The Cayley Contest is the Grade 10 paper among the Waterloo math contests, written each February in schools across Canada and abroad. It sits between Pascal in Grade 9 and Fermat in Grade 11 and looks exactly like them: an hour, twenty-five questions, multiple choice until the last five. What changes is the math, which now reaches quadratics, coordinate geometry and similar triangles, and the final five questions, where there are no options to fall back on. This guide is drawn from the University of Waterloo CEMC Cayley pages for 2027, the cover of the 2025 Cayley booklet and the 2026 results booklet, and it adds eight problems of our own with every step of the solution shown.
Teaching live since 2020 · 10,000+ students taught · problems on this page are our own, written for Grade 10
The answer in brief
The Cayley Contest is the University of Waterloo CEMC math contest for Grade 10: one hour, 25 questions, 150 marks, written at school in a two-day window, Tuesday 23 or Wednesday 24 February 2027 in the Americas. Entry is only through the school. Modern Age Coders runs live online Cayley preparation, in a batch or one to one, inside its Waterloo Math Contests course, Gauss to Euclid.
Teaching for Grade 10
The three courses a Cayley student usually takes
One is built for the Waterloo contests themselves. One is for the student who wants the hardest questions on the paper. One makes sure the Grade 10 curriculum underneath is solid.
Cayley 2027: who writes it, when, and in what form
CEMC prints the details for Pascal, Cayley and Fermat on a single page. The rows below are the Cayley ones, as that page and the 2025 Cayley booklet showed them on 22 October 2026.
The Cayley Contest, from CEMC's contest page and the 2025 Cayley booklet cover (both read 22 October 2026)
Question families ask
The published answer
Where it comes from
Who is it for?
"Students in Grade 10 are eligible to write the Cayley Contest." Officially, it "may be written by any participant in Grade 10 or below."
CEMC contest page
When in 2027?
Tuesday, February 23, 2027 or Wednesday, February 24, 2027 in North and South America; the school runs it on one of the two days.
CEMC contest page
Deadline for the school?
Tuesday, February 9, 2027 for schools in Canada.
CEMC contest page
How long, how many?
Sixty minutes for twenty-five questions, marked out of 150.
CEMC contest page
What kind of answers?
"Part A and Part B of this contest are multiple choice." In Part C, "The correct answer to each question in Part C is an integer from 0 to 99, inclusive."
2025 Cayley booklet
Is guessing punished?
"There is no penalty for an incorrect answer." Right answers earn 5, 6 or 8 marks by part, and a blank earns 2, for at most ten blanks.
2025 Cayley booklet
What can be on the desk?
"You may use rulers, compasses and paper for rough work." Some calculators are permitted; the rest of the aids rule is CEMC's.
2025 booklet and contest page
Paper or screen?
Either: the school decides when it orders, and in both cases the student writes at school, alone.
CEMC contest page
What math does it use?
CEMC bases the questions on "curriculum common to all Canadian provinces".
CEMC contest page
One of three, once a year
A student may write only one of Pascal, Cayley and Fermat in a school year. So a Grade 10 student cannot sit Cayley for the real result and Pascal for practice in the same February.
A Grade 9 student who is far ahead may choose Cayley instead of Pascal; CEMC says "Exceptional participants may elect to write a contest beyond their actual grade level." The reverse is not allowed: a Grade 10 student cannot drop down to Pascal.
Not the UK Cayley
The name is shared with a Cayley paper set by the UK Mathematics Trust, which is a different paper from a different organizer. Everything on this page is the Waterloo contest.
Sources: CEMC, Pascal, Cayley and Fermat Contests and CEMC's 2025 Cayley Contest booklet, front cover only, read 22 October 2026. We are an independent teaching company with no tie to CEMC or the University of Waterloo.
The last five questions
Part C on Cayley: an answer from 0 to 99, and no options to lean on
In Parts A and B a student can test the five options, rule some out, or estimate their way to a letter. Part C takes that away. The answer is a whole number, and the 2025 booklet explains the convention used when a natural answer would be bigger than 99.
The Part C conventions, quoted from the 2025 Cayley booklet
Rule
Booklet wording
Range
"an integer from 0 to 99, inclusive"
Large answers
"The integer formed by the rightmost two digits of 12 345 is 45."
Small answers
"The integer formed by the rightmost two digits of 6307 is 7, coded 07."
Marks
8 for each correct Part C answer, the highest of the three parts
Cayley 2026, Part C: share of contestants giving the correct answer and share leaving it blank, from CEMC's 2026 PCF Results booklet (Cayley statistics page: 16368 contestants, average mark 90.66). The answers themselves are not printed here.
Question
Correct answer given
Left blank
21
53.55%
24.1%
22
14.72%
32.99%
23
13.28%
67.9%
24
1.92%
74.79%
25
2.11%
80.47%
One line in that booklet deserves a second look. On question 22 of the 2026 Cayley, the single most popular answer was a wrong one: 15.51% of contestants gave it, slightly more than the 14.72% who gave the correct answer. A wrong number written confidently in Part C earns nothing, and in 2026 more students did that on question 22 than got it right. That is what Part C asks of a student: to check an answer when no list of options can confirm it.
Put it back in. If the question gives a condition, test your number against it. Roots go back into the equation; a count goes back into the rule.
Try a smaller case. Shrink the question (20 terms to 4, 30 to 9) and solve the small version by listing. If your method gives the wrong small answer, it is wrong for the big one too.
Check size and parity. Should the answer be even? Divisible by 3? Smaller than some obvious bound? A count of pairs that comes out odd when the pairs are symmetric is a warning sign.
Read the last line again. Many Part C answers are "the rightmost two digits of" something, a sum, or a count of objects. Make sure the number you code answers that, and not a step on the way.
Code it correctly. A one-digit answer needs a leading zero, so 7 is coded 07, and an answer of 0 is coded 00.
Those five checks are ours, not CEMC's. We teach them from the first week and use them on every Part C question in class, including the last three problems in the practice set below.
Eight practice problems
Eight Cayley-style problems we wrote, with every step shown
These are our practice problems, not CEMC problems. Our teachers wrote all eight for Grade 10 students, and nothing in them is taken or reworked from an old Cayley paper. Two are at Part A level, three at Part B level and three are Part C questions with an answer from 0 to 99. Every answer was confirmed by a brute-force check in Python before this page went up.
Cayley practice 1 (our practice problem, not a CEMC problem; Part A standard). The line y = 3x - 6 meets the x-axis at P and the y-axis at Q. If O is the origin, what is the area of triangle OPQ? (A) 3 (B) 6 (C) 9 (D) 12 (E) 18
Solution. At P, y = 0, so 3x = 6 and P is (2, 0). At Q, x = 0, so Q is (0, -6). The triangle has a right angle at O, with legs of length 2 and 6, so its area is (2 x 6) / 2 = 6, answer (B). A length is never negative: the leg along the y-axis is 6, not -6.
Cayley practice 2 (our practice problem, not a CEMC problem; Part A standard). If x + y = 11 and x - y = 3, what is the value of x2 - y2? (A) 24 (B) 30 (C) 33 (D) 36 (E) 49
Solution. There is no need to find x and y. Factor: x2 - y2 = (x + y)(x - y) = 11 x 3 = 33, answer (C). Check by solving: adding the equations gives 2x = 14, so x = 7 and y = 4, and 49 - 16 = 33. The factoring takes ten seconds, the solving a minute, and on a 60-minute paper that difference adds up.
Cayley practice 3 (our practice problem, not a CEMC problem; Part B standard). The product of two consecutive positive odd integers is 323. What is their sum? (A) 32 (B) 34 (C) 36 (D) 38 (E) 40
Solution. Call the even number between them n, so the odd integers are n - 1 and n + 1. Their product is n2 - 1 = 323, so n2 = 324 and n = 18. The integers are 17 and 19, and their sum is 2n = 36, answer (C). Check: 17 x 19 = 323. Choosing the middle number as the unknown turns a messy quadratic into a perfect square.
Cayley practice 4 (our practice problem, not a CEMC problem; Part B standard). A student 1.8 m tall stands 6 m from the foot of a lamp post. The student's shadow, cast by the lamp, is 3 m long on level ground. How tall, in metres, is the lamp post? (A) 3.6 (B) 4.8 (C) 5.4 (D) 6.0 (E) 7.2
Solution. The tip of the shadow is 6 + 3 = 9 m from the post. The light ray from the top of the post to the tip of the shadow passes over the student's head, so the small triangle (student and shadow) is similar to the large one (post and full distance). Heights compare as bases: h / 9 = 1.8 / 3, so h = 9 x 0.6 = 5.4, answer (C). The usual slip is using 6 m as the base of the large triangle, which gives 3.6, option (A).
Cayley practice 5 (our practice problem, not a CEMC problem; Part B standard). How many three-digit positive integers have three odd digits that strictly increase from left to right, such as 139? (A) 6 (B) 10 (C) 20 (D) 60 (E) 125
Solution. The odd digits are 1, 3, 5, 7 and 9. Any choice of three different odd digits can be arranged in increasing order in exactly one way, so the count equals the number of ways to choose 3 digits from 5, which is (5 x 4 x 3) / (3 x 2 x 1) = 10, answer (B). Option (D), 60, counts every order instead of only the increasing one, and (E), 125, allows repeats.
Cayley practice 6 (our practice problem, not a CEMC problem; Part C standard). The equation x2 - 24x + k = 0 has two different positive integer roots. How many different values can k take? Give a whole-number answer between 0 and 99.
Solution. If the roots are r and s, then r + s = 24 and rs = k. With r smaller than s and both positive, r can be 1, 2, 3 and so on up to 11 (r = 12 would make the roots equal). Each r gives a different product r(24 - r): 23, 44, 63, 80, 95, 108, 119, 128, 135, 140 and 143. That is 11 values, so the answer is 11. Check with the first value: x2 - 24x + 23 = (x - 1)(x - 23), whose roots, 1 and 23, are different positive integers.
Cayley practice 7 (our practice problem, not a CEMC problem; Part C standard). Let S = 3 + 32 + 33 + ... + 320. What is the integer formed by the rightmost two digits of S?
Problem 7 · our workingtext
3^10 = 59049, so 1 + 3^10 = 59050
Pair each term with the one ten places later:
3^1 + 3^11 = 3^1 x 59050
3^2 + 3^12 = 3^2 x 59050
...
3^10 + 3^20 = 3^10 x 59050
An odd number times 59050 ends in 50.
Ten numbers that each end in 50 add to a number ending in 00.
Rightmost two digits: 00, so the answer is 0, coded 00.
The answer is 0, coded 00. A zero answer makes students doubt themselves, so this is a good place to use check 2 from the list above: the sum 3 + 9 + 27 + 81 = 120 is a small version of the same idea, pairing 3 with 27 and 9 with 81, and it also ends in a zero.
Cayley practice 8 (our practice problem, not a CEMC problem; Part C standard). In how many ways can 30 be written as a + b + c, where a, b and c are positive integers with a ≤ b ≤ c, and a, b and c can be the side lengths of a triangle?
Problem 8 · our workingtext
Largest side c, with a <= b <= c, a + b + c = 30 and a + b > c
a + b > c means 30 - c > c, so c < 15; and c >= 10, as c is the largest
c = 10 a + b = 20 (10,10) 1
c = 11 a + b = 19 (9,10) (8,11) 2
c = 12 a + b = 18 b from 9 to 12 4
c = 13 a + b = 17 b from 9 to 13 5
c = 14 a + b = 16 b from 8 to 14 7
total 19
The answer is 19. The triangle condition only needs checking for the largest side, because if a + b is greater than c, the other two inequalities follow automatically. Fixing the largest side first and counting what remains is a method worth practising: it keeps a long count in order.
Timing the set
Sixteen minutes for all eight matches the pace of a real Cayley. Part A problems should take a minute or two each, which leaves more time for the Part C ones.
Mark yourself with the booklet's scale (5, 6, 8, and 2 for a blank) before checking the solutions, so the score means what a real Cayley score means.
The ideas to carry forward
Choosing a good unknown (problem 3), factoring instead of solving (problem 2), and fixing the largest piece first (problem 8) all appear again and again in Fermat and Euclid.
Problem 7 is the one to redo in a month. Pairing terms that sit a fixed distance apart is an idea students forget, and it unlocks whole families of last-two-digit questions.
The math behind it
What Grade 10 math the Cayley Contest draws on
CEMC ties the questions to the math common to every province, not to one curriculum. Below is how we map that to the Grade 10 courses Canadian students actually take; the table is our own analysis.
Topics that carry a Cayley paper, and what we teach for each (ours, not a CEMC syllabus)
Topic
What it looks like on the paper
What we teach
Quadratics
Factoring, roots, and the link between roots and coefficients
Sum and product of roots, so many questions need no formula at all
Coordinate geometry
Lines, intercepts, midpoints, distances and areas on a grid
Sketching first, then one equation per fact
Similar triangles
Shadows, ladders, nested shapes, scale
Marking matching angles before writing any ratio
Algebraic manipulation
Expressions to simplify, identities like a difference of squares
Looking for the factored form the question is hiding
Counting
Digits with rules, arrangements, selections
Choosing whether order matters before choosing a formula
Number theory
Remainders, last digits, divisibility, often in Part C
Small cases by hand, then the pattern, then a proof of the pattern
Ontario and British Columbia
In Ontario, the Grade 10 academic math course is MPM2D, Principles of Mathematics, and it is the prerequisite for Grade 11 Functions (MCR3U). Its quadratics, analytic geometry and trigonometry of right triangles line up well with the Cayley paper.
In British Columbia the Grade 10 course is Foundations of Mathematics and Pre-calculus 10. Students elsewhere in Canada meet similar content in their own Grade 10 course, which is why CEMC builds the paper on what the provinces share.
Where school stops and the contest starts
Very little on a Cayley paper is outside the school course. What is outside is the way it is used: a school question tells you to factor, a contest question expects you to notice that factoring helps.
That is why our classes spend less time on new content than families expect, and more on recognizing which tool a question needs.
The 2026 figures
How the 2026 Cayley went, in CEMC's numbers
CEMC shows an average and two cutoffs on its contest page, and a table for every question in its results booklet. These are the Cayley rows only, read 22 October 2026.
Cayley 2026 averages and cutoffs, from CEMC's contest page
Figure
Cayley 2026
Average (mean)
90.0 out of 150
Certificate of Distinction cutoff
106
School Medal Champion cutoff
90
Number of contestants
19 953
Selected Cayley 2026 questions in Parts A and B: share of contestants choosing the correct option and share leaving it blank, from CEMC's 2026 PCF Results booklet. No answer letters are given.
Question
Chose the correct option
Left blank
1
97.89%
under 1%
9
59.6%
7%
13
53.81%
4.56%
16
45.14%
13.17%
18
19.42%
35.04%
19
22.49%
44.43%
20
20.33%
42.85%
Our reading, labelled as ours: a Cayley writer in 2026 needed 106 for the Certificate of Distinction, about 71 percent of the 150 marks (106 / 150). The question table shows a pattern we see in class too. Students leave the end of Part B blank far more often than the start of Part C: in 2026, question 20 was left blank by 42.85% of the counted contestants, while question 21 was left blank by only 24.1%. Our explanation, which CEMC does not give, is that students who are short of time skip ahead to Part C's first question, which is often more approachable than the last questions of Part B. We teach students to look at question 21 before they get stuck on question 19.
Sources: CEMC contest page, averages and cut-offs, and the 2026 PCF Results booklet linked there. CEMC reminds schools that its contests have "no passing or failing score".
Getting a seat
How a Grade 10 student gets into the Cayley Contest
Only a school can order the paper. CEMC's own figure for how far this reaches: "Each year, educators order CEMC contests for over 300 000 students worldwide". A family's job is to make sure their child is one of them.
If the school already orders
Ask the math teacher in November 2026 whether Cayley will be ordered, on which of the two February days it will run, and whether the school has chosen paper or the online platform.
The student then writes at that school on that day. CEMC's standards say "Participants may only write the contest at the school which ordered the contest."
If it does not
A school without a CEMC account needs one, and CEMC reviews new applications by hand, which can take three weeks. That is a conversation for the head of math before the winter break.
Where a school declines, CEMC points students to other local schools that might take them in for the sitting. A homeschooling family would take the same route; we found no separate home-education process on the CEMC pages we read.
On the day, the rules about aids come from the paper itself: CEMC says "Only permitted aids indicated on the Contest Question Booklet or Supervisor Instructions Document may be used while writing the contest." From CEMC's How contests work page, as it read on 22 October 2026.
A countdown
Counting back from the February 2027 Cayley
Our plan, written as weeks before Tuesday 23 February 2027. A school that writes on Wednesday 24 February simply gains a day. It assumes two live classes a week from the start of November.
Our countdown to the 2027 Cayley Contest (a plan, not a CEMC schedule)
Weeks to go
Dates
Focus
Checkpoint
16 to 13
2 to 29 November 2026
Algebra fluency: factoring, identities, linear systems; Parts A and B of one past paper, untimed
Every Part A question attempted with working shown
12 to 10
30 November to 20 December 2026
Coordinate geometry and similar triangles; problems 1 and 4 above
A sketch drawn before every geometry answer
9 to 8
The holiday fortnight, 21 December to 3 January
Holiday weeks: one Topic generator set a day, chosen by weak topic
A personal list of the five topics that cost the most marks
7 to 5
4 to 24 January 2027
Quadratics and roots, counting, last digits; problems 2 and 3, then 5, 6, 7 and 8
The five Part C checks used on every answer
4 to 2
25 January to 14 February 2027
Two full past papers under contest conditions, reviewed line by line
School order confirmed before Tuesday, February 9, 2027
1
15 to 22 February 2027
No new material: redo the questions missed in the timed papers
A calm, early night before the contest
First contest in Grade 10
Aim to finish Part A and most of Part B cleanly. Part C is a bonus this year.
Wrote Pascal last year
Start from your Pascal mistakes list; the target is all of Part B and two Part C questions.
Grade 9, writing up
Only if past Pascal papers feel easy. Spend the extra weeks on quadratics and similar triangles.
The Grade 10 contest year
Cayley in a Grade 10 year: CIMC in November, Galois in April
CEMC offers a Grade 10 student three individual math contests in 2026 to 2027. They test different things, and a student who writes all three gets a full picture of where they stand.
CEMC's individual math contests open to a Grade 10 student in 2026 to 2027 (dates for the Americas, from CEMC's pages read 22 October 2026)
Contest
Date
Shape, in CEMC's words
What it adds
CIMC (Canadian Intermediate Mathematics Contest)
Wednesday, November 18, 2026
"9 questions; 6 are answer only and 3 are full-solution." "Time allowed for completion is 2 hours."
An early full-solution taste; CEMC says most problems use topics "commonly taught up to and including Grade 10"
Cayley
Tuesday 23 or Wednesday 24 February 2027
Multiple choice and number answers, one hour
Speed and range across the Grade 10 course
Galois
Wednesday, April 7, 2027
"4 questions; a mix of final answer-only and full-solution."; 75 minutes, marked out of 40
Written explanations, marked for how clearly the reasoning is set out
The CIMC ordering deadline on CEMC's page was Thursday, October 22, 2026, the day this page was written, so most families reading it now will plan around Cayley and Galois. For a student who likes Cayley, Galois is the more important second contest: the same grade, the same curriculum, but every step of the working is read by a marker. After Grade 10 come Fermat and Hypatia in Grade 11, and the Euclid in the final year, which our CEMC contests overview covers alongside the rest of the family.
Practice material
CEMC's past Cayley papers and the Topic generator
The best practice for Cayley is old Cayley papers, and CEMC gives them away. We send students to CEMC's site for them rather than copying any onto ours.
Whole past papers
CEMC's past contests page holds the Cayley paper, its solutions and its results for each recent year, filtered by contest and grade.
We use them in two ways: untimed and part by part in the autumn, then whole and timed in February. A paper used as a timed rehearsal is never one the student has seen before.
Sets by topic
The Problem Set Generator has a Topic generator that "Randomly chooses up to 30 problems according to contest, level of difficulty (Part A/B/C), and topic."
That makes it the right tool for a weak spot. A student who keeps losing marks on counting can ask it for Part B counting questions from past Cayley papers, and work through a set in an evening.
For the curriculum itself, CEMC's free Courseware covers school math lesson by lesson. Our only rule for any of this material: try the question properly first, and read the official solution afterwards.
Placing a Grade 10 student
Where a Cayley student usually starts
Our four levels, used in the first class.
Levels for a Cayley writer (ours)
Level
What we see
1. Course-ready
Comfortable with Grade 10 homework, new to contest questions
2. Part A secure
Clears the first ten questions of a past Cayley in about twenty minutes
3. Part B solver
Finds the idea in most Part B questions, sometimes too slowly
4. Part C checker
Attempts Part C with time left and checks each answer before coding it
For parents
Level 2 by Christmas 2026 is a good sign. The step to level 3 is where steady weekly practice shows.
A student who wrote Pascal last year often starts at level 2 or 3. Ask them for their old Pascal score and their mistakes; both help us.
For students
When a Part C answer comes out, stop and check it before moving on. A checked answer is worth 8; an unchecked one may be worth nothing.
Look at question 21 before you get stuck at the end of Part B.
More courses
Nine courses for a Grade 10 student who likes math
Live and online, taught by one teacher who knows the student, in a group or privately, for families in every province.
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Fees
Fees
Ordering the paper from CEMC is the school's cost. Ours is the teaching, billed monthly, and a family can stop at the end of any month, including straight after the Cayley.
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Grade 10 families ask
Cayley Contest: questions and answers
What is the Cayley Contest?
It is the University of Waterloo CEMC math contest for Grade 10 students. The paper has twenty-five questions, takes an hour and is marked out of 150, and it is written at school in February.
When is the Cayley Contest in 2027?
In North and South America CEMC lists Tuesday, February 23, 2027 and Wednesday, February 24, 2027, and the school picks one of those days. Schools in Canada must have their order in by Tuesday, February 9, 2027.
Who can write the Cayley Contest?
CEMC says students in Grade 10 are eligible, and that it may be written officially by any participant in Grade 10 or below. Pascal, Cayley and Fermat count as a set, so a student writes just one of them each school year.
What is Part C on the Cayley Contest?
It is the last five questions. Instead of choosing a letter, the student gives a whole number from 0 to 99. When a natural answer is larger, the question usually asks for the integer formed by its rightmost two digits, and a one-digit answer is coded with a leading zero, such as 07.
Is there a penalty for wrong answers on Cayley?
No. Under the 2025 booklet rules a wrong answer simply scores zero. Correct answers earn 5, 6 or 8 marks by part, and each unanswered question earns 2, up to ten unanswered questions.
What was the Cayley average in 2026?
CEMC lists a 2026 Cayley average of 90.0 out of 150, a Certificate of Distinction cutoff of 106 and a School Medal Champion cutoff of 90, with 19 953 contestants.
Is the Waterloo Cayley Contest the same as the UK Cayley?
No. The Waterloo Cayley is CEMC's Grade 10 paper. The UK Cayley belongs to the UK Mathematics Trust and is a different paper, with its own route to entry and its own date.
How does my child enter the Cayley Contest?
Through their school, which orders the contest from CEMC. Students cannot enter on their own. If the school does not take part, CEMC suggests looking for another school nearby that will let the student write there.
Should a Grade 10 student also write Galois?
We think so, if they enjoy Cayley. Galois is CEMC's Grade 10 full-solution contest, on Wednesday, April 7, 2027 in the Americas, with four questions in 75 minutes, and it rewards clear written reasoning.
When are your Cayley classes held?
Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays. Times are set in Eastern time and shown to each family in its own time zone.
Keep reading
Related pages
Pages a Grade 10 contest writer in Canada may find useful next.
Breaking a problem into steps, the same skill Part C needs.
Start here
Book a free first class
In the first class the student works through a short Cayley-style set we wrote, with a teacher, on screen. You come away knowing which of the four levels your child is at and what we would do with them between now and late February 2027.
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