08
Grade 8
A fun, low stakes introduction to problem solving. Numbers, geometry, some equations, combinatorics, and the first steps into logic and proofs.
- 12 multiple-choice questions
- 60 minutes
- Difficulty: entry to intermediate
The November round
Unexpectedly satisfying maths exam.
Written by students, written for students.
Delivered by undergraduate mathematicians at Cambridge, Prism Maths Challenge aims to provide enjoyable problems for all, real opportunities for creative thinkers, and, after all, an experience worth remembering.
Every round, a question or two is written by students like you; top scorer... or just a submission.
Publish your own question
Rounds in Nov & Mar
Certificates & Book prizes
School / Home divisions
Why participate in PMC?
We built the competition we wished we'd sat as students:
fair enough that hard work is rewarded, generous enough that anyone can get something out of it.
Have fun!
Every paper mixes accessible questions and puzzles with a few that require genuine insight. Questions are designed to train mathematical thinking, while staying connected to the students' real interests.
Something to show for it.
Most participants receive a certificate, and top scorers earn recognition worth mentioning; on a university application, or just to yourself. Moreover, the grand prize...
Become the examiner!
Top scorers in each grade are invited to co-author a question for the next round, credited on the paper. A few creative submissions might also be selected. Better than a trophy, a mathematical footprint.
There are three of them...
Meet our papers!
Every paper is based on concepts taught at each grade level. Pick the one that matches your grade, or a higher one if you are ambitious.
10
Grade 10
Number theory, more geometry, more equations, inequalities, combinatorics and probability — we are now mixing them with longer, multiple step problems.
- 12 multiple-choice questions
- 60 minutes
- Difficulty: intermediate
12
Grade 12
We draw problems from everything that you would have met in your school so far, including differential / integral calculus, and vectors.
- 12 multiple-choice questions
- 60 minutes
- Difficulty: intermediate to advanced
Here it is,
Flip through a few sample questions
Six questions straight out of a Grade 10 paper,
from puzzles and brainteasers to multi-staged reasoning problems.
Grade 10 paper · Sample
Sheet 1 of 6
Lantern
Two lamp posts stand upright on level ground, one 6 m tall and the other 12 m tall. A cable runs from the top of each post to the foot of the other. A lantern hangs at the point where the two cables cross.
How high above the ground is the lantern?
- A3 m
- B4 m
- C4.5 m
- D9 m
- EIt depends on how far apart the posts are
Grade 10 paper · Sample
Sheet 2 of 6
You are playing with a square paper napkin that has an area of 32 square cm. You fold one corner over so that its tip lands exactly on the centre of the square.
Once the napkin is pressed flat, what is the area of the shape you now see from above?
- A20 cm2
- B24 cm2
- C26 cm2
- D28 cm2
- E30 cm2
Grade 10 paper · Sample
Sheet 3 of 6
How many ordered pairs of positive integers (a, b) satisfy
1/a + 1/b = 1/12 ?
- A8
- B12
- C14
- D15
- Einfinitely many
Grade 10 paper · Sample
Sheet 4 of 6
Questions 10 to 12 · Doomscroll
You are doomscrolling on your phone, bouncing between exactly 3 apps: TikTok, Instagram, and YouTube. You start your session by opening TikTok (this is app #1).
To keep your brain stimulated, you follow one strict rule: you never view the exact same app twice in a row.
Let's imagine you have scrolled through a sequence of exactly 20 apps. Suppose there are exactly A different valid sequences of 20 apps that end on TikTok, and exactly B different valid sequences that end on either Instagram or YouTube.
If you scroll to a 21st app, how many valid sequences of 21 apps will end on TikTok?
- AA
- BB
- C2A
- D2B
- EA + B
Grade 10 paper · Sample
Sheet 5 of 6
Doomscroll · continued
The same three apps, starting on TikTok, and still never the same app twice in a row.
Notice that after your first app, you always have exactly 2 choices for the next one. Using the trick you discovered in Question 10, you can easily build a step-by-step list to find out how many sequences of 9 apps end on TikTok. What is that exact number?
- A42
- B86
- C128
- D170
- E256
Grade 10 paper · Sample
Sheet 6 of 6
Doomscroll · continued
The same three apps, starting on TikTok, and still never the same app twice in a row.
You keep scrolling until you have viewed a valid sequence of exactly 10 apps (starting with TikTok as app #1). You realise that out of these 10 apps, exactly 4 of them were TikTok. How many different valid sequences of 10 apps fit this description?
- A120
- B160
- C200
- D240
- E320
I once left a maths competition having solved almost nothing.
About us
Prism is the competition I wish I'd had at twelve
Started by a Cambridge maths undergrad, written by a small group of friends. Our goal is to make the joy of problem solving open to everyone, whether or not you are already good at it.
Read more? →
Mark the calendar
The November round
November 2026
- Grade 8Friday 20 NovemberSchool Division
- Grade 10Monday 23 NovemberSchool and Home Divisions
- Grade 12Tuesday 24 NovemberSchool and Home Divisions
Quick to join, easy to sit
Registration for the November round is open. Reserve your school's slot, or sit the Home Division as an individual student.
Register