The Junior Maths Challenge doesn't test advanced content — it tests whether a small set of number, angle and counting facts are so familiar that your child can reach for them without thinking. Below is a curated list of around two dozen of the strongest, pulled from the patterns we've noticed across a decade of JMC-style questions, organised by topic and explained in one or two sentences each. It's a companion to our full JMC guide, not a repeat of it — that piece covers dates, format and how to enter; this one is about what actually shows up.

Number: divisibility and factors

These are the facts that let a child rule out answer choices in seconds, without doing any division.

A number is a multiple of 3 exactly when its digit sum is. Add the digits; if that total is a multiple of 3, so is the original number. The same trick works for 9 — just check whether the digit sum is a multiple of 9 instead.

A number is a multiple of 4 exactly when its last two digits are. You never need to look further back than that, because 100 itself is a multiple of 4. The equivalent test for 8 uses the last three digits.

A number is a multiple of 5 only if it ends in 0 or 5, and a multiple of 15 needs both the 3-test and the 5-test to pass. Because 3 and 5 share no common factor, checking both separately is exactly the same as checking for 15 directly.

Even numbers aren't automatically multiples of 4. This is one of the most common slips: an even number only divides by 4 if its last two digits do too, so 5678 is even but not a multiple of 4 (78 isn't).

1 is not a prime number, and 2 is the only even prime. A prime needs exactly two factors, and 1 only has one. Because 2 is the odd one out among primes, questions about sums of primes often hinge on whether a 2 is involved — two odd primes can never sum to a prime, since odd plus odd is always even.

The prime digits are 2, 3, 5 and 7 — four out of the ten. Worth knowing by heart, because "prime digit" clues turn up in crossnumber-style puzzles regularly.

To test whether a number is prime, you only need to try dividing by primes up to its square root. Any factor larger than the square root would have to pair with a smaller one you'd already have found. Numbers between 120 and 170 are a favourite trap for this — 143, 161 and 169 all look prime and none of them is.

Number: squares, cubes and useful shortcuts

Know your squares up to 15² (225) and your cubes up to 6³ (216) automatically. Recognising 169 as 13² or 125 as 5³ on sight saves real time, especially in questions that describe a number rather than naming it.

A square number never ends in 2, 3, 7 or 8. Working out the last digit of 1² through 9² shows why: the only possible last digits for a square are 0, 1, 4, 5, 6 and 9. This alone eliminates most options in a "which of these is a square?" question.

Squaring a number between 0 and 1 makes it smaller, not bigger. A half squared is a quarter — smaller than the number you started with. "Squaring always makes things bigger" is only true above 1, and JMC setters know it's a common assumption to lean on.

The gap between two consecutive square numbers is always odd. Specifically, the gap between n² and (n+1)² is 2n + 1, so the differences run 1, 3, 5, 7, 9… Square numbers are not evenly spaced, even though it can feel that way.

Algebra and sequences

The sum 1 + 2 + 3 + … + n equals n(n + 1) ÷ 2. Pairing the first term with the last, the second with the second-last and so on is the fastest way to see why it works, and it turns up constantly in number-pyramid and grid-total questions.

The exterior angle of a triangle equals the sum of the two interior angles that aren't next to it. It follows from the triangle angle sum and the straight-line angle sum, and it often lets you skip straight to an answer instead of working out every angle in a diagram.

When a long sum shares a common factor across every term, pull the factor out first. 3 + 6 + 9 + 12 becomes 3 × (1 + 2 + 3 + 4) — and when that kind of bracket sits over a matching one in a fraction, it frequently cancels completely.

Geometry: angles

The exterior angles of any polygon always add up to 360°, whatever the number of sides. This holds for a triangle just as much as a fifty-sided shape, and it's often a faster route to a missing angle than the interior-angle formula.

The interior angles of an n-sided polygon add up to (n − 2) × 180°. A quadrilateral is 360°, a pentagon 540°, a hexagon 720°. Splitting the shape into triangles from one corner is the reason this works.

Angles around a point add up to 360°, and angles on a straight line add up to 180°. Together with vertically opposite angles being equal (the "X" shape where two lines cross), these three facts solve most angle-chasing questions in the paper.

In parallel lines, alternate ("Z") angles and corresponding ("F") angles are equal, and co-interior ("C") angles add to 180°. Any trapezium, parallelogram or rhombus hands you at least one of these pairs for free, because two of its sides are always parallel.

An isosceles triangle's two equal sides sit opposite two equal angles. Spotting the equal sides in a diagram is the first move — the equal angles follow immediately, and most JMC angle-chase questions are really a chain of isosceles triangles linked together.

Geometry: shapes and 3D solids

A cube has 6 faces, 12 edges and 3 edges meeting at every vertex. It's easy to undercount the edges from a drawing — a typical sketch only shows nine of the twelve, because three run away from view.

A cube has 11 different nets, and a face diagonal is not the same thing as a space diagonal. A cube has 12 face diagonals (one pair per face) but only 4 space diagonals running corner-to-corner through the middle — a distinction worth keeping straight.

Regular shapes give you equal sides and equal angles for free — that's what "regular" means. It's the single word that unlocks a diagram: equal sides let you build isosceles triangles from the centre, and equal angles mean the exterior angle is simply 360° divided by the number of sides.

Ratio, proportion and quick estimation

To turn a share of a ratio into a fraction, the denominator is the total of every part, not the other share. In a 2 : 7 ratio, the first share is 2 out of 9 — not 2 out of 7. This is one of the most common ratio slips on the paper.

When two people move towards each other, the gap between them closes at the sum of their speeds; moving the same way, it changes at the difference. Getting this the wrong way round is an easy trap under time pressure.

When answer choices are spread far apart, round and estimate rather than calculating exactly. 97 × 31.8 is close enough to 100 × 30 = 3000 to separate options that differ by a thousand — and reading the spread of the choices before deciding how precisely to work is a genuinely useful competition habit.

Why knowing these cold is worth it

None of this is a shortcut around understanding — every fact above comes with the one-line reason it's true, because a fact a child only half-remembers is more dangerous than one they never learned: it gets misapplied with confidence. What these facts buy is speed. On a 60-minute, 25-question paper, a child who has to re-derive the exterior-angle rule from scratch every time it appears is spending minutes that a child who just knows it gets to spend on the harder questions further down the page.

We picked these particular two dozen because they're the ones that showed up again and again as we built our JMC flashcard library — read what keeps coming up every year for the actual numbers behind that claim, and how many of the mistakes above turn out to trace back to the same handful of misconceptions in common JMC mistakes.

FAQ

Do I need to memorise all of these facts before the Junior Maths Challenge?

No. Think of this as a map of what tends to matter, not a checklist to tick off. A child who knows a handful of these cold — the digit-sum tests, the exterior-angle rule, the squares up to 15² — already has a real edge, because they stop spending working time on things that should be instant.

Why do facts like this help on a multiple-choice paper?

Because the Junior Maths Challenge rewards speed and recognition as much as method. A child who instantly knows that a square number never ends in 2, 3, 7 or 8 can eliminate answer choices without calculating anything — which leaves more of the 60 minutes for the questions that actually need working out.

Where do these facts come from?

From our own JMC flashcard decks at aitutors.me, covering the toolkit deck and every year from 2015 to 2026 — 156 cards written and checked in-house, not reproduced from any UKMT paper.

Duke Harewood built aitutors.me's KS3 maths tutor (Professor Pi) for his Year 8 daughter, who has sat her share of maths challenges. This list grew out of the flashcard decks Professor Pi wrote for JMC revision. Updated 26 August 2026.