The Intermediate Maths Challenge doesn't test obscure tricks — it tests whether a fixed set of number, algebra and geometry facts come to mind instantly, under a 60-minute clock, across 25 questions. We went back through thirteen years of our own IMC-style practice material — the toolkit deck plus one deck for every sitting from 2015 to 2026 — and pulled out the facts that keep doing the work. Here they are, organised by topic, in plain English. For the full picture of how the IMC works — dates, entry, awards — see our parent's guide.

Number

A prime number has exactly two factors — itself and 1. That single rule is why 1 is not prime: it only has one factor. It sounds obvious written down, but under time pressure "is 1 prime?" trips up more candidates than any other single fact on this list.

The last digit of a power runs in a short, repeating cycle. Powers of 3 go 3, 9, 7, 1, then repeat; powers of 2, powers of 7, powers of 8 each have their own four-step cycle. To find the units digit of something like 3¹⁰⁰, work out which position in the cycle 100 lands on — and if it divides exactly, that's the last entry in the cycle, not the first.

Divisibility by 8 only needs the last three digits. By 4, only the last two. By 3 or 9, add up all the digits. These shortcuts turn a "test this huge number" question into arithmetic on two or three digits.

Subtracting a negative number behaves exactly like adding. 5 − (−3) is 5 + 3 = 8, not 5 − 3. It's a rule everyone can state and still slip on mid-calculation when a chain of signs builds up.

A prime factorisation tells you how many factors a number has, without listing them. Write 360 as 2³ × 3² × 5, add one to each exponent, and multiply: (3+1)(2+1)(1+1) = 24 factors. It's faster than any list, and it's the same idea behind checking whether a number is a multiple of something like 24 — you check its prime factorisation contains 2³ × 3, not just "a 2 and a 3 somewhere".

A recurring decimal like 0.1̇ is exactly a ninth — not almost a ninth. Single recurring digits are always that digit over 9 (0.7̇ = 7/9); the temptation to round them off, or to write 0.7̇ as 7/10, loses marks on questions that need the exact fraction.

Standard form's number part must be at least 1 and less than 10. 0.6 × 10⁴ isn't finished standard form — it needs rewriting as 6 × 10³. And a million is 10⁶, a billion (the modern one) is 10⁹ — not the old British billion, which was a million million.

Algebra

The laws of indices decide between close answer choices fast. If five options are all powers of 2, or if a question compares 4ⁿ with 8ᵐ, the winning move is to rewrite everything in the same base first — usually the smallest prime involved — rather than trying to evaluate anything.

A power of a power multiplies the indices; two powers of the same base multiplying together add them. (aᵐ)ⁿ = aᵐⁿ, but aᵐ × aⁿ = aᵐ⁺ⁿ. Mixing these two up the wrong way round is one of the most common algebra slips on the paper.

When a question gives two equations in three unknowns and only asks for one combination of them, you don't need to "solve" anything. Subtracting or adding the two equations directly often produces exactly the combination being asked for — no third equation required.

Geometry and measures

The interior angles of any polygon with n sides add up to 180(n − 2) degrees, whatever shape it is — regular or not, as long as it's convex. A hexagon's angles always total 720°; a pentagon's, 540°. Learn the formula, not a table of individual totals.

The exterior angles of any convex polygon always add up to 360°, regardless of how many sides it has — unlike the interior sum, which depends on n. It's the cleaner of the two rules and the one that's easy to forget exists.

A right-angled triangle with shorter sides 5 and 12 has a hypotenuse of 13 — instantly, no calculation. The 5-12-13 triple (and its scaled versions, like 6-8-10) turns up constantly in diagrams, and recognising it on sight saves a square root every time.

When two shapes are similar, lengths scale by a factor k, but areas scale by k². Double every length and the area goes up by four, not two — a trap that catches almost every candidate the first time they meet it, and almost none the second.

When one geometry fact leads to another, write down every angle you can find on the diagram — not just the one asked for. Angles on a straight line, in a triangle, between parallel lines: each new angle you mark is a stepping stone toward the one the question actually wants, several steps away.

A rhombus's diagonals bisect each other at right angles — but they are not necessarily equal in length. That last part is the property of a rectangle, not a rhombus, and the two get muddled constantly.

Reflecting a point (x, y) in the x-axis sends it to (x, −y). Reflecting in the line y = x swaps the coordinates, sending (a, b) to (b, a). Both are one-line rules, not something to re-derive from a sketch each time.

Probability and statistics

When two choices are made independently, you multiply the number of options, not add them. A starter chosen from three options and a main from four gives twelve possible meals, not seven — the single most common counting mistake on the whole paper.

The range is simply the largest value minus the smallest — nothing more. It measures spread, not a "typical" value, and it's frequently confused with the mean or asked for as a trap alongside the mode and median in the same question.

If you're told the mean of a set and how many numbers are in it, you can write down their total straight away: total = mean × count. Most "mean" questions are really total questions in disguise — you rarely need to find the individual numbers at all.

Reaching beyond KS3

A few facts on the paper sit just past the KS3 curriculum, on the edge of GCSE content — a deliberate feature of the IMC's slightly-older pitch, not a mistake. Two worth knowing: a² − b² factorises as (a + b)(a − b), the difference of two squares; and a sector's share of a circle's area is exactly its angle's share of 360° — the same fraction whether you're working out area or arc length. Neither needs GCSE-level fluency to use once you've seen the pattern.

Where this comes from, and what to do with it

None of these facts were plucked from a textbook index — every one of them is drawn from real IMC-style practice cards our maths tutor, Professor Pi, has built and refined across thirteen years of papers, from the IMC toolkit deck through each individual year. If a fact here catches you out, it's worth a look at which mistakes we see most often around it, and at which of these topics come up almost every single year — some of what's above is far more load-bearing than the rest. For a practical way to actually get this stuck, rather than just read once, see how spaced repetition fits the IMC.

FAQ

Do I need to memorise all of these facts before sitting the IMC?

No single fact here is compulsory, and no child needs the full list to enjoy the paper or do well on it. But each one removes a small delay on a question where the clock matters, and together they cover the ground that keeps reappearing across IMC papers.

Is this list official UKMT content?

No. These are facts and habits our own tutors built from working through thirteen years of IMC-style practice questions, written in our own words. For the official syllabus and past papers, see the UK Mathematics Trust site.

My child is in Year 9 — is some of this too advanced?

A little of it is, deliberately. The IMC is open to Years 7 to 9 but pitched a bit older, so a handful of these facts (marked above) sit just past standard KS3 content, on the GCSE horizon. Seeing them once, in context, is enough — nothing here needs to be mastered to GCSE depth.

Duke Harewood built aitutors.me's KS3 maths tutor (Professor Pi) for his Year 8 daughter. Updated 26 August 2026.