Competitions · Mathematics · Gauss
The Gauss Mathematics Contests.
Twenty-five multiple-choice questions in an hour, written by Waterloo’s Centre for Education in Mathematics and Computing for students in grades 7 and 8. It carries an unusual rule that changes how the paper should be attempted: an unanswered question scores two marks, and a wrong one scores nothing.
At a glance
What the contest is, and who sits it.
Gauss is run by the Centre for Education in Mathematics and Computing at the University of Waterloo - the only Faculty of Mathematics in Canada, and one of the largest in the world. It is among the most senior contests available at Canadian middle-school level.
- Organiser
- The Centre for Education in Mathematics and Computing (CEMC) at the University of Waterloo
- Two levels
- A Grade 7 paper, open to grade 7 and below, and a Grade 8 paper, open to grade 8 and below
- One entry a year
- A student may sit only one of the two levels in a given year. A grade 8 student cannot enter both papers
- Format
- 25 multiple-choice questions with five options each. Sat on paper or online
- Duration
- 60 minutes
- Marks
- 150 in total, distributed unevenly across three sections
- Syllabus
- Based on the curriculum common to every Canadian province, covering algebra, geometry, probability, statistics and number theory. The last few questions are set to test insight rather than coverage
- Calculators
- Permitted for numerical work. Excluded are any with internet access, the ability to communicate with other devices, previously stored information such as formulas or notes, a computer algebra system, or dynamic geometry software
- Standing
- Widely recognised by universities in Canada and the United States, to the point that it is sometimes described as a mathematics equivalent of a language test
Marking
Three sections, and marks for leaving a question alone.
The scoring is what sets Gauss apart from almost every other multiple-choice contest. A blank answer is worth two marks, up to a maximum of ten blanks, while a wrong answer is worth nothing at all.
| Section | Questions | Marks each | Section total | What it tests |
|---|---|---|---|---|
| Section A | 10 | 5 | 50 | Broadly accessible questions that a prepared candidate should bank |
| Section B | 10 | 6 | 60 | The mathematical foundation, tested directly |
| Section C | 5 | 8 | 40 | The hard end, where insight matters more than technique |
| Unanswered | Up to 10 | 2 | Up to 20 | A deliberate reward for knowing what you cannot do |
A wrong answer scores zero. Because a blank is worth two marks, the arithmetic of guessing is unusual on this paper - see below.
Strategy
When guessing is worse than leaving it blank.
With five options and two marks guaranteed for a blank, a completely blind guess loses money on every section of this paper. The question is how much you have to eliminate before it stops doing so.
A blind guess never pays
One chance in five of 5 marks averages 1.0 in Section A, 1.2 in Section B and 1.6 in Section C. All three are below the 2 marks a blank is worth, so answering at random costs you marks on every part of the paper.
Section A: eliminate three
At 5 marks a question you need better than a two-in-five chance to beat a blank. That means ruling out three of the five options - with only two left standing, a guess averages 2.5 against the 2 you would keep.
Sections B and C: eliminate two, or one
At 6 marks, three remaining options break even exactly. At 8 marks, four remaining options break even - so on the hardest questions, ruling out a single option is already enough to make an attempt worthwhile.
Ten blanks, no more
Only the first ten unanswered questions earn the 2 marks, which caps the tactic at 20 marks. Beyond ten blanks there is nothing to protect, so the remaining questions should all carry an answer.
Awards
Individual recognition only.
Gauss has no team standings. Everything it awards is to a named student.
Certificate of Participation
Issued to every student who sits the paper, regardless of score. For many entrants at this age this is the first internationally issued academic certificate they hold, which is part of the point of entering.
Certificate of Distinction
Awarded to the top 25% of entrants. It is the same threshold CEMC uses on Euclid and its other contests, which makes results across the Waterloo family directly comparable to one another.
The school champion
The highest-scoring student at each participating school receives a separate award for it. In a school entering a large cohort this is a genuinely scarce distinction, independent of the global percentile.
Named on the contest website
Students achieving high scores globally have their names published on the official contest site. It is the one form of recognition here that reaches beyond the candidate’s own school and country.
Season
How the year runs.
Gauss sits late in the school year, well after the autumn contests, which makes it a natural finish to a season rather than a start.
Registration closes
Schools register their candidates. For the 2027 cycle the deadline was 22 April.
Contest window opens
The paper is sat within a two-week window - 10 to 21 May for the 2027 cycle - in both the Americas and other regions.
Marking and results
Papers are marked by CEMC and results returned to the registering school.
Preparation
How Hanlin prepares students for Gauss.
CEMC states plainly that most Gauss questions test logical thinking and problem solving rather than content. Preparation that treats it as a syllabus exam misses what the paper is doing.
Insight over coverage
The final questions are written to test ingenuity, not whether a topic has been taught. Practice concentrates on approaching an unfamiliar problem methodically, which is a habit rather than a body of knowledge.
Rehearsing the blank rule
Students are trained to decide, per question, whether they have eliminated enough to beat two marks. This is not intuitive and it is worth several marks a paper, so it is practised explicitly under timed conditions.
Past papers at both levels
CEMC publishes papers for both grades going back years. Working the Grade 7 paper before the Grade 8 one shows how the same ideas are escalated, and settles which level a borderline student should enter.
Pacing across three sections
Sixty minutes for twenty-five questions of rising value means Section A has to be banked quickly. Most marks are lost by spending too long early and arriving at Section C with no time to think.
Why Hanlin
Preparation for Gauss at Hanlin is taught by full-time subject tutors, on a course tier chosen by placement paper rather than by year group.
Programme formats
Five ways this is taught.
Group size is the whole difference between these. It decides how much of the tutor's attention a student gets and how far the course can bend to them, so it is worth choosing rather than accepting.
| Format | Group size | What it suits |
|---|---|---|
| One-to-one | One student | Content, pace and emphasis set by what this student actually needs rather than by a syllabus. The only format that can be rebuilt mid-course. |
| Small group | Three to eight | Opens at three. Frequent discussion, and the tutor can slow down or move on according to what the group has absorbed. Most students chasing a higher award are taught in this one. |
| Class | Eight to twelve | Taught by the subject lead or a medal coach. Less back-and-forth, so it suits a student who will say when something is unclear. |
| Large class | Ten to twenty | For students in the middle of the placement range, and a way to try a competition before committing. Students often move to a small group or to one-to-one afterwards. |
| Past-paper class | No cap | Past papers worked through in sequence in the weeks before the sitting. Free to students already enrolled. |
Course length for Gauss is set after a placement paper. The gap between where a student is and what the paper asks for is what the course has to close, and that gap is not the same for two students in the same year group.
How it runs
Six steps, four people on your side.
The same process for every student, so that nothing depends on one tutor remembering to do it.
Step 1Pre-course assessment
A placement paper before anything is booked, so the course tier is chosen on evidence rather than on a year group.
Step 2Tutor matching
A subject tutor is matched to the result, and their full background is provided before you agree to it.
Step 3Plan and group
A study group opens with four people on one student: the tutor, a planner, a supervisor and an academic manager.
Step 4Scheduled teaching
The teaching office sets the timetable around the competition date and school term, and teaching begins.
Step 5After-class reports
What was covered, and how the student handled it, reported after every lesson.
Step 6Homework follow-up
The teaching assistant and the form teacher both check that homework is done, which is where most preparation quietly fails.
Free resource pack
Past papers and preparation pack
The same material our own Gauss students work from. Scan the code, say what you need, and an advisor sends it.
Tell an advisor which competition and year group, and the material comes back the same day. Free, and you do not have to enrol in anything. If we do not already hold what you need, we will go and find it.
- PDFPast papers, every year the organiser has released
- PDFWorked solutions, where the organiser publishes them
- PDFThe syllabus on one page - every topic, and how heavily it is examined
- PDFEntry checklist and the dates for the current cycle
Teaching team
Who would teach it.
Three of the twenty-two tutors on the published Hanlin roster - the ones whose subject this is.
Dr Zhang
PhD in theoretical mathematics from the University of Rochester and a postdoctoral fellow at the Shanghai Center for Mathematical Sciences at Fudan. An AMC-accredited coach, ASDAN-accredited for DMM and ITCCC, and an accredited UKMT BMO2 and PUMaC coach. His students have gone on to read mathematics and physics at Harvard, MIT, Princeton, Caltech and Cambridge.
Dr Zhuang
PhD in mathematics from the University of Denver, working in mathematical logic, after a first degree in applied mathematics at Southeast University. A repeat conference speaker on his research, with three years lecturing and assisting on undergraduate mathematics at Denver.
He
A Cambridge master's after a computer science degree with a mathematics minor at the University of Nebraska-Lincoln, and a US high school GPA of 4.13. Six years overseas and teaches entirely in English, across AMC, Math League, AP Computer Science and AP Calculus.
Student results
What Hanlin students got.
Taken from Hanlin's published record of student honours and rewritten in English. Figures are as the organisers reported them.
Gauss
- 2022Grade 7: 22 students entered and 12 scored above 100. Grade 8: 6 entered and 3 scored above 120
The full record, across every subject and year, is on the student honours page.
Where it is taught
Hanlin learning centres.
Teaching runs from Hanlin's own centres in Shanghai, Shenzhen, Chengdu and Hangzhou, and online for students elsewhere.
Talk to an advisor
Everything Hanlin does for Gauss, in one conversation.
Entry, planning, coaching and the past papers. An advisor will tell you which of these a student actually needs, including when the answer is none of them.
Competition entry and planning
Which competitions suit this student, in what order, and by when - then we handle the registration, including the ones that can only be entered through a school or a centre.
Competition coaching
Taught by full-time subject tutors, on a course tier chosen by placement paper rather than by year group.
International curriculum tutoring
IB, AP, A-Level, IGCSE and the US high school curriculum, taught alongside school and timed around the exam calendar.
Competition past papers, free
A large library of past papers, mark schemes and syllabus breakdowns across every competition on this site. No charge and no enrolment.
Scan to add an advisor
Say which competition or course you are asking about. Replies in Chinese or English.
Or call +86 21 6352 6630FAQ
Gauss, briefly answered.
What students and families ask before entering.
Who can sit the Gauss contest?
The Grade 7 paper is open to students in grade 7 and below, and the Grade 8 paper to students in grade 8 and below. Younger students with an interest are welcome to enter the level appropriate to them.
Can a grade 8 student sit both papers?
No. A student may enter only one level in a given year, so a grade 8 candidate chooses between the two rather than sitting both.
Do I lose marks for a wrong answer?
You score nothing for it, but nothing is deducted either. The unusual part is the other way round: an unanswered question earns 2 marks, for up to ten blanks, so a wrong answer costs you the 2 you would have had.
Should I guess?
Not blindly - at every section value, a one-in-five guess averages less than the 2 marks a blank is worth. Guess once you have eliminated three options in Section A, two in Section B, or one in Section C.
Can I use a calculator?
Yes, for numerical work. Calculators are excluded if they have internet access, can communicate with other devices, hold stored formulas or notes, run a computer algebra system, or include dynamic geometry software.
What is on the paper?
Questions are based on the curriculum common to all Canadian provinces and cover algebra, geometry, probability, statistics and number theory. Most are written to test reasoning rather than recall, and the last few specifically test insight.
What awards are available?
Every entrant receives a Certificate of Participation. The top 25% receive a Certificate of Distinction, each school’s highest scorer receives a separate award, and high global scorers are named on the contest website.
Is there a team competition?
No. Gauss awards individual results only, which is a difference from the senior Waterloo contests where schools may enter team standings.
Related
The rest of the Waterloo family.
CEMC runs a contest for nearly every school year, and they share a house style.
Gauss enquiries
Choose the level, then learn the blank rule.
A grade 8 student can only sit one paper, and the scoring rewards judgement as much as mathematics. Tell us the student’s year group and an advisor will confirm the level and the registration route through a school.