin@abdn.ac.uk
The Scottish Mathematical Council
www.scot-maths.co.uk
MATHEMATICAL CHALLENGE
2024−2025
Four explorers wish to get one of their number as far
as possible into the wilderness from their base. Each
explorer can carry supplies for up to 10 days. At any
time supplies can be transferred between explorers
and individual explorers can return to base, provided
they have sufficient supplies for the return journey.
Supplies cannot be left unattended in the wilderness.
What is the greatest number of days the lead explorer
can travel from the base so that all return safely?
Ask your teacher for details of the new competition problems.
The Scottish Mathematical Council
www.scot-maths.co.uk
MATHEMATICAL CHALLENGE
2023−2024
Did you solve it?
One of the highlights of the local village social life is the stage
production organised by the Amateur Youth Players and the
rehearsals are in full swing for The Gondoliers. When I called the
treasurer the other day he was estimating the costs. The first
scene, in case you have forgotten, shows 24 maidens of Venice
making up small bunches of red and white roses. He had
intended that each girl would have three red and two white roses
until he realised that the red roses cost twice as much each as
the white ones. He decided to give half the girls three red and two
white roses each and the remainder two red and three white
roses. He had cut the cost by £3. How much is a red rose?
Solution
Let the cost of a red rose be £R and of white rose be £W .
12 girls had three red roses and two white roses so the cost of
these is £(36R + 24W).
The other 12 girls had two red roses and three white roses so the
cost of these is £(24R + 36W).
Originally the cost would have been £(72R + 48W) and the new
cost is £(60R + 60W).
So the saving is (72R + 48W) − (60R + 60W) = 12R − 12W = 3.
1
But W = 2R so
12R − 12 ( 21R) = 3
6R = 3.
Hence the cost of a red rose is 50 pence.
Archives and Books
The books listed below and there are still limited stocks. However, since 2006, an online
archive has been developed. It covers all Challenges, with sheets with questions and separate
sheets which include solutions. These are all available at
www.wpr3.co.uk/MC-archive/
The books which were published are as follows:
Price Number Cost
Mathematical Challenges VI 2003 to 2006 £8.00
Mathematical Challenges V 2000 to 2003 £8.00
Mathematical Challenges IV 1997 to 2000 £8.00
Mathematical Challenges III 1994 to 1997 £7.50
Primary Mathematical Challenges up to 2001 £5.00
Total
Copies of any of these can be obtained from
Bill Richardson, Kintail, Longmorn, Elgin IV30 8RJ
Cheques with order are preferred.
Cheques should be made payable to 'SMC Mathematical Challenge'.
A few copies of Mathematical Challenges II and also the original
Blackie Mathematical Challenges are still available.
email enquiries:- wpr3145@gmail.com
