Almost none of the mistakes that cost marks on the Junior Maths Challenge come from not knowing enough maths. They come from a correctly-learned rule being reached for in the wrong moment. Below are the families of misconception that show up again and again across our JMC card library — grouped not by topic, but by the shape of the mistake, because the same habit of mind trips a child up in number, geometry and probability alike. This pairs with the facts worth knowing cold — that piece is what to know; this one is what tends to go wrong when applying it.

Family 1: a test that works for one number, borrowed for another

Divisibility tests are a JMC staple, and they're also where a huge share of mistakes cluster — because there are several different tests and it's tempting to assume they all work the same way.

The digit-sum test genuinely works for 3 and for 9: add the digits, and if that total is a multiple of 3 (or 9), so is the original number. But it does not work for 11 — that needs an alternating sum of digits instead, added and subtracted in turn from the right. And it doesn't work for 4 at all, where only the last two digits matter, because 100 itself is already a multiple of 4. A child who checks a number's last digit to test for divisibility by 3 (the way they correctly would for 2 or 5) is borrowing the wrong test entirely.

There's a subtler version of this trap too: assuming a multiple of 9 has something to do with the digit 9 physically appearing in the number, rather than the digit sum being a multiple of 9. The rule is about arithmetic, not appearance.

The fix: before applying a shortcut, name which number you're testing divisibility by, and recall the specific test for that number — last digit, last two digits, plain digit sum, or alternating digit sum. There are four different tests in regular use, not one universal trick.

Family 2: "always" statements that are only true some of the time

A cluster of common mistakes comes from over-generalising a rule that's true in the everyday range of numbers a child has practised with, but false at the edges.

Squaring doesn't always make a number bigger — it does above 1, but a positive number between 0 and 1 gets smaller when squared (a half squared is a quarter). Dividing doesn't always make a number smaller — dividing by something smaller than 1 makes a number bigger. Multiplying doesn't always give a bigger result than adding — for two positive whole numbers, adding beats multiplying in exactly one case: when either number is 1 (1 + 7 = 8, but 1 × 7 = 7). And a matched pair: an increase and a decrease of the same percentage do not cancel out, because the second percentage change is calculated on a different starting amount than the first.

Even the lowest common multiple has one of these traps: it is not always the product of two numbers — only when the two numbers share no common factor. The LCM of 4 and 6 is 12, not 24, because both share a factor of 2.

The fix: treat any "always" belief about squaring, dividing, multiplying or percentage changes as something to test on a small example first — a half, a fraction, a 1 — before trusting it in the actual question.

Family 3: losing track of a quantity that's been reused

Several mistakes come from double-counting or forgetting that the same thing has already been included once.

Two readings of a container — a paint pot half full, then a quarter full — both include the weight of the pot itself, so subtracting one reading from the other cancels the pot out and leaves pure paint. Doubling the half-full reading to estimate the full weight is a common error, because it counts the pot's own weight twice. A similar trap appears with sheep-and-hens-style "how many of each" questions: dividing the total number of legs by the total number of heads doesn't work, because heads and legs aren't reused at the same rate per animal.

Results tables carry the same idea in disguise: every goal scored by one team is a goal conceded by another, so the two columns must balance across the whole table — treating each team's row as independent of the others throws that connection away.

The fix: before subtracting or dividing two related totals, ask explicitly what's shared between them and what changes — that's usually the entire content of the question.

Family 4: fencepost and counting errors

This family shows up in completely different topics but is always the same underlying slip: counting the gaps instead of the things, or vice versa.

Counting whole numbers from a up to b inclusive gives b − a + 1, not b − a — subtracting the ends alone forgets to add back the number you started counting from. A square grid of n-by-n cells needs n + 1 lines in each direction to draw it, not n. And a solid's edges are easy to undercount straight from a drawing: a cube has 12 edges, but a typical sketch only shows 9 of them, because three run away from view and get missed.

The same instinct causes people counting squares hidden in a grid to stop at the smallest size, missing the larger ones layered on top — a systematic count needs one pass per size of square, not just the obvious cells.

The fix: for any "how many" question involving a range, a grid, or a 3D shape, count on your fingers with a tiny example first (3 to 5 inclusive is obviously 3 numbers, not 2) before trusting a formula.

Family 5: geometry assumptions that feel obvious but aren't

Diagrams invite assumptions that feel like common sense but don't actually hold.

Cutting a piece out of a shape feels like it should shrink the perimeter — but an L-shape cut from a rectangle has exactly the same perimeter as the original rectangle, because the two new edges of the notch are just the missing pieces of the old sides, moved inward. A single flat cut through a cube feels like it should always leave a square or rectangular cross-section — it can produce anything from a triangle up to a hexagon. And the exterior angles of a polygon are often assumed to grow as the number of sides grows, the way interior angles do — they don't; exterior angles always sum to exactly 360°, whatever the shape.

Reflections carry their own trap: a vertical mirror line reverses the order of digits or letters and flips each one individually — forgetting the second half of that (that each digit is also flipped, not just reordered) is an easy slip.

The fix: for any geometry question, draw the actual diagram rather than reasoning about it in your head, and treat any "obviously true" geometric intuition as worth a thirty-second sanity check against a simple example.

Family 6: reading the wording too fast

A handful of mistakes are pure reading-speed traps, where the question is worded precisely to catch a quick skim.

"Different integers" (rather than "different positive integers") is a deliberate signal that negative numbers and zero are back in play — reading "integer" as if it meant "positive whole number" throws away exactly the values the question is testing. A ratio written as "one part to three parts" describes four total parts, not three — reading it as a straightforward third is a very natural but wrong shortcut. And a rope described as "21 cm plus a quarter of its own length" has 21 cm as three quarters of the total, not one quarter — the fixed amount is whatever fraction is left over, not the fraction that's named.

The fix: underline or restate in your own words exactly what's being asked before starting to calculate — the JMC's wording is rarely accidental.

Why understanding the family matters more than memorising the example

None of the specific traps above is likely to appear on any future paper in exactly this form — new questions get written every year. What repeats is the shape of the mistake: an over-generalised rule, a double-counted quantity, an off-by-one, an assumption dressed up as common sense. A child who's spent time noticing these families catches the next unfamiliar example of one; a child who's only memorised "143 isn't prime" has learned a single fact that won't come up again. If you're building a revision routine around this kind of pattern-spotting, spaced repetition is a genuinely good fit for it — and it pairs naturally with the facts most worth knowing cold in the first place.

FAQ

Are these mistakes about not knowing the maths?

Mostly not. Nearly every mistake here comes from a rule that's correctly known but misapplied — the digit-sum test used for the wrong divisor, an "always" belief that's only true some of the time, a step in the working that got double-counted. That's actually good news: they're fixed by a habit of checking, not by learning new content.

Is it worth drilling these mistakes directly?

Better to notice the family a mistake belongs to than to memorise each example. A child who understands "always statements about squaring and dividing are usually false outside a narrow range" will catch the next one they haven't seen before, where a memorised example wouldn't help.

Why do so many of these come from reading the question quickly?

A 60-minute paper with 25 questions leaves under two and a half minutes each, so there's real pressure to move fast. The trap is that many JMC questions are worded precisely so that a fast, careless read gives a different (wrong) problem than the one actually being asked.

Duke Harewood built aitutors.me's KS3 maths tutor (Professor Pi) for his Year 8 daughter, who has sat her share of maths challenges. These families of mistake come from the misconception notes Professor Pi writes into every JMC flashcard. Updated 26 August 2026.